Novel techniques for efficient quantum state tomography and quantum process
tomography and their experimental implementation

Akshay Gaikwad
A thesis submitted for the partial fulfillment of
the degree of Doctor of Philosophy

Department of Physical Sciences

Indian Institute of Science Education and Research Mohali

Knowledge city, Sector 81, SAS Nagar, Manauli PO, Mohali 140306, Punjab, India

December 2023

Declaration

The work presented in this thesis has been carried out by me under the
guidance of Prof. Kavita Dorai and Prof. Arvind at the Indian Institute of
Science Education and Research Mohali.

This work has not been submitted in part or in full for a degree, diploma or a
fellowship to any other University or Institute. Whenever contributions of
others are involved, every effort has been made to indicate this clearly, with
due acknowledgment of collaborative research and discussions. This thesis is a
bonafide record of original work done by me and all sources listed within have
been detailed in the bibliography.

Akshay Gaikwad

Place :
Date :

In our capacity as supervisors of the candidate’s PhD thesis work, we certify
that the above statements by the candidate are true to the best of our
knowledge.

Dr. Kavita Dorai Dr. Arvind
(Supervisor) (Co-supervisor)
Professor of Physics Professor of Physics
Department of Physical Sciences Department of Physical Sciences
IISER Mohali IISER Mohali

Place : Place :
Date : Date :

Acknowledgments

I wish to begin by expressing my profound gratitude to my thesis supervisors,
Prof. Kavita Dorai and Prof. Arvind, for their steadfast support, guidance,
and encouragement during my entire PhD journey. Their mentorship not only
involved teaching and directing various projects within this thesis but also
instilled in me the mindset to comprehend, develop, and accomplish any given
task. I am truly thankful for their unwavering assistance and mentorship. I
would also like to extend my thanks to Dr. Sandeep Goyal, Dr. K. P. Singh, Dr.
Manabendra Nath Bera, and Dr. Vishal Bhardwaj, members of my doctoral
committee, for their support and guidance. Special appreciation goes to Dr.
Paramdeep Singh Chandi, the scientific officer, for his invaluable help in
resolving software issues. Additionally, my gratitude extends to Mr. Balbir
Singh, a member of the NMR lab scientific staff, for his generous and
consistent support.

I would like to acknowledge the present and past members of the research
group, including Rajbinder, Jyotsana, Sumit, Dileep, Akanksha, Vaishali,
Krishna, Gayatri, Arshdeep, Matreyee, Shaileyee, Ram Sagar, Rithu, Sneha,
Anupama, and past members Amandeep Singh, Harpreet Singh, Shruti Dogra, Satnam
Singh, Rakesh Sharma, Navdeep Gogna, for their valuable insights shared during
our meetings, which have greatly benefited me. Special thanks are reserved for
Amandeep Singh for his guidance in NMR experiments.

My deep gratitude extends to the NMR research facility at IISER Mohali for
their tremendous support in facilitating my experimental work. I also
appreciate the research fellowship and financial assistance provided by IISER
Mohali, enabling me to attend conferences during my PhD.

The accomplishment of obtaining a PhD is not solely attributed to individuals
encountered during this journey but also to everyone encountered throughout my
life. I express gratitude to all my teachers, with special thanks to my
childhood friends Tushar Pawar and Nitin Shivthare for their steadfast support
during my formative years. My gratitude extends to Sameer Shah, my friend from
senior secondary education, who played a crucial role in helping me with math
during entrance exam preparations. I attribute my success in entering IISER
for undergraduate and graduate studies to Sameer’s assistance. Subsequently, I
made numerous friends during my undergraduate studies, and I express my thanks
to Abhinav Kala, Joydeep, Akhil, Love Grover, Rajendra Bhati, Shubham,
Gyanendra, Manvendra Rajwanshi, Manvendra Singh, Abhishek Singh for five years
of wonderful memories.

During my PhD program at IISER Mohali, I forged new friendships with Sandeep,
Mandeep, Mamta, Amreen, Yattu, Sheru, and Chitra. I express my gratitude to
Sheru for introducing me to table tennis and Yattu for engaging in humorous
chit-chats. I also spent very good time playing table tennis game with Mamta
and Mandeep. Special thanks to Mandeep for tolerating my behavior and
imparting valuable habits. I consider myself lucky to have friend like Mandeep
in my life. I also appreciate the companionship of my video game and trekking
partners Bhati, Jassi, Love, Abhishek, Krishna, and Sandeep, making my PhD
journey less stressful. I cherish the delightful moments I shared with my
lovely friend Tarang during my final days at IISER Mohali.

I extend my sincere appreciation to my father Ramdas Gaikwad, my mother Padma,
and my brother Ajinkya for their unwavering support and faith in me. I would
also like to mention my late grandfather Shankar Gaikwad, whom I called
’Aaba,’ as the main source of my inspiration, and I dedicate this thesis to
his memory. Their encouragement has been a significant source of motivation
throughout my endeavors.

Akshay Gaikwad

## Abstract

The study carried out in this thesis focuses on designing and experimentally
implementing various quantum tomography protocols to efficiently characterize
and reconstruct unknown quantum states and processes using spin ensemble based
nuclear magnetic resonance (NMR) quantum processors and superconducting
technology-based IBM quantum processors. The task of reconstructing quantum
states is achieved with the help of quantum state tomography (QST) protocols
while quantum processes are characterized using quantum process tomography
(QPT) protocols. Both QST and QPT are essential to check the reliability and
to evaluate the performance of a quantum processor. However, both QST and QPT
are cursed with a fundamental difficulty, i.e., the computational complexity
increases exponentially with the size of the system which makes them
infeasible to perform experimentally, even for smaller dimensional systems.
Besides this, having finite size of ensembles and inevitable systematic errors
will lead to unphysical density matrices and process matrices. To tackle such
issues, numerous QST and QPT protocols have been proposed. However, most of
them are yet to be experimentally demonstrated. The prime objective of the
study undertaken in this thesis is to design experimental strategies to
efficiently implement tomography protocols on NMR and IBM quantum processors.
Generalized quantum circuits are proposed to efficiently acquire experimental
data to perform QST and QPT and further demonstrated for two- and three-qubit
quantum states and quantum processes.

To tackle the issue of the unphysicality of experimentally reconstructed
quantum states and processes using standard tomography techniques, the tasks
of QST and QPT are converted into a constrained convex optimization (CCO)
problem and the CCO problem is solved to reconstruct valid quantum states and
processes which in case of QPT allows us to compute the complete set of Kraus
operators corresponding to a given quantum process. Further, the compressed
sensing (CS) and artificial neural network (ANN) techniques have also been
employed to perform tomography of quantum states and gates from a heavily
reduced data set as compared to standard methods. CS and ANN based tomography
methods are promising techniques to deal with complexity issue to characterize
higher-dimensional quantum gates. Moreover, the problem of selective and
direct estimation of desired elements of process matrix characterizing quantum
process has also been explored, where partial knowledge about underlying
unknown quantum process can be acquired efficiently using selective and
efficient quantum process tomography protocol (SEQPT). A generalized quantum
algorithm and quantum circuit to perform SEQPT has been proposed and
successful experimental demonstration has been shown on NMR and IBM quantum
processors. In addition to that, we also proposed an efficient direct QST and
QPT scheme based on weak measurement approach and demonstrated experimentally
using a three-qubit NMR system. The thesis also investigates the problem of
experimentally simulating dynamics of open quantum systems based on dilation
techniques. To show the efficacy of above-mentioned quantum tomography and
simulation protocols, experimental results are compared with theoretically
predicted results in case of several two-and three-qubit quantum systems. The
content of this thesis has been divided into eight chapters as described
below:

##### Chapter 1

The initial section of this chapter presents an overview of quantum computing
and information processing. It encompasses essential principles and the
physical realization of quantum processors using NMR. Additionally, it
provides a brief introduction to various tomography and simulation protocols.
Finally, it concludes with closing remarks that outline the objectives and
motivations behind the research conducted in this thesis.

##### Chapter 2

This chapter focuses on the problem of invalid experimental density and
process matrices. It introduces the constrained convex optimization (CCO)
method for QST and QPT which allow us to reconstruct valid (positive semi-
definite) density and process matrices charactering unknown quantum states and
processes respectively. It also improves the fidelity of density and process
matrix characterization. The chapter discusses the NMR quantum information
processor-based implementation of QST and QPT using the CCO method.

##### Chapter 3

This chapter addresses scalability issues in QST and QPT by employing the
application of compressed sensing (CS) algorithms. The CS algorithm allows for
full as well as valid QST and QPT from incomplete data sets, resulting in high
fidelity estimates of density and process matrices. The chapter also discusses
the characterization of small-dimensional quantum gates in higher-dimensional
systems. The experimental demonstration of CS based QST and QPT for 2- and
3-qubit system is given using NMR ensemble quantum processor and
superconducting based IBM cloud quantum processor.

##### Chapter 4

This chapter explores the application of artificial neural network (ANN)
techniques in QST and QPT to attempt to overcome scalability issues. The Feed-
Forward Neural Network (FFNN) architecture is used to reconstruct density and
process matrices from noisy experimental data obtained from NMR quantum
processor. The results show efficient as well as high fidelity QST and QPT as
compared to the standard linear inversion method.

##### Chapter 5

This chapter introduces a scheme for selective and efficient quantum process
tomography (SEQPT) using local measurements without ancilla. The method
estimates specific elements of the process matrix by a restrictive set of
subsystem measurements, reducing the experimental resources required. The
efficacy of the scheme is demonstrated experimentally on NMR and IBM
processors for 2- and 3-qubit systems.

##### Chapter 6

This chapter presents an efficient weak measurement (WM) scheme for direct
quantum state tomography (DQST) and direct quantum process tomography (DQPT)
without projective measurements. A generalized quantum circuit is proposed and
implemented on an NMR ensemble quantum information processor to directly
measure multiple selective elements of density and process matrices in a
single experiment which enable us to efficiently extract desired information
from the system.

##### Chapter 7

This chapter investigates the experimental simulation of open quantum system
dynamics using dilation techniques. The Sz-Nagy’s dilation (SND) algorithm is
experimentally implemented on an NMR quantum information processor to simulate
the action of a 2-qubit pure phase damping channel, correlated amplitude
damping channel and magnetic field gradient pulse (MFGP). The algorithm
successfully simulates the dynamics using only one ancilla qubit, and the
experimental fidelity is assessed using CCO-QPT.

##### Chapter 8

This chapter describes the summary of the thesis and some future directions.

List of Publications

  1. 1.

Akshay Gaikwad, Omkar Bihani, Arvind and Kavita Dorai
Neural network assisted quantum state and process tomography using limited
data sets
Phys. Rev. A 109, 012402, (2024)

  2. 2.

Akshay Gaikwad, Arvind and Kavita Dorai
Direct tomography of quantum states and processes via weak measurements of
Pauli spin operators on an NMR quantum processor
Eur. Phys. J. D (2023) 77: 209, (2023)

  3. 3.

Akshay Gaikwad, Krishna Shende, Arvind and Kavita Dorai
Implementing efficient selective quantum process tomography of superconducting
quantum gates on IBM quantum experience
Scientific reports 12 (1), 1-11, (2022)

  4. 4.

Akshay Gaikwad, Arvind and Kavita Dorai
Experimental simulation of open quantum dynamics using Sz-Nagy’s dilation
algorithm using NMR
Phys. Rev. A 106, 022424, (2022).

  5. 5.

Akshay Gaikwad, Arvind and Kavita Dorai
Efficient characterization of quantum processes from reduced data set via
compressed sensing using NMR
Quant. Inf. Proc. 21, (12), (2022)

  6. 6.

Akshay Gaikwad, Arvind and Kavita Dorai
True experimental reconstruction of quantum states and processes via convex
optimization using NMR
Quant. Inf. Proc. 20, (19), (2021)

  7. 7.

Akshay Gaikwad, Krishna shende and Kavita Dorai
Experimental demonstration of optimized quantum process tomography on the IBM
quantum experience
Int. J. Quantum Inf. 19 (07), 2040004, (2021)

  8. 8.

Akshay Gaikwad, Diksha Rehal, Amandeep Singh, Arvind and Kavita Dorai
Experimental demonstration of selective quantum process tomography on NMR
Phys. Rev. A 97, 022311, (2018)

###### Contents

  1. Abstract
  2. 1 Introduction
    1. 1.1 Quantum Mechanics as a Computational Paradigm
    2. 1.2 Quantum Computing and Information Processing
      1. 1.2.1 Quantum bit
      2. 1.2.2 Quantum gates
      3. 1.2.3 Quantum Measurement
      4. 1.2.4 Quantum ensemble and density matrix formalism
    3. 1.3 Theory of NMR
    4. 1.4 NMR quantum computing
      1. 1.4.1 Well characterized qubits in NMR
      2. 1.4.2 Initializing quantum circuits in NMR
      3. 1.4.3 Quantum gate implementation in NMR
      4. 1.4.4 Measurement in NMR
      5. 1.4.5 Open quantum dynamics
    5. 1.5 Quantum state and process tomography
      1. 1.5.1 Dealing with invalid density and process matrices via convex optimization methods
      2. 1.5.2 Data driven based QST and QPT via machine learning methods
      3. 1.5.3 Selective and direct QST and QPT protocols
      4. 1.5.4 Simulation and characterization of open quantum dynamics
    6. 1.6 Organization of the thesis
  3. 2 Achieving valid quantum states and processes on an NMR quantum information processor through convex optimization
    1. 2.1 Introduction
    2. 2.2 QST via standard and CCO method
      1. 2.2.1 Linear inversion based standard QST on NMR
      2. 2.2.2 Fidelity measure for quantum states
      3. 2.2.3 Convex optimization based QST
    3. 2.3 QPT via standard and CCO method
      1. 2.3.1 Standard QPT
      2. 2.3.2 Convex optimization based QPT
      3. 2.3.3 Markovian Quantum Process Tomography
      4. 2.3.4 Comparison of CCO QPT with standard Protocols
    4. 2.4 Conclusions
  4. 3 Scalable characterisation of noisy quantum gates using compressed sensing and reduced data set
    1. 3.1 Introduction
    2. 3.2 Compressed sensing algorithm for QPT
    3. 3.3 Experimental implementation of CS-QPT on NMR
      1. 3.3.1 CS-QPT of two-qubit gates
      2. 3.3.2 CS-QPT of three-qubit gates
      3. 3.3.3 CS-QPT of two-qubit processes in a three-qubit system
      4. 3.3.4 Comparison of CS-QPT and LS-QPT protocols
    4. 3.4 CS-QPT on IBM cloud quantum computer
      1. 3.4.1 IBM experimental results and analysis
    5. 3.5 Conclusions
  5. 4 Predicting quantum states and processes with incomplete measurements using feed-forward neural network
    1. 4.1 Introduction
    2. 4.2 ANN based QST and QPT
      1. 4.2.1 Artificial Neural Networks (ANNs)
      2. 4.2.2 Training FFNN for QST and QPT
        1. 4.2.2.1 Data generation and training FFNN for QST and QPT
    3. 4.3 Experimental results and analysis
      1. 4.3.1 2- and 3-qubit QST on NMR via FFNN
      2. 4.3.2 2-qubit QPT on NMR via FFNN
        1. 4.3.2.1 Experimental simulation of fully correlated error channels and their reconstruction via ANN on NMR
    4. 4.4 Conclusions
  6. 5 Generalized algorithm and experimental demonstration of selective quantum process tomography on NMR and IBM quantum processor
    1. 5.1 Introduction
    2. 5.2 Protocol for selective and efficient quantum process tomography using local measurements
      1. 5.2.1 Standard SEQPT protocol
      2. 5.2.2 Modified SEQPT
    3. 5.3 NMR implementation of MSQPT
    4. 5.4 IBM implementation of MSQPT protocol
      1. 5.4.1 Initial state preparation for efficient MSQPT
        1. 5.4.1.1 Generalized algorithm for preparation of input states for n-qubit system
      2. 5.4.2 MSQPT of 2-qubit quantum gates on IBM QX2 processor
      3. 5.4.3 MSQPT of 3-qubit toffoli gate on IBM QX2 processor
      4. 5.4.4 IBM Experimental Results and analysis
        1. 5.4.4.1 2-qubit MSQPT results and analysis on IBM
        2. 5.4.4.2 3-qubit MSQPT results and analysis on IBM
    5. 5.5 Conclusions
  7. 6 Direct tomography of quantum states and processes using weak measurements on NMR
    1. 6.1 Introduction
    2. 6.2 General scheme for direct QST and QPT via weak measurement
    3. 6.3 Efficient experimental implementation of weak measuremnt scheme on NMR
      1. 6.3.1 Weak measurement of Pauli spin operators on NMR
        1. 6.3.1.1 Measurement of $O_{x}^{\phi}$ and $O_{y}^{\phi}$ on NMR
      2. 6.3.2 Experimentally demonstrating WM of $\sigma_{1z}$
      3. 6.3.3 Experimental DQST and DQPT using WM scheme
        1. 6.3.3.1 Extension to $n$-qubit system
    4. 6.4 Conclusions
  8. 7 Experimental simulation of non-unitary quantum processes on NMR
    1. 7.1 Introduction
    2. 7.2 Time evolution of open quantum systems and the Sz.-Nagy algorithm
    3. 7.3 Experimentally simulating two-qubit non-unitary quantum processes
      1. 7.3.1 Simulating a two-qubit phase damping channel
      2. 7.3.2 Kraus operators & unitary dilation operators for phase damping channel
      3. 7.3.3 Simulation of 2-qubit correlated amplitude damping channel
      4. 7.3.4 Kraus operators & unitary dilation operators for fully correlated amplitude damping channel
      5. 7.3.5 Simulating a magnetic field gradient pulse
      6. 7.3.6 Kraus operators and unitary dilation operators for shaped MFGP
    4. 7.4 Conclusions
  9. 8 Summary and Future Outlook
    1. 8.1 Summary
    2. 8.2 Future Scope

###### List of Figures

  1. 1.1 A logarithmic graph showing the timeline of how transistor counts in microchips are almost doubling every two years from 1970 to 2020; (Moore’s Law). Source: ourworldindata.org
  2. 1.2 Qubit representation on bloch sphere, source: Google
  3. 1.3 Implementation of 3-qubit Toffoli gate using Hadamard (H), phase gate (S), $\pi/8$ gate (T) and CNOT gate.
  4. 1.4 Common quantum logic gates by name, circuit form(s) and matrices. source: Google
  5. 1.5 Allowed transitions and NMR spectrums in coupled two spin-1/2 nuclei.
  6. 1.6 Signal acquisition process in NMR.
  7. 2.1 Schematic of standard QST based on linear inversion.
  8. 2.2 Real (left) and imaginary (right) parts of experimental tomographs of two qubit states, constructed via standard (1st row) and CCO (2nd row) method are shown.
  9. (a) $|01\rangle$
  10. (b) $|0+\rangle$
  11. (c) $(|00\rangle+|11\rangle)/\sqrt{2}$
  12. (d) $(|01\rangle+|10\rangle)/\sqrt{2}$
  13. 2.3 Schematic of standard QPT based on linear inversion.
  14. 2.4 The tomographs in the row denote the real and imaginary parts of the process matrix for 2-qubit quantum gates. The tomographs in the first column represents the theoretically constructed process matrix while those in second and third represent the experimentally measured linear inversion based and CCO method based process matrices respectively.
  15. (a) CNOT gate
  16. (b) C-$R_{x}^{\pi}$ gate
  17. 2.5 The tomographs in the row denote the real and imaginary parts of the process matrix for the decoherence process at various times. The tomographs in the first and second column represents the experimentally constructed process matrix via CCO method and process matrix obtained by numerically simulating the decoherence model respectively.
  18. (a) $t=0.05s$
  19. (b) $t=0.5s$
  20. (c) $t=5s$
  21. (d) $t=15s$
  22. 3.1  Graphical depiction illustrating the operational functionality of the compressed sensing algorithm.
  23. 3.2 (Color online) (a) Quantum circuit comprising all settings to acquire data to perform CS-QPT of a CNOT gate. (b) NMR implementation to perform CS-QPT of CNOT gate. All rectangles filled with different colours denotes hard NMR pulses to perform various single qubit rotation gates. (c) 13C-enriched chloroform molecule with 1H and 13C labeling the first and second qubits, respectively. (d) and (e) depict the NMR spectra of 13C and 1H, respectively, corresponding to the configuration $\\{|++\rangle\langle++|,IX\\}$.
  24. 3.3 (Color online) General quantum circuit to obtained data for implementation of CS-QPT of 3-qubit Control-NOT-NOT ($U_{\rm{CNN}}$) gate. (b) NMR implementation of CS-QPT of 3-qubit Control-NOT-NOT ($U_{\rm{CNN}}$) gate. (c) ${}^{13}C$-labeled diethyl fluoromalonate with ${}^{1}H$, ${}^{19}F$ and ${}^{13}C$ nuclei labeled as the first, second and third qubits, respectively. NMR spectra depicted in (d), (e) and (f) correspond to 1H, 19F and 13C nuclei respectively, for the configuration $\\{|11+\rangle\langle 11+|,R_{x}^{3}R_{y}^{2}R_{x}^{1}\\}$.
  25. 3.4 (Color online)The first column represents the average gate fidelity $\overline{\mathcal{F}}$ corresponding to (a) a three-qubit $U_{\rm{CNN}}$ gate, (b) a CNOT gate and (c) $U^{J}_{23}$ vs number of data points $m_{\text{data}}$ while second column represents standard deviation ($\sigma$) in average fidelity corresponding to (d) a three-qubit $U_{\rm{CNN}}$ gate, (e) a CNOT gate and (f) $U^{J}_{23}$ vs number of data points $m_{\text{data}}$. The plots in red, blue, green and magenta correspond to CS-PEB, CSPB, LSPEB and LSPB methods, respectively.
  26. 3.5 Two-qubit quantum circuit to acquire experimental data for QPT of CNOT gate.
  27. 3.6  (a) IBM quantum circuit for implementing QPT of a CNOT gate, corresponding to configuration $\\{|1+\rangle\langle 1+|,\sigma_{1x}\sigma_{2y}\\}$. Bit flip gate $X\equiv R_{y}(\pi)$. (b) Histogram of the probabilities $p_{0}$ and $p_{1}$, of obtaining the two-qubit states $|0\rangle$ and $|1\rangle$ respectively, after running the quantum circuit (given in (a)) 4096 times. We obtain $p_{0}=0.51831$ and $p_{1}=0.48169$.
  28. 3.7 Experimental plots (a), (b), and (c) were obtained using the IBM QX2 processor, while corresponding simulation results (d), (e), and (f) are provided for the QPT of the Identity gate, CNOT gate, and SWAP gate, respectively. The average gate fidelity $\mathcal{F}$ is represented on the $y$-axis, while the size of the data set $m_{{\rm conf}}$ is depicted on the $x$-axis. The CS method is represented by red diamonds and green stars, while the LS method is indicated by pink triangles and blue squares for the PEB and PB basis, respectively. Error bars in all cases are determined by calculating the standard deviation $\sigma$ through the random selection of $m_{{\rm conf}}$ configurations, repeated 50 times.
  29. 4.1  Schematics representation of 2-qubit QST using FFNN model.
  30. 4.2  (Color online) The graph illustrates the experimental state fidelity ($\bar{\mathcal{F}}$) of the FFNN model and the linear inversion method with respect to the dataset size ($M_{data}$). The panel (a) represents 100 2-qubit states, while (b) represents 128 3-qubit states. The x-axis represents the numbering of the states, while the y-axis corresponds to the dataset size ($M_{data}$). The legend on the graph provides the fidelity values for reference.
  31. 4.3  (Color online) In the first row graphs represents average state fidelity ($\bar{\mathcal{F}}$) obtained via FFNN method vs size of dataset ($M_{data}$) while in second row corresponding standard deviation $\Delta\bar{\mathcal{F}}$ are given. Plots (a) and (c) given in first column correspond to 2-qubit system while plots (b) and (d) correspond to 3-qubit system. For test data-set, average is calculated on 3000 states while for experimental data-set $\bar{\mathcal{F}}$ and $\Delta\bar{\mathcal{F}}$ are calculated on randomly considering $M_{data}$ elements 50 times for 100 2-qubit states and 128 3-qubit states each.
  32. 4.4 (Color online) Experimental state fidelity ($\bar{\mathcal{F}}$) vs size of dataset ($M_{data}$) of entangled states is given. Part (a) corresponds to 2-qubit bell states while part (b) corresponds to 3-qubit GHZ and bi-separable states. $\bar{\mathcal{F}}$ is calculated between respective density matrices predicted via FFNN and density matrices obtained via standard linear inversion for reduced data set of a size $M_{\rm data}$
  33. 4.5  (Color online) Experimental process fidelity ($\bar{\mathcal{F}}$) vs data elements ($M_{data}$). $M_{data}$ is given on x-axis while experimental quantum processes/gates are given on y-axis. The legend represents the fidelity value.
  34. 4.6  (Color online) Panel (a) represents average process fidelity ($\bar{\mathcal{F}}$) obtained via FFNN method vs size of dataset ($M_{data}$) while panel (b) corresponds to standard deviation $\Delta\bar{\mathcal{F}}$. For test data-set, denoted by solid blue circle, average is calculated on 3000 processes while for experimental unitary and non-unitary processes, denoted by solid red triangle and solid magenta star respectively, $\bar{\mathcal{F}}$ and $\Delta\bar{\mathcal{F}}$ are calculated on randomly considering $M_{data}$ elements 300 times for 4 unitary quantum gates and 6 non-unitary processes.
  35. 4.7  (Color online) General quantum circuit to simulate the action of error channels: correlated bit flip, correlated phase flip and correlated bit+phase flip.
  36. 5.1 The figure presents a block diagram illustrating the Multi-Step Quantum Process Tomography (MSQPT) protocol. The protocol involves several steps. Step 1 involves the preparation of the basis operator $E_{i}$ state. In Step 2, a quantum channel $\Lambda$ is applied. Step 3 consists of mapping the basis operator to local measurements, followed by the measurement of the expectation values of Pauli $z$-operators on the subsystems. Finally, in Step 4, local detection takes place.
  37. 5.2 (a)The provided quantum circuit depicts the implementation of the MSQPT. The circuit incorporates local unitary operations denoted as $R_{\phi}^{\theta}$, which are accomplished through rotations by an angle $\theta$ and a phase $\phi$. This particular quantum circuit utilises non-unitary operation illustrated by rectangle filled with gray color. (b)NMR implementation of MSQPT. The non-unitary operation in implemented using magnetic field gradient pulse.
  38. 5.3 (Color online) (a) Molecular structure of 13C labeled chloroform, which serves as a two-qubit quantum system. In this system, the first qubit is encoded as the nuclear spin of 1H, while the second qubit is encoded as the nuclear spin of 13C. The values of the scalar coupling $J_{\rm CH}$ (in Hz), relaxation times $T_{1}$ and $T_{2}$ (in seconds), and chemical shifts $\nu_{i}$ are indicated alongside. (b) Thermal equilibrium NMR spectrum after a $\frac{\pi}{2}$ detection pulse.
  39. 5.4 (Color online). The tomographs in the columns denote the real and imaginary parts of the $\chi$ matrix respectively, for the identity operator. The tomographs on the top represent the theoretically constructed while those on the bottom represent the experimentally measured $\chi$ matrix of the identity operator. The fidelity of the identity operator turned out to be 0.98.
  40. 5.5 (Color online). The tomographs in the columns denote the real and imaginary parts of the $\chi$ matrix respectively, for the CNOT gate (control-$R_{x}^{\pi}$). The tomographs on the top represent the theoretically constructed while those on the bottom represent the experimentally measured $\chi$ matrix of the CNOT operator. The fidelity of the CNOT operator turned out to be 0.93.
  41. 5.6 (Color online). The tomographs in the columns denote the real and imaginary parts of the $\chi$ matrix respectively, for the controlled-Hadamard gate (control-$R_{y}^{-\frac{\pi}{2}}$). The tomographs on the top represent the theoretically constructed while those on the bottom represent the experimentally measured $\chi$ matrix of the CH operator. The fidelity of the CH operator turned out to be 0.92.
  42. 5.7 The generalized quantum circuit to acquire data to perform n-qubit MSQPT is given. The symbol ’/’ through the input wire represents multiqubit quantum register. The first quantum register contains single qubit while second and third quantum registers comprise $n-1$ qubits. The first and the second quantum registers collectively represent system qubits and denoted by $|0\rangle_{s}$ where third quantum register represents ancilla qubits denoted by $|0\rangle_{a}$. The quantum circuit below is the elaborated version where blue color denotes the system qubits and red color denotes the ancilla qubits.
  43. 5.8 (a) The IBM quantum circuit to perform MSQPT of SWAP gate is given where in the first block the 3-qubit input state $|\psi_{6}\rangle$ is prepared using $U^{6}=S_{2}.CNOT_{12}.CNOT_{23}.H_{2}.H_{1}$ and in the second block quantum process corresponding to 2-qubit SWAP gate is applied and in the last block quantum map $U_{13}=CNOT_{12}.R_{y}(-\frac{\pi}{2})$ is applied to efficiently compute $\textbf{Tr}(\sigma_{z}\bigotimes\sigma_{x}\Lambda(\tilde{M_{6}}))$ by detecting system’s second qubit in $\sigma_{z}$ basis. (b) Histogram represents statistical results after running quantum circuit given in (a) 4096 times. It gives the probability $p_{0}=0.4502$ and $p_{1}=0.5498$ of obtaining second qubit in $|0\rangle$ and $|1\rangle$ state.
  44. 5.9 (a) The IBM quantum circuit to perform MSQPT of 3 qubit Toffoli gate is given where in the first block the 5-qubit input state $|\Omega_{50}\rangle$ is prepared and in the second block quantum process corresponding to 3-qubit Toffoli gate is applied on the system qubits and in the last block quantum map $U_{15}=CNOT_{23}$ is applied to efficiently compute $\textbf{Tr}(I\otimes\sigma_{z}\otimes\sigma_{y}\Lambda(\tilde{M}_{50}))$ by detecting system’s third qubit in $\sigma_{z}$ basis. (b) Histogram represents statistical results after running quantum circuit given in (a) 4096 times. It gives the probability $p_{0}=0.51489$ and $p_{1}=0.48511$ of obtaining third qubit in $|0\rangle$ and $|1\rangle$ state respectively.
  45. 5.10 Matrix plots are shown corresponding to (a) real part of $\chi$ matrix for CNOT gate and (b) real part of $\chi$ matrix for SWAP gate. The first column represents theoretically constructed process matrix $\chi_{\rm the}$ while second and third column represent $\chi_{\rm msqpt}$, and $\chi_{\rm cco}$ respectively obtained by implementing MSQPT protocol on IBM QX2 processor.
  46. 5.11 Tomographs are shown corresponding to 3 qubit toffoli gate. The first column represents the real part of $\chi$ matrix and second column represents the imaginary part of $\chi$ matrix corresponding to Toffoli gate. The first row denotes the theoretically constructed $\chi$ matrix while second and third row represent experimentally constructed $\chi$ matrix obtained by implementing MSQPT protocol and CCO protocol on IBM QX2 processor respectively.
  47. 6.1 Schematic of WM based DQST and DQPT. (a) General quantum circuit for DQST of initial unknown state $|\psi\rangle_{s}$ and DQPT of quantum channel $\Lambda$ using WM method. (b) GRAPE optimised pulses to implement blocks of quantum circuit. (c) 3-qubit quantum system.
  48. 6.2 In the part (a), all the NMR spectra are obtained by acquiring third nuclear spin $\rm F_{3}$ corresponding to meter qubit. The thermal spectra is depicted in blue color while reference spectra is depicted in red color. The first five spectrums (from top) are obtained by implementing quantum circuit given in Fig.6.1 for different values of $g$ corresponding to initially prepared input state $|00\rangle_{s}|0\rangle_{m}$ and weak interaction $U^{\rm weak}_{\rm SM}$ corresponding to Pauli operator $\sigma_{1z}$ followed by $90^{\circ}$ phase shift on $\rm F_{3}$. In part the (b), the theoretically and experimentally obtained quantities $\langle O^{00}_{x}\rangle,\langle O^{00}_{y}\rangle$ and $\langle A\rangle^{00}_{w}\Pi^{00}_{00}$ are compared for different values of $g$.
  49. 6.3  NMR spectra shown in part (a) are obtained by implementing quantum circuit given in Fig.6.1, for initial state of the form $|\psi\rangle_{s}=Cos(\frac{n\pi}{20})|00\rangle+Sin(\frac{n\pi}{20})|10\rangle$ for fixed value of $g=0.1$ and observable $\sigma_{1z}$. The spectrum green and red correspond to $n=10$ and $n=0$ respectively. In part (b), experimental quantity $\langle\sigma_{1z}\rangle_{w}^{\phi}\Pi_{\psi}^{\phi}$ is compared with theoretical value as function of initial state $|\psi\rangle_{s}=Cos(\frac{n\pi}{20})|00\rangle+Sin(\frac{n\pi}{20})|10\rangle$.
  50. 6.4 Experimental readouts demonstrating DQST of Bell state $|\psi\rangle_{s}=\frac{1}{\sqrt{2}}(|00\rangle+|11\rangle)$ via WM technique on NMR. NMR spectra are obtained by implementing quantum circuit given in Fig.6.1 followed by $90^{\circ}$ phase shift for initial state $|\psi\rangle_{s}=\frac{1}{\sqrt{2}}(|00\rangle+|11\rangle)$, value of $g=0.2$ and observables $\sigma_{1z}$ (red), $\sigma_{1x}$(purple), $\sigma_{2x}$(green), and $\sigma_{1x}\sigma_{2x}$ (blue).
  51. 6.5 In part (a) theoretical and experimental density matrices corresponding to quantum state $|\psi_{2}\rangle=(|01\rangle+|10\rangle)/\sqrt{2}$ are given whereas in part (b) theoretical and experimental process matrix corresponding to rotation operation $R_{x}(\frac{\pi}{2})$ is given.
  52. 7.1  (a) Molecular structure of diethyl fluoromalonate labeled with 13C, used as the three-qubit quantum system. The spectra presented in (b), (c), and (d) correspond to the 1H, 19F, and 13C spins, respectively, obtained after applying a $90^{\circ}$ readout pulse on the initial state $|000\rangle$. The $J$-couplings between different nuclei are as follows: $J_{HF}=47.5$ Hz, $J_{HC}=161.5$ Hz, and $J_{FC}=-191.7$ Hz. The spin-lattice relaxation times measured for the different nuclei are $T^{H}_{1}=3.0\pm 0.34$ s, $T^{F}_{1}=3.3\pm 0.15$ s, and $T^{C}_{1}=3.2\pm 0.38$ s, while the spin-spin relaxation times are $T^{H}_{2}=1.3\pm 0.24$ s, $T^{F}_{2}=1.4\pm 0.22$ s, and $T^{C}_{2}=1.2\pm 0.18$ s.
  53. 7.2  (a) Quantum circuit diagram illustrating the simulation of the Kraus operator $A_{1}$ of the phase damping channel on the initial state $|00\rangle$ using the Sz.-Nagy algorithm. The input state $|00\rangle$ is encoded as $|000\rangle_{\rm HFC}=|0\rangle_{\rm H}\otimes|00\rangle_{\rm FC}$ to comply with the requirements of the algorithm. (b) NMR-based implementation of the quantum circuit depicted in (a). (c) The corresponding pulse phases are indicated below. The intervals of free evolution time are specified as $\tau=0.0078$ s, $\tau_{1}=0.0105$ s, and $\tau_{2}=0.0031$ s, respectively.
  54. 7.3 The impact of an independent phase damping channel on the state $|++\rangle$ is illustrated through an experimental simulation using the Sz-Nagy algorithm. Figure (a) demonstrates the damping effect on the real part of the off-diagonal element $\rho_{12}$, while Figure (b) depicts the damping of the real part of $\rho_{13}$. The $y$-axis represents the values of the real part of $\rho_{12(13)}$, and the $x$-axis represents the time intervals in seconds. The black bars correspond to theoretical values, while the red bars correspond to experimental values.
  55. 7.4 The bar plot illustrates the average normalized trace distance between the experimentally obtained output Hermitian matrix $(A_{i}\rho_{j}A_{i}^{\dagger}){{\rm exp}}$ and the theoretically expected matrix $(A_{i}\rho_{j}A_{i}^{\dagger}){{\rm the}}$ for the phase damping channel. The x-axis represents the numbered two-qubit states $\rho_{j}$. The bars are color-coded, with black, red, gray, and blue bars corresponding to Kraus operators $A_{1}$, $A_{2}$, $A_{3}$, and $A_{4}$, respectively.
  56. 7.5 The process matrices obtained from theoretical simulations $(\xi_{\rm sim}^{\rm the})$ and experimental simulations $(\xi_{\rm sim}^{\rm exp})$ of phase damping channels acting independently on each qubit in a two-qubit NMR system. The first column of bar plots displays the real part of the process matrices for the theoretically simulated channel ($\rm{Re}(\xi_{\rm sim}^{\rm the})$) and the experimentally obtained phase damping channel computed using convex optimization-based QPT ($\rm{Re}(\xi_{\rm sim}^{\rm exp})$, respectively). The second column of bar plots represents the imaginary part of the respective process matrices. The average process fidelity of the experimentally simulated channel is found to be $0.8883\pm 0.0375$.
  57. 7.6 The process matrices obtained from theoretical simulations $(\xi_{\rm sim}^{\rm the})$ and experimental simulations $(\xi_{\rm sim}^{\rm exp})$ of a 2-qubit fully correlated amplitude damping channel on a two-qubit NMR system. The first column of bar plots depicts the real part of the process matrices for the theoretically simulated channel ($\rm{Re}(\xi_{\rm sim}^{\rm the})$) and the experimentally obtained fully correlated amplitude damping channel computed using convex optimization-based QPT ($\rm{Re}(\xi_{\rm sim}^{\rm exp})$, respectively). The second column of bar plots represents the imaginary part of the respective process matrices. The average process fidelity of the experimentally simulated channel is determined to be $0.9216\pm 0.0017$.’
  58. 7.7 Bar plot showing the average normalized trace distance between the experimentally obtained output Hermitian matrix $(A_{i}^{CAD}\rho_{j}{A_{i}^{CAD}}^{\dagger})_{{\rm exp}}$ and the theoretically expected matrix $(A_{i}^{CAD}\rho_{j}{A_{i}^{CAD}}^{\dagger})_{{\rm the}}$ for a correlated amplitude damping channel. The x-axis represents the numbered two-qubit states $\rho_{j}$. The black and red bars represent the corresponding Kraus operators $A_{1}^{CAD}$ and $A_{2}^{CAD}$.
  59. 7.8 Experimental simulation of the correlated amplitude damping channel’s effect on the $|11\rangle$ state using the Sz-Nagy algorithm. Part (a) represents the ground state population ($P_{00}$), and part (b) represents the excited state population ($P_{11}$). The y-axis represents population, and the x-axis represents the damping parameter ($p$). The black bars indicate theoretical values, while the red bars indicate experimental values.
  60. 7.9 The bar plot illustrates the average normalized trace distance between the experimentally simulated output Hermitian matrix $(A_{i}\rho_{j}A_{i}^{\dagger})^{\rm{exp}}_{\rm{sim}}$, obtained using the SND algorithm, and the experimentally obtained matrix $(A_{i}\rho_{j}A_{i}^{\dagger})^{\rm{exp}}_{\rm{qpt}}$ through quantum process tomography. The simulation involves the implementation of MFGP on two qubits. The x-axis represents the numbered two-qubit states $\rho_{j}$, while the black, red, gray, and blue bars correspond to the Kraus operators $A_{1}$, $A_{2}$, $A_{3}$, and $A_{4}$, respectively.
  61. 7.10 In the top panel, $\rm{Re}(\eta^{\rm exp}_{\rm qpt})$ and $\rm{Im}(\eta^{\rm exp}_{\rm qpt})$ represent the real and imaginary parts of the experimental process matrix obtained from QPT of the shaped MFGP applied on two qubits. In the bottom panel, $\rm{Re}(\eta^{\rm exp}_{\rm sim})$ and $\rm{Im}(\eta^{\rm exp}_{\rm sim})$ depict the real and imaginary parts of the process matrix for the same MFGP, experimentally simulated using the Sz.-Nagy algorithm. The process fidelity between $\eta^{\rm exp}_{\rm qpt}$ and $\eta^{\rm exp}_{\rm sim}$ is found to be $0.8687\pm 0.0192$.

###### List of Tables

  1. 2.1 Eigenvalues for the two-qubit density matrix, obtained from experimentally reconstructed density matrices via standard and CCO QST.
  2. 2.2  Two-qubit quantum state fidelity obtained from standard and CCO method.
  3. 2.3  Eigenvalues obtained from experimental $\chi$ matrices constructed via standard and CCO QPT.
  4. 2.4  Two-qubit gate fidelities obtained via standard QPT and CCO QPT.
  5. 2.5  Difference in fidelity between ideal and predicted gate output (from experimentally constructed $\chi$ matrix) for all 16 input states, obtained via standard QPT and CCO QPT.
  6. 2.6  Calculated state deviation for each input state (using experimentally reconstructed $\chi$ matrix) via standard QPT and CCO QPT.
  7. 2.7  Average state deviation computed from standard ($\Delta_{avg}^{std}$) and from CCO ($\Delta_{avg}^{cco}$) methods.
  8. 2.8  The fidelity difference between an actual evolved Bell state (denoted by $|\rm{B_{i}}\rangle$) computed via CCO QPT, and predicted output state QPT. The first column represents the different time intervals for which evolution under the decoherence process was considered.
  9. 3.1  Relation between readout positions $\rho_{ij}^{\prime}$ of subsystem’s reduced density matrix with readout positions $\rho_{mn}$ of system’s full density matrix is given.
  10. 3.2  For given quantum gate, the minimum value of $m_{\rm{data}}$ at which the experimental average gate fidelity $\overline{\mathcal{F}}$ turns to be $\geq$ 0.9 computed via CS and LS algorithm with corresponding standard deviation $\sigma$ for PEB and PB is given.
  11. 3.3  Experimental quantum gate fidelities obtained via CS and LS method using full data set $m^{\rm{full}}_{\rm{data}}$.
  12. 3.4  The first row represents the qubit index q[i] within the 5-qubit IBM QX2 quantum processor. The second row displays the resonance frequencies $\omega_{R}$ of the corresponding read-out resonators. Qubit frequencies $\omega$ are provided in the third row. The anharmonicity $\delta$ is given in the fourth row, gauges the extent of information leakage from the computational space. The fifth and sixth rows correspond to the qubit-cavity coupling strengths $\mathcal{C}$ and the coupling of the cavity to the environment $\kappa$ for the respective qubits. The seventh and eighth rows present the longitudinal relaxation time $T_{1}$ and transverse relaxation time $T_{2}$, respectively.
  13. 4.1 The average state fidelity over 3000 2-qubit test states for $M_{data}=20$ for different training size and different numbers of epochs.
  14. 4.2 The average state fidelity over 3000 3-qubit test states for $M_{data}=120$ for different training size and different numbers of epochs.
  15. 4.3 The average process fidelity over 3000 2-qubit test processes for $M_{data}=200$ for different training size and different numbers of epochs.
  16. 4.4  Experimental fidelities $\mathcal{F(\chi_{\rm\small FFNN},\chi_{\rm\small STD})}$ are given. $\chi_{\rm\small FFNN}$ is process matrix corresponding given 2-qubit quantum process predicted via FFNN using full data set and $\chi_{\rm\small STD}$ is process matrix corresponding given quantum process obtained via standard linear inversion method.
  17. 5.1  Comparison of experimental resources for different protocols for the determination of a specific element of the $\chi$ matrix for a two-qubit system.
  18. 5.2  Fifteen observables for two qubits mapped to local z-magnetization of one of the qubit. $\rho_{k}=U_{k}\Lambda(E_{i})U_{k}^{\dagger}$ in order to calculate $\langle E_{k}\rangle=\textbf{Tr}\left[E_{k}\Lambda(E_{i})\right]$.
  19. 5.3  Parameters chosen to implement different unitary quantum processes.
  20. 5.4  The experimental complexity and the extra ancillary qubits required for 3-qubit MSQPT is compared with linear inversion based standard QPT and standard SQPT method.
  21. 6.1  The list of all unitary operators $\mathcal{U}_{k}$ and $U_{ij}^{J}(t)$ to implement weak interaction operation ${U}^{weak}_{SM}$ corresponding to all 2-qubit Pauli operators $P_{k}$.
  22. 6.2  The list of $\langle O_{x(y)}^{\phi}\rangle$ corresponding to energy transitions and readout elements for post selected state $|\phi\rangle$ being one of the computation basis vector.
  23. 6.3  Experimental state and process fidelities obtained using WM based DQST and DQPT method respectively.
  24. 7.1  Fidelity between the experimentally and theoretically simulated two-qubit states evolving under independent phase damping channels.
  25. 7.2  Fidelity between the experimentally and theoretically simulated two-qubit states evolving under correlated amplitude damping channels.
  26. 7.3  Fidelity between experimentally simulated and experimentally implemented two-qubit state under the action of a magnetic field gradient pulse.
  27. 7.4  The values of $\theta_{i}$s (Eq.7.9) required to the implement unitary dilation operators $U_{A_{j}}$.

##  Chapter 1 Introduction

###  1.1 Quantum Mechanics as a Computational Paradigm

Modern computing is based on the laws of classical physics and mathematical
logic. Though the functioning of electronic components is based on quantum
mechanical principles, the logic they follow is classical. The classical
computer is tailored for serial computation, whereby algorithms progress
sequentially from one point to another over time. This sequential nature
necessitates the completion of specific operation before subsequent ones can
commence, thus rendering classical computation time-intensive. It is important
to note, however, that parallel computing offers an alternative approach:
Parallel computing involves the dissection of computational problems or tasks
into autonomous logical units that can be executed concurrently. For instance,
the multiplication of two matrices, denoted as $AB=C$. Elements such as
$C_{11}$ and $C_{21}$ within matrix $C$ can be computed in parallel fashion.
By employing the first row of matrix $A$ and the first column of matrix $B$ to
calculate $C_{11}$, and the second row of matrix $A$ and the first column of
matrix $B$ for $C_{21}$, a simultaneous and independent computation of all
elements $C_{ij}$ within matrix $C$ becomes achievable.

Moreover, classical computers utilize irreversible gates which generate heat
during their operational processes. This phenomenon arises when information is
erased while operating irreversible gates, leading to the dissipation of
energy, as observed by Landauer [1]. The act of deleting information
consistently, necessitates work and the energy consumption [2]. In order to
render classical computation reversible, supplementary bits must be
incorporated, thereby giving rise to the predicament of excess waste commonly
referred to as the "garbage problem" in classical reversible computation [3,
4]. The amount of heat produced by a computer is contingent upon the scale and
number of gates (electronic components), and if these components are
positioned too closely, the heat generated by one component can potentially
harm neighboring components, posing a challenge in the miniaturization of
classical computers. This challenge pertaining to the growth of computer
hardware was projected and formulated by Intel co-founder Gordon Moore in
1965, famously known as Moore’s Law, which states-

  * -

The number of transistors on a microchip doubles every two years that we can
expect the speed and capability of our computers to increase every couple of
years.

Figure 1.1: A logarithmic graph showing the timeline of how transistor counts
in microchips are almost doubling every two years from 1970 to 2020; (Moore’s
Law). Source: ourworldindata.org

The conventional approaches to developing computer technology will inevitably
encounter challenges associated with size limitations. As devices continue to
shrink, the impact of quantum effects on their functionality becomes a
concern. The question arises: will Moore’s Law become obsolete? The answer is
no. An alternative solution exists - transitioning to quantum computation!
Quantum computation is an immensely captivating and rapidly advancing field of
research [5, 6, 7]. This novel computing paradigm leverages the principles of
quantum mechanics instead of classical physics to execute computations, a
concept initially conceived by Feynman [8, 9] and Deutsch [10].

  * -

"How can we simulate the quantum mechanics? Can you do it with a new kind of
computer - a quantum computer? It is not a Turing machine, but a machine of a
different kind." \- R. P. Feynman, 1981.

The operation of quantum computers is rooted in the principles of quantum
mechanics, harnessing various quantum features such as unitary evolution,
quantum superposition, and quantum entanglement. These elements provide a
fundamental speed advantage over classical computers [11, 12]. This speed
advantage is so substantial that many researchers believe that no feasible
advancements in classical computation could bridge the gap between the power
of a classical computer and that of a quantum computer. Quantum computing
represents a complete paradigm shift in the functioning and operation of
computers. By leveraging these novel quantum properties, we can develop
innovative types of software and hardware. It is anticipated that explorations
in this field may eventually yield information processing devices with
capabilities surpassing those of today’s computing and communication systems.

###  1.2 Quantum Computing and Information Processing

The domain of quantum computing and quantum information (QCQI) has undergone
substantial expansion during the preceding two decades. This field encompasses
the study and execution of information processing tasks that can be
effectively conducted employing a quantum mechanical framework. Quantum
computers possess the capacity to undertake computational operations that lie
beyond the capabilities of classical computers. The encoding of $n$-classical
information bits mandates no less than $n$ classical resources. However, due
to the principle of quantum superposition, quantum mechanical systems can
theoretically exhibit a more efficient encoding efficiency compared to
classical systems[13].

In 1981, R. Feynman introduced the concept of a ‘quantum computer’ and
demonstrated that a classical computer would encounter an exponential
deceleration while emulating a quantum phenomenon, whereas a quantum computer
would not be subject to such constraints[14]. Subsequently, in 1985, D.
Deutsch, took Feynman’s ideas further and defined two models of quantum
computation; he also devised the first quantum algorithm. One of Deutsch’s
ideas is that quantum computers could take advantage of the computational
power present in many ”parallel universes” and thus outperform conventional
classical algorithms[15]. In 1994, P. Shor showcased the resolution of two
significant problems determining the prime factors of an integer and the
discrete logarithm problem both of which could be efficiently solved on a
quantum computer, underscoring the prowess of quantum computing[13, 16].
Furthermore, in 1996, L. Grover demonstrated that a search algorithm for an
unsorted database executed on a quantum computer exhibits quadratic speed-up
relative to its classical counterpart[17].

In 2000, theoretical physicist David P. DiVincenzo proposed a set of
prerequisites for the practical realization of an operational quantum
computer, later known as DiVincenzo criteria[18]. These prerequisites
encompass a scalable physical framework, the capability to initialize the
system to any quantum state, a comprehensive list of quantum gates amenable to
implementation, qubit-specific measurement procedures, and sufficiently long
coherence times relative to the durations of gate implementations. To this
day, no quantum hardware has been found to comprehensively satisfy these
specified criteria. Howwver, numerous experiments in the realm of quantum
computing have been conducted employing diverse technologies such as optical
photons[19, 20], ion traps[21, 22], superconducting qubits[23], nitrogen-
vacancy (NV) centers[24, 25], and nuclear magnetic resonance (NMR)
techniques[26, 27, 28, 29].

In optical photon-based quantum computers, qubits are encoded in the
polarization state of photons. The initial state is prepared by generating
single-photon states through light attenuation. Quantum gates are implemented
using beam-splitters, phase shifters, and nonlinear Kerr media. Measurement is
conducted by detecting individual photons utilizing a photomultiplier
tube[19]. Similarly, trapped ion quantum computers rely on ions that are
cooled to a state where their vibrational energy approaches zero, thereby
enabling the realization of qubits via the hyperfine state of an atom combined
with the lowest-energy vibrational modes of the trapped atoms[22]. Quantum
gates in this system are constructed using laser pulses, and measurements are
derived from population measurements of hyperfine states[21]. In the realm of
superconducting quantum computers, qubits are denoted by phase, charge, and
flux qubits[23]. For instance, in the charge qubit, different energy levels
correspond to integral numbers of Cooper pairs residing on a superconducting
island. Quantum gates for such systems are realized through microwave pulses.

The NV-center, a point defect in diamond, presents a controlled, isolated
quantum system that can be manipulated at room temperature. With electronic
and nuclear spin components of $S=1$ and $I=1$, respectively, the ${}^{14}N$
NV- center exhibits nine eigenvalue corresponding different spin levels that
can be harnessed to realize qubits. Employing resonant microwave pulses,
comprehensive quantum control on quantum state is achieved. Measurement
methods encompass optical and electrical detection. Although the NV center is
susceptible to the effects of absolute temperature and temperature variations,
its characteristics at room temperature make it highly appropriate for a
diverse range of applications, encompassing quantum sensors and quantum
computing, as indicated in [24, 25]. In May 2016, IBM Corporation introduced a
quantum computer comprising five qubits onto the IBM Cloud platform. This
initiative aimed to facilitate the execution of algorithms, conduct
experiments, and facilitate the exploration of tutorials and simulations to
gauge the potential of quantum computing. This five-qubit quantum computer is
of a universal nature and is constructed upon superconducting transmon
qubits[30]. Additionally, IBM is actively engaged in the ongoing enhancement
of qubit capabilities. Thus, the concept of a quantum computer has evolved
beyond the realm of theory and materialized as a tangible computational
apparatus. In the foreseeable future, quantum computers are poised to address
genuine problems effectively.

This thesis employs NMR as a tool to undertake tasks associated with quantum
information processing. NMR-based quantum computing has established itself as
a robust platform for the practical implementation of a diverse array of
quantum information processing protocols[31]. Within the NMR context, the
chemical shifts of different spins are leveraged to individually address these
spins in frequency space, and external radio frequency pulses are harnessed
for quantum control[29]. Quantum information processing necessitates the
utilization of pure quantum states. However, the NMR spin system, when
operating at room temperature, deviates significantly from this ideal due to
the fact that the energy gap between spin levels $\hbar\omega$ is
substantially smaller than $k_{\beta}T$. As a result, the initial state of an
ensemble of nuclear spins is mixed. Nonetheless, for computational purposes,
it becomes possible to initialize the system into a pseudo pure state (PPS)
[29] that emulates a true pure state. By employing radio frequency pulses and
leveraging the interactions among spins, any unitary operator can be executed.
Furthermore, the mitigation of errors arising from pulse imperfections and
offset errors can be achieved through the utilization of numerically optimized
pulses employing techniques like Gradient Ascent Pulse Engineering (GRAPE) and
genetic algorithms [32, 33]. This feature solidifies NMR as an ideal
experimental domain for the realization of quantum algorithms[34, 35, 36, 37].

The research conducted in this thesis is centred around the designing and
empirical implementation of a range of quantum tomography protocols. These
protocols are devised to efficiently characterize and reconstruct unknown
quantum states and processes, employing both spin ensemble-based NMR quantum
information processors and IBM quantum processors built on superconducting
technology. The primary goal is to achieve the reconstruction of quantum
states through quantum state tomography (QST) protocols and the
characterization of quantum processes via quantum process tomography (QPT)
protocols. Both QST and QPT play pivotal roles in assessing the reliability
and performance of quantum processors. However, the exponential increase in
computational complexity with system size poses a fundamental challenge,
rendering their experimental execution unfeasible, particularly for systems of
greater dimensions. The presence of finite ensemble sizes and inherent
systematic errors further leads to the emergence of unphysical density
matrices and process matrices. To address these issues, a multitude of QST and
QPT protocols have been proposed, albeit many remain untested in experimental
settings.

The core objective of this study is to outline pragmatic strategies for the
effective implementation of tomography protocols on NMR and IBM quantum
processors. To mitigate the unphysicality concern associated with
experimentally reconstructed quantum states and processes using standard
tomography techniques, the tasks of QST and QPT are reformulated as
constrained convex optimization (CCO) problems. This conversion facilitates
the reconstruction of valid quantum states and processes. Additionally, for
QPT, it permits the computation of a comprehensive set of Kraus operators
corresponding to a given quantum process. The application of compressed
sensing (CS) and artificial neural network (ANN) techniques is also explored,
enabling tomography of quantum states and gates from a significantly reduced
dataset compared to conventional methods. CS and ANN-based tomography methods
hold promise for handling complexity issues inherent in characterizing higher-
dimensional quantum gates.

Furthermore, the investigation delves into the challenge of selectively and
directly estimating desired elements within the process matrix that
characterizes a quantum process. This exploration results in the formulation
of a selective and efficient quantum process tomography protocol (SEQPT),
presenting a generalized quantum algorithm and circuit for SEQPT
implementation. Successful experimental demonstrations are showcased using NMR
and IBM quantum processors. The research also introduces an efficient direct
approach to QST and QPT, employing the weak measurement technique, with
empirical validation accomplished through a three-qubit NMR system. Moreover,
the thesis also studies the simulation and characterization of open quantum
system dynamics via dilation techniques. To underscore the efficacy of the
proposed quantum tomography and simulation protocols, empirical findings are
juxtaposed with theoretically anticipated outcomes across various two- and
three-qubit quantum systems.

####  1.2.1 Quantum bit

A quantum bit, generally termed as Qubit (this term was coined by
Schumacher[38]) is the quantum analogue of classical bit and is the most
fundamental and the smallest unit of information used in QCQI . In the case of
qubit, logical states 0 and 1 are represented by $|0\rangle$ and $|1\rangle$.
A single qubit is a two-level quantum system represented by a vector in two
dimensional Hilbert space spanned by $|0\rangle=\begin{pmatrix}1\\\
0\end{pmatrix}$ and $|1\rangle=\begin{pmatrix}0\\\ 1\end{pmatrix}$ basis
vectors and can be physically realized using the spin-1/2 particle.

Figure 1.2: Qubit representation on bloch sphere, source: Google

The most general state $|\psi\rangle$ of the qubit in polar form, also
referred as ‘Bloch sphere representation of a qubit’ is given as,

$|\psi\rangle=\cos\left(\frac{\theta}{2}\right)|0\rangle+e^{i\phi}\sin\left(\frac{\theta}{2}\right)|1\rangle$
(1.1)

Note that the global phase is ignored in the above representation of
$|\psi\rangle$ as it does not have over all observable effect on measurement
outcome. The state $|\psi\rangle$ given in Eq.1.1 can be visualized using
Bloch sphere of unit radius as given in Fig.1.2.

One can also construct multi-qubit quantum register of length $n$. A $n$-qubit
quantum register comprises of $n$ number of qubits and the state of such
$n$-qubit composite system can be represented by vector in $2^{n}$ dimensional
Hilbert space. The most general representation of $n$-qubit quantum state is
given as,

$|\Psi\rangle=\sum_{i=1}^{2^{n}}c_{i}|e_{i}\rangle$ (1.2)

where $|e_{i}\rangle=\\{|0\rangle,|1\rangle\\}^{\otimes n}$ is the $n$-qubit
basis vector constructed by taking tensor product of basis vectors of
individual qubits and $c_{i}\in\mathbb{C}$ is the corresponding coefficient
such that $\sum_{i}|c_{i}|^{2}=1$. As an example, the 2-qubit quantum state is
given as,

$|\Psi\rangle=c_{00}|00\rangle+c_{01}|01\rangle+c_{10}|10\rangle+c_{11}|11\rangle$
(1.3)

where $|00\rangle=|0\rangle\otimes|0\rangle$,
$|01\rangle=|0\rangle\otimes|1\rangle$, and so on.

If one could able to write the $n$-qubit state given in Eq.1.2 as tensor
product of individual qubit states $|\psi_{i}\rangle$ as given below,

$|\Psi\rangle=\left|\psi_{1}\right\rangle\otimes\left|\psi_{2}\right\rangle\otimes\ldots\otimes\left|\psi_{n}\right\rangle$
(1.4)

then the state $|\Psi\rangle$ is said to be separable state else it is a
entangled state. Entangled states don’t have any classical analogue and play
very crucial role in QCQI[39, 40, 41, 42].

Figure 1.3: Implementation of 3-qubit Toffoli gate using Hadamard (H), phase
gate (S), $\pi/8$ gate (T) and CNOT gate.

####  1.2.2 Quantum gates

An $n$-qubit quantum gate is represented by $2^{n}\times 2^{n}$ dimensional
unitary matrix which acts on $n$-qubit input state vector and gives $n$-qubit
output state vector. The following are some of the important quantum gates
(matrix representation is given in Fig.1.4):

1-qubit QGs $\displaystyle:\textit{X, Y, Z, H, S, T}$ 2-qubit QGs
$\displaystyle:\textit{CNOT, CZ, SWAP}$ 3-qubit QGs
$\displaystyle:\textit{Toffoli}$

Arbitrary $n$-qubit quantum gate can be decomposed into set of 1 and 2-qubit
quantum gates called universal quantum gates. One of the widely used universal
set of quantum gates is: $\\{\textit{H, S, T, CNOT}\\}$. As an example, the
decomposition of 3-qubit Toffoli gate into universal set of quantum gates is
given in the Fig.1.3. However, in general, finding decomposition using
universal quantum gates in order to create a given unitary matrix that can be
implemented efficiently is itself a challenging task.

####  1.2.3 Quantum Measurement

Consider the Hermitian observable represented by operator $A$ with
eigenvectors $|\lambda_{n}\rangle$ and corresponding eigenvalues
$\lambda_{n}$. Using spectral theorem, operator $A$ can be rewritten as,

$A=\sum_{n}\lambda_{n}P_{n}$ (1.5)

where $P_{n}=|\lambda_{n}\rangle\langle\lambda_{n}|$ is a projection operator
onto eigenstate $|\lambda_{n}\rangle$ satisfying following properties: i)
$P_{m}P_{n}=\delta_{mn}P_{n}$, ii) $P_{n}^{\dagger}=P_{n}$ and iii)
$\sum_{n}P_{n}=I$.

If a measurement of $A$ is made on quantum state $|\Psi\rangle$ given in
Eq.1.2 then the state of the system will collapse onto one of the eigenstates
of $A$ and the result will be the corresponding eigenvalue $\lambda_{n}$ with
probability $|c_{n}|^{2}=\langle\Psi|P_{n}|\Psi\rangle$. The post-measurement
state $|\Psi\rangle_{final}$ of the system is given as,

$|\Psi\rangle_{final}=\frac{P_{n}|\Psi\rangle}{\sqrt{\langle\Psi|P_{n}|\Psi\rangle}}$
(1.6)

The expectation value $\langle A\rangle$ of observable $A$ is defined as,

$\langle A\rangle=\langle\Psi|A|\Psi\rangle$ (1.7)

Figure 1.4: Common quantum logic gates by name, circuit form(s) and matrices.
source: Google

####  1.2.4 Quantum ensemble and density matrix formalism

A collection of independent, isolated (not interacting) and identical systems
is called an ensemble. For example, the collection of proton spins in tube of
water can be treated (up to a very good approximation) as ensemble of
spin-$\frac{1}{2}$ particles. Consider the ensemble of size $N$ out of which
$n_{1}$ number of individual systems are prepared in the state
$|\psi_{1}\rangle$, $n_{2}$ number of individual systems are prepared in the
state $|\psi_{2}\rangle$, and so on such that $\sum_{i}n_{i}=N$. The density
matrix $\rho$ describing the state of such ensemble system is given as,

$\rho=\sum_{i}p_{i}|\psi_{i}\rangle\langle\psi_{i}|$ (1.8)

where $p_{i}=\frac{n_{i}}{N}$ is classical probability of having individual
system in the state $|\psi_{i}\rangle$ such that $\sum_{i}p_{i}=1$. The $\rho$
has following three properties: i) $\rho$ is Hermitian matrix, i.e.
$\rho=\rho^{\dagger}$, ii) $\rho$ is positive semi-definite, i.e. $\rho\geq
0$. In other words, all the eigenvalues of $\rho$ are non-negative and iii)
$\rho$ has a unit trace, i.e. $\rm{Tr}(\rho)=1$. It turns out that for pure
ensemble we have $\rm{Tr}(\rho^{2})=1$ while in the case of mixed ensemble we
have $\rm{Tr}(\rho^{2})<1$. The density matrix elements in given basis set
$\\{|e_{i}\rangle\\}$ can be written as,

$\rho_{mn}=\langle e_{m}|\rho|e_{n}\rangle$ (1.9)

The diagonal elements $\rho_{mm}$ represents the Born probability of getting
state $|e_{m}\rangle$ or sometimes also interpreted as the population of state
$|e_{m}\rangle$ while the off-diagonal elements $\rho_{mn}$ ($m\neq n$) are
called coherences between state $|e_{m}\rangle$ and $|e_{n}\rangle$.

###  1.3 Theory of NMR

The term ’NMR’ stands for ’Nuclear Magnetic Resonance’. It is a natural
phenomenon observed in the atomic nuclei having non-zero nuclear spin (or
magnetic dipole moment). If such nucleus is concomitantly placed in the
presence of a static magnetic field and an oscillating electromagnetic field
with appropriate frequency then an absorption or emission can occur, the
phenomenon is termed as ’nuclear magnetic resonance’. In NMR, the Zeeman
Hamiltonian of single nuclear spin having magnetic dipole moment
$\boldsymbol{\mu}$ placed in external magnetic field
($\mathbf{B}=B_{0}\hat{z}$) is given as,

$H=-\boldsymbol{\mu}.\mathbf{B}=-\mu_{z}B_{0}=-\gamma\hbar
B_{0}I_{z}=-\omega_{0}\hbar I_{z}$ (1.10)

where $\gamma$ is called the gyromagnetic ratio of the nucleus, $I_{z}$ is a
dimensionless operator representing z-component of angular momentum or nuclear
spin and $\omega_{0}=\gamma B_{0}$ is known as ’Larmor frequency’. The
eigenvectors of $I_{z}$ are denoted by $|m\rangle$ where
$m=-I,-I+1,-I+2,...,I-1,I$. The eigenvalues of the Hamiltonian are directly
proportional to eigenvalues of $I_{z}$, given as,

$E_{m}=-m\hbar\omega_{0}$ (1.11)

In the case where given molecule has more than one NMR active nuclei placed in
magnetic field then the total NMR Hamiltonian $H_{\rm tot}$ is given as,

$H_{\rm tot}=\sum_{i}H_{z}^{i}+H_{\rm int}$ (1.12)

where $H_{z}^{i}$ is Zeeman Hamiltonian of $i$th nucleus and $H_{\rm int}$ is
inter-spin interaction Hamiltonian. The general form of interaction
Hamiltonian $H_{\rm int}$ for a system of coupled spins is given as,

$H_{\rm int}=H_{\rm cs}+H_{\rm J}+H_{\rm Q}+H_{\rm D}$ (1.13)

where $H_{\rm cs}$ is chemical shift interaction, $H_{\rm J}$ is scalar or
J-coupling, $H_{\rm Q}$ is quadrupolar interaction and $H_{\rm D}$ is dipolar
interaction. The $H_{\rm cs}$ is due to orbital motion of the surrounding
electrons, $H_{\rm J}$ is the electron-mediated interaction between nuclei,
$H_{\rm Q}$ is the quadrupolar interaction between a nucleus with spin $>$ 1/2
and the electric field gradient at the nuclear position and $H_{\rm D}$ is the
direct (through space) dipolar interaction between nuclei. However, in liquid
state NMR, $H_{\rm Q}$ and $H_{\rm D}$ get averaged to zero and only surviving
terms are $H_{\rm cs}$ and $H_{\rm J}$.

##### Interaction with radio frequency field: The NMR phenomenon

The transitions between energy eigenstates of Zeeman Hamiltonian can be
induced using a radio-frequency pulse. The Hamiltonian associated with RF
pulse is known as RF Hamiltonian $H_{RF}$. For RF pulse applied along x
direction with oscillating magnetic field
$\boldsymbol{B}_{1}(t)=2B_{1}\cos(\Omega t+\phi)\hat{x}$, RF Hamiltonian has
the form[29],

$H_{RF}=-\boldsymbol{\mu}\cdot\boldsymbol{B}_{1}(t)=-\hbar\gamma
I_{x}[2B_{1}\cos(\Omega t+\phi)]$ (1.14)

On resonance ($\omega_{0}\approx\Omega$), the transition rate between spin
state $|m\rangle$ and $|n\rangle$ is given by Fermi golden rule[29],

$P_{m\longrightarrow n}\propto\gamma^{2}\hbar^{2}B_{1}^{2}|\langle
m|I_{x}|n\rangle|^{2}.$ (1.15)

The allowed transitions are given by selection rule: $\triangle m=\pm 1$. The
amplitude of oscillating magnetic field $B_{1}$ is very small compared to
$B_{0}$. So, $H_{RF}$ can be treated as a small perturbation to the Zeeman
Hamiltonian and the evolution of nuclear spin in presence of RF pulse can be
obtained using time-dependent perturbation theory. However, for simplicity the
evolution of spin states under $H_{RF}$ can also be visualized using rotating
frame approximation via semi-classical treatment known as ’vector model’. The
effective Hamiltonian $H_{eff}$ of single spin in rotating frame of frequency
$\Omega$ can be approximated as[29],

$H_{eff}=\omega^{0}I_{z}+w_{1}\left[\cos(\phi)I_{x}+\sin(\phi)I_{y}\right]$
(1.16)

where $\omega^{0}=(\omega_{0}-\Omega)$ is called resonance offset and
$\omega_{1}=\gamma B_{1}$ is known as Nutation frequency. The resonance occurs
at $\omega_{0}=\Omega$.

##### Thermal density matrix

In reality, no system is completely isolated, there is always some interaction
present with environment which leads system to go to thermal equilibrium with
it’s environment. The thermal density matrix $\rho_{0}$ is simply related to
system’s Hamiltonian $H$ as,

$\rho_{0}=\frac{e^{-H/k_{B}T}}{\sum_{m}e^{-E_{m}/k_{B}T}}$ (1.17)

where sum in denominator is called the partition function $Z$ of the system
extended over all Hamiltonian eigenstates and $E_{m}$ represents the
eigenvalues of $H$. In the basis formed by Hamiltonian eigenstates, the
thermal density matrix is given as,

$\rho_{0}=\frac{1}{Z}\left(\begin{array}[]{cccccc}e^{\frac{-E_{1}}{k_{B}T}}&0&0&\cdot&\cdot&\\\
0&e^{\frac{-E_{2}}{k_{B}^{T}}}&0&0&\cdot&\cdot\\\
0&0&e^{\frac{-E_{3}}{k_{B}T}}&0&0&\cdot\\\ \cdot&\cdot&\cdot&\cdot&&\\\
\cdot&\cdot&\cdot&&\cdot&\\\ \cdot&\cdot&\cdot&&&.\end{array}\right)$ (1.18)

where $E_{m}=m\hbar\omega_{o}$. In high-temperature limit
($k_{B}T\gg\hbar\omega_{0}$) we get,

$\displaystyle e^{m\hbar\omega_{0}/k_{B}T}\cong
1+\frac{m\hbar\omega_{0}}{k_{B}T}$ (1.19)
$\displaystyle\sum_{s=-I}^{I}e^{s\hbar\omega_{0}/k_{B}T}\cong 2I+1$

Using Eq.1.18 and 1.19, thermal density matrix $\rho_{0}$ for spin-1/2
ensemble can be written as,

$\rho_{0}=\left(\frac{1}{2I+1}\right)\mathbf{1}+\left(\frac{1}{2I+1}\Delta\right)I_{z}$
(1.20)

where $\mathbf{1}$ is the identity matrix and
$\Delta=\frac{\hbar\omega_{0}}{k_{B}T}$. The first term in the Eq.1.20 is
treated as uniform background hence it does not contribute to NMR signal. On
the other hand the second term is called as deviation density matrix
($\Delta\rho_{0}$) which constitutes the starting point for all NMR
experiments.

##### Relaxation phenomenon

NMR spin ensemble regain the thermal equilibrium state through mainly two
types of relaxation processes: i) Transverse relaxation ($T_{2}$) affecting
$M_{x,y}$, and ii) Longitudinal relaxation ($T_{1}$) impacting $M_{z}$. The
transverse and longitudinal damping rates, $\gamma_{2}$ and $\gamma_{1}$, are
defined by $\gamma_{2}=1/T_{2}$ and $\gamma_{1}=1/T_{1}$ respectively. These
processes are governed by the Bloch equations, encompassing the decay of
transverse components ($M_{x,y}$) and recovery of the thermal magnetization
along z-axis ($M_{z}=M_{0}$).

Transverse relaxation
$\displaystyle:\frac{dM_{x,y}}{dt}=\frac{-M_{x,y}}{T_{2}}\Longrightarrow
M_{x,y}=M_{0}e^{-t/T_{2}}$ Longitudinal relaxation
$\displaystyle:\frac{dM_{z}}{dt}=\frac{M_{0}-M_{z}}{T_{1}}\Longrightarrow
M_{z}=M_{0}\left(1-e^{-t/T_{1}}\right)$

##### NMR spectrum

In signal acquisition, an RF pulse flips (also referred as readout pulse), the
magnetization vector from equilibrium to the transverse plane which precesses
at Larmor frequency in the xy-plane. This precession induces a magnetic field
change, leading to electromagnetic induction and voltage signal emergence,
known as free induction delay (FID). The total FID signal is generally
represented by complex number $S(t)$ as,

$\displaystyle S(t)$ $\displaystyle=(S_{x}(t)+iS_{y}(t))e^{-t/T_{2}}$ (1.21)
$\displaystyle=(S_{0}\cos\omega_{0}t+iS_{0}\sin\omega_{0}t)e^{-t/T_{2}}$
$\displaystyle=S_{0}\exp(i\omega_{0}t)e^{-t/T_{2}}.$

where $S_{0}$ is total magnetization while $S_{x}=S_{0}\cos\omega_{0}t$ and
$S_{y}=S_{0}\sin\omega_{0}t$ are the $x$ and $y$ components of the total
magnetization and $e^{-t/T_{2}}$ is transverse magnetization damping factor.
Fourier transformation (FT) of $S(t)$ gives the frequency domain signal
$S(\omega)$, a Lorentzian function, called ‘NMR spectrum’ given as,

$\displaystyle S(t)$ $\displaystyle\xrightarrow[\text{}]{\text{FT}}S(\omega)$
(1.22) $\displaystyle S_{0}\exp(i\omega_{0}t)e^{-\gamma_{2}t}$
$\displaystyle\xrightarrow[\text{}]{\text{FT}}\underbrace{\frac{S_{0}\gamma_{2}}{\gamma_{2}^{2}+(\omega-\omega_{0})^{2}}}_{\text{real
}}+\mathrm{i}\underbrace{\frac{-S_{0}(\omega-\omega_{0})}{\gamma_{2}^{2}+(\omega-\omega_{0})^{2}}}_{\text{imaginary
}}$

In the NMR language, the real part of spectrum is called ‘absorption mode’
while the imaginary part of spectrum is called ‘dispersion mode’. Generally,
an NMR spectrum is a combination of the absorption and dispersion modes.

###  1.4 NMR quantum computing

In the year 1997, D. G. Cory and I. L. Chuang independently introduced a
quantum computer model based on Nuclear Magnetic Resonance (NMR), capable of
being programmed akin to conventional quantum computers[43, 44]. Their model
involves an ensemble quantum computer framework where measurement outcomes
manifest as the anticipated values of observables. This computational paradigm
can be actualized through NMR spectroscopy on extensive ensembles of nuclear
spins. The NMR has been used to experimentally demonstrate several quantum
algorithms like Grover’s search algorithm[45], Deutsch-Jozsa algorithm[35],
Order-Finding algorithm[46], Shor’s algorithm[47] and many more[29].

The NMR spectrometer encompasses a superconducting magnet inducing a robust
magnetic field along the z-direction, along with RF coils for both exciting
spins and capturing NMR signals from the relaxing spin ensemble. Upon sample
placement within the magnetic field, spin interactions trigger energy level
splitting contingent on the spin system size. At room temperature, energy
level populations adhere to the Boltzmann distribution, yielding a mixed state
at thermal equilibrium. This scenario presents a challenge for quantum
computing, which mandates pristine initial quantum states. This challenge is
surmounted in NMR quantum computing by initiating a "pseudopure" state that
mimics a pure state. Quantum gates are executed using RF pulses and inter-spin
interactions, with computation outcomes recorded as NMR signals representing
average magnetization in the x and y directions. These signals correspond
directly to expectation values of certain qubit basis set elements. By
applying RF pulses on individual spins, the anticipations for all basis set
elements can be computed, enabling density matrix reconstruction.

Recent advancements in NMR, specifically in the domain of spin dynamics
control through RF pulses, have facilitated the high-fidelity implementation
of quantum gates for NMR quantum computing. Nuclear spins exhibit extended
coherence times due to their isolation from the environment. Despite these
merits, a notable constraint of liquid state NMR quantum computers remains
scalability. Subsequent sections delve into topics including state
initialization, quantum gate implementation, and measurement within NMR
quantum computing.

####  1.4.1 Well characterized qubits in NMR

A Qubit is a two level quantum system which can be realized in NMR by
spin-$\frac{1}{2}$ nucleus placed in static magnetic field with corresponding
Zeeman Hamiltonian $H=\nu I_{z}$, yielding two energy eigenstate:
$|\frac{1}{2}\rangle$ and $|-\frac{1}{2}\rangle$ with corresponding energy
eigenvalues $+\frac{1}{2}\nu$ and $-\frac{1}{2}\nu$ respectively where
$\nu=\frac{\gamma B_{0}}{2\pi}$ is Larmor frequency (in Hertz (Hz)) of given
nucleus.

The Hamiltonian for a system of $n$-interacting spin-1/2 nuclei in a magnetic
field, is given as:

$H_{0}=\sum_{i=1}^{n}\omega_{i}I_{z}^{i}+2\pi\sum_{i<j}^{n}J_{ij}I^{i}\cdot
I^{j}$ (1.23)

where $J_{ij}$ is scalar coupling between the $i$th and $j$the coupling.
Multi-qubit system can be realized using such $n$-interacting spin-1/2 nuclei.

Figure 1.5: Allowed transitions and NMR spectrums in coupled two spin-1/2
nuclei.

Apart from qubit one can also have other units of information like:

Qutrit $\displaystyle:\text{3-level quantum system}\quad(\text{spin-1 nuclei
like: ${}^{2}H$, ${}^{14}N$})$ Ququart $\displaystyle:\text{4-level quantum
system}\quad(\text{spin-3/2 nuclei like: ${}^{11}B$, ${}^{75}As$,
${}^{79}Br$})$ Qudit $\displaystyle:\text{d-level quantum system}\quad(\text{A
quadrupolar nuclei})$

Nuclei with spin $>1/2$ are called quadrupolar nuclei.

####  1.4.2 Initializing quantum circuits in NMR

The NMR apparatus operates with a spin ensemble at ambient temperature. In
liquid-state NMR experiments, the sample volume is typically 400-600 $\mu l$,
containing approximately $10^{17}-10^{18}$ nuclear spins. At thermal
equilibrium, Boltzmann distribution governs energy eigenstate population,
resulting in a mixed thermal density matrix of the form:

$\rho_{\mathrm{eq}}\approx\frac{1}{2^{\mathrm{N}}}\left({I}+\epsilon\Delta\rho_{\mathrm{eq}}\right)$
(1.24)

where $I$ is ${2^{n}\times 2^{n}}$ identity matrix , $\epsilon$ is associated
with NMR signal strength, The first term in the $\Delta\rho_{\mathrm{eq}}$ is
deviation density matrix. Using the fact that NMR is only responsive to
deviation part of the density matrix which provides the base of the concept of
Pseudo-pure state (PPS)[29]. The PPS ($\rho_{pps}$) of an n-qubit system
corresponding to $|\psi\rangle$ is given as,

$\rho_{pps}=\frac{1-\epsilon}{N}I+\epsilon|\psi\rangle\langle\psi|$ (1.25)

The first term in the $\rho_{pss}$ corresponds to uniform background while the
second term is the deviation density matrix. All the computational tasks and
calculations are analysed by considering only deviation term.

In NMR, the preparation of PPS from thermal density matrix is done using
several techniques, out of which two methods used in study are briefly
described here: i) Temporal averaging method and ii) Spatial averaging method.

##### Temporal averaging method

In temporal averaging method, results from multiple experiments are added
together to get PPS. E.g, in the case of two-qubit PPS state preparation,
three unitary gates $\\{U_{0},U_{1},U_{2}\\}$ are applied on initial thermal
density matrix $\rho_{th}$ given as,

$\rho_{th}=\left(\begin{array}[]{llll}a&0&0&0\\\ 0&b&0&0\\\ 0&0&c&0\\\
0&0&0&d\end{array}\right)$ (1.26)

The three unitary matrices are given as,

$U_{0}=\left(\begin{array}[]{llll}1&0&0&0\\\ 0&1&0&0\\\ 0&0&1&0\\\
0&0&0&1\end{array}\right),U_{1}=\left(\begin{array}[]{llll}1&0&0&0\\\
0&0&1&0\\\ 0&0&0&1\\\ 0&1&0&0\end{array}\right)\text{ and
}U_{2}=\left(\begin{array}[]{llll}1&0&0&0\\\ 0&0&0&1\\\ 0&1&0&0\\\
0&0&1&0\end{array}\right)$ (1.27)

After adding all output density matrices
$\rho_{i}=U_{i}\rho_{th}U_{i}^{\dagger}$, the PPS can be obtained as:

$\rho_{pps}=\rho_{0}+\rho_{1}+\rho_{2}=\left(\begin{array}[]{cccc}3a&0&0&0\\\
0&b+c+d&0&0\\\ 0&0&b+c+d&0\\\ 0&0&0&b+c+d\end{array}\right)$ (1.28)

The above equation can be rewritten as,

$\rho_{pps}=(1-a)\mathbf{1}+(4a-1)|00\rangle\langle 00|$ (1.29)

The $\rho_{pps}$ represents the PPS corresponding to state $|00\rangle$ and
will evolve exactly as pure state $|00\rangle\langle 00|$.

##### Spatial averaging method

The spatial averaging method is based on spatially separated sub-ensembles. In
NMR, sub-ensemble can be accessed using combination of RF pulses and pulsed
magnetic field gradients (helps to dephase spins so that off-diagonal terms
vanishes). The PPS is then average over all sub-ensembles. So, using both RF
and gradient pulses together the PPS can be prepared in a single experiment.
In case of two-qubit system, the pulse sequence used in spatial averaging
method to prepare PPS corresponding to $|00\rangle$ from thermal density
matrix is given below,

$\left[G_{z}(\tau)\left(\frac{\pi}{4}\right)_{-y}^{1}U_{J}\left(\frac{1}{2J}\right)\left(\frac{-\pi}{4}\right)_{x}^{1}G_{z}(\tau)\left(\frac{\pi}{3}\right)_{x}^{2}\right]$
(1.30)

where $G_{z}(\tau)$ represents a gradient pulse of duration $\tau$, and
$U_{J}\left(\frac{1}{2J}\right)$ is evolution under J-coupling and
$\theta^{i}_{\pm x(y)}$ is local rotation about $\pm x(y)$ on $i$th qubit.
Generally spatial averaging method of PPS preparation suffer magnetization
loss due to non-unitary evolution achieved by gradient pulses.

####  1.4.3 Quantum gate implementation in NMR

The quantum gates are unitary operators which are used to process the quantum
information encoded in the state of the quantum system. They are broadly
classified as: single qubit gates and multi-qubit gates. As mentioned earlier
in this chapter, any $n$-qubit arbitrary quantum gate can be decomposed using
universal set of quantum gates comprising the set of single qubit gates and
two-qubit CNOT gate. In this section we briefly describe the NMR
implementation of universal quantum gates which further can be used to
implement arbitrary high dimensional gates.

##### Single qubit gates

In NMR rotation operators are implemented using appropriate radio frequency
(RF) pulses. From Eq.1.16, the effective RF Hamiltonian (on resonance) is
given as:

$H_{eff}=w_{1}\left[\cos(\phi)I_{x}+\sin(\phi)I_{y}\right]$

If RF pulse of length $t$ is applied with appropriate phase $\phi$, then
rotation operators can be implemented as follows:

$\displaystyle R_{x}(\theta)=e^{-i\theta I_{x}}=(\theta)_{x},\quad\text{such
that $\phi=0$ and $\theta=\omega_{1}t$}$ $\displaystyle
R_{y}(\theta)=e^{-i\theta I_{y}}=(\theta)_{y},\quad\text{such that
$\phi=\frac{\pi}{2}$ and $\theta=\omega_{1}t$}$ $\displaystyle
R_{z}(\theta)=e^{-i\theta
I_{z}}=(\frac{\pi}{2})_{x}(\theta)_{y}(\frac{-\pi}{2})_{x}$

It has been shown that, an arbitrary single qubit unitary matrix can be
decomposed and implemented on NMR using following pulse sequence:

$U=e^{i\alpha}\left[\begin{array}[]{cc}e^{-i\beta/2}&0\\\
0&e^{i\beta/2}\end{array}\right]\left[\begin{array}[]{cc}\cos\frac{\gamma}{2}&-\sin\frac{\gamma}{2}\\\
\sin\frac{\gamma}{2}&\cos\frac{\gamma}{2}\end{array}\right]\left[\begin{array}[]{cc}e^{-i\delta/2}&0\\\
0&e^{i\delta/2}\end{array}\right]=e^{i\alpha}(\beta)_{z}(\gamma)_{y}(\delta)_{z}$
(1.31)

where $\alpha$ is overall phase and $\beta$, $\gamma$ and $\delta$ are real
numbers whose values depend on given unitary matrix $U$.

##### 2-qubit CNOT gate

The two-qubit CNOT gate is in general entangling gate which can be implemented
with set of local rotations and J-coupling ($U_{J}$). The exact pulse sequence
is given below:

$\text{CNOT}=(\frac{\pi}{2})^{1}_{x}(\frac{\pi}{2})^{1}_{y}(\frac{\pi}{2})^{1}_{\bar{x}}(\frac{\pi}{2})^{2}_{x}(\frac{\pi}{2})^{2}_{\bar{y}}U_{J}(\frac{1}{2J})(\frac{\pi}{2})^{2}_{y}$
(1.32)

where $U_{J}(\frac{1}{2J})=e^{-i2\pi JI_{1z}I_{1z}t}$ is free evolution under
J-coupling for time $t=\frac{1}{2J}$. Note that all pulse sequences are to be
read from right to left.

##### Gate time vs decoherence time

The typical gate implementation time lies in the range of microseconds (local
rotation gates) to milliseconds (high dimensional entangling gates like CNOT
or Toffoli) in contrast to decoherence time which lies in the range of few
seconds, in some cases it is in hours.

####  1.4.4 Measurement in NMR

The conventional NMR signal detection involves an ensemble weak measurement,
wherein the spin’s interaction with the radio-frequency coil does not
considerably alter the quantum state during the measurement of total spin
magnetization. As detailed in Sec.1.3, when nuclear spins are placed in
magnetic field $B_{0}$ along z-axis, the bulk magnetization is produced alined
along z-axis. Application of rf pulse rotates this magnetization to the xy
plane, initiating precession around the z-axis at a Larmor frequency
$\omega_{0}$. The resulting precessing magnetization introduces flux changes
in the rf coils, generating a signal voltage (as depicted in Fig.1.6). This
precession induces a magnetic field change, leading to electromagnetic
induction and voltage signal emergence, known as free induction delay (FID),
illustrated in Fig.1.6. The temporal transverse magnetization signal in the
time domain is expressed as follows:

$S(t)=(S_{x}(t)+iS_{y}(t))e^{-t/T_{2}}\propto
S(t)\propto\operatorname{Tr}\left\\{\rho(t)\sum_{k}\left(\sigma_{kx}+i\sigma_{ky}\right)\right\\}$
(1.33)

where $\sum_{k}(\sigma_{kx}+i\sigma_{ky})$ is detection operator,
$\sigma_{kx}$ and $\sigma_{ky}$ are Pauli spin operators proportional to x and
y component of kth spin. Fourier transformation (FT) of $S(t)$ gives the
frequency domain signal $S(\omega)$ and the spectral intensity gives the
expectation values of detection operators which further can be used to perform
QST.

Figure 1.6: Signal acquisition process in NMR.

####  1.4.5 Open quantum dynamics

In the case of closed quantum system, the time evolution of a density matrix
$\rho$ is given by Liouville-Von Neumann equation:

$\frac{\partial\rho}{\partial t}=-\frac{i}{\hbar}[H,\rho]$ (1.34)

where $[H,\rho]=H\rho-\rho H$, is the commutator between operators $H$ and
$\rho$.

However, in more general scenario where the system under consideration may not
be isolated and may have interaction with it’s surrounding, generally referred
as system’s environment, in such case the total Hamiltonian is given as,

$H_{tot}=H_{sys}+H_{env}+H_{int}$ (1.35)

where $H_{sys}$ and $H_{env}$ represent the system’s and environment’s
Hamiltonian while $H_{int}$ represents system-environment interaction
Hamiltonian. One can still treat such composite system (main system +
environment) as whole and the time evolution of corresponding composite
density matrix $\rho_{comp}$ is still given by Eq.1.34. However, if one is
only interested in the time evolution of reduced density matrix $\rho_{sys}$
describing the dynamics of system alone then one has to trace out the
environment, i.e. $\rho_{sys}(t)={\rm Tr}_{env}(\rho_{comp}(t))$. In this
scenario system is generally termed as open quantum system and the
differential equation describing the time evolution of such system is given by
the ’Lindblad master equation’[48],

$\frac{\mathrm{d}\rho_{sys}}{\mathrm{d}t}=-i[H_{sys},\rho_{sys}]+\frac{1}{2}\sum_{k}\left(\left[L_{k}\rho_{sys},L_{k}^{\dagger}\right]+\left[L_{k},\rho_{sys}L_{k}^{\dagger}\right]\right)$
(1.36)

where $L_{k}$ are called Lindblad operators generally describing the effect of
environment on the system. The Lindblad master equation can be separated into
components which cause unitary and non-unitary evolution of the system:

Unitary evolution $\displaystyle:-i[H_{sys},\rho_{sys}]$ Non-unitary evolution
$\displaystyle:\frac{1}{2}\sum_{k}\left(\left[L_{k}\rho_{sys},L_{k}^{\dagger}\right]+\left[L_{k},\rho_{sys}L_{k}^{\dagger}\right]\right)$

Another equivalent representation describing the dynamics of open quantum
system is given via super operator method which generally referred as Kraus
operator representation. Consider a quantum system undergoing a general
quantum evolution represented by a completely positive (CPTP) map. The action
of such a map on a quantum state $\rho_{sys}$ via the superoperator $\Lambda$
in the Kraus operator representation is given as[49, 50],

$\rho_{sys}(t)=\Lambda(\rho_{sys}(0))=\sum_{i}A_{i}(t)\rho_{sys}(0)A_{i}^{\dagger}(t),\quad
s.t\quad\sum_{i}A_{i}(t)A_{i}^{\dagger}(t)=I\quad\forall\hskip 5.69054ptt$
(1.37)

where $\rho_{sys}(0)$ is initial density matrix at $t=0$ and $A_{i}(t)$’s are
Kraus operators at particular time instant $t$. The Kraus operator
representation is usually helpful when dealing with system evolution for fixed
time interval. The utilization of Kraus operator representation becomes
evident during the execution of quantum process tomography (QPT) task which
will be elaborated later in the subsequent chapters. Note that the time
evolution of density matrix given in Eq.1.36 and 1.37 are equivalent and
equations can be inter-converted using techniques given in [51, 52].

###  1.5 Quantum state and process tomography

The reliability of quantum devices primarily relies on two factors: the
initial states in which information is encoded and the quantum gates used to
process that information. Additionally, the measurement process plays a
crucial role in extracting information from an unknown quantum state.
Therefore, it is essential to characterize quantum states and processes
(gates) using efficient measurement techniques. This evaluation is typically
achieved through QST and QPT methods.

QST allows us to estimate the unknown density matrix, while QPT enables us to
estimate a quantity known as the process matrix, which characterizes the
unknown quantum process. Both QST and QPT are statistical processes that
consist of two key components: a set of measurements and an estimator that
maps measurement outcomes to an estimate of the unknown state or process. Due
to the finite size of the ensemble and the presence of systematic errors,
there is always some degree of ambiguity associated with estimating an
experimentally created or implemented state or process. This ambiguity often
results in unphysical density or process matrices.

Moreover, both tasks face a common fundamental challenge: as the size of the
quantum system grows, the required resources and computational complexity
increase exponentially. This exponential growth makes the implementation of
QST and QPT protocols infeasible, even for systems with only a few qubits.
Therefore, it is crucial to design efficient QST and QPT protocols that yield
physically valid density or process matrices.

On an ensemble quantum computer such as NMR, standard QST is carried out by
measuring the expectation values of a fixed set of basis operators, with the
n-qubit density operator $\rho$ being represented in the tensor product of the
Pauli basis:

$\rho=\sum_{i=0}^{3}\sum_{j=0}^{3}\ldots\sum_{n=0}^{3}c_{ij\ldots
n}\sigma_{i}\otimes\sigma_{j}\otimes\ldots\sigma_{n}$ (1.38)

where $c_{00...0}=1/2^{n}$ and $\sigma_{0}$ denotes the $2\times 2$ identity
matrix and $\sigma_{i}$, $i=1,2,3$ are single qubit Pauli operators. The
standard procedure to estimate all the coefficients $c_{ij...n}$ typically
involves solving linear system of equations of the form[53]:

$\mathcal{A}x=\mathcal{B}$ (1.39)

where matrix $\mathcal{A}$ is referred to as a fixed coefficient matrix, the
vector $x$ contains elements of the density matrix which needs to be
reconstructed and vector $\mathcal{B}$ contains actual experimental data.

To formulate the QPT task mathematically, consider the quantum state $\rho$ is
undergoing through general quantum evolution given by completely positive
trace preserving (CPTP) map $\Lambda$. The action of such map on input state
$\rho$ is described by operator sum representation given in Eq.1.37 as:

$\Lambda(\rho)=\sum_{i=1}^{4^{n}}A_{i}\rho A_{i}^{\dagger}$ (1.40)

In the fixed set of basis operator $\\{E_{i}\\}$ the Kraus operators can be
expanded as $A_{i}=\sum_{m}a_{im}E_{m}$ and Eq.1.40 can be rewritten as,

$\Lambda(\rho)=\sum_{m,n}\chi_{mn}E_{m}\rho E_{n}^{\dagger}$ (1.41)

where $\chi_{mn}=\sum_{i}a_{im}a_{in}^{*}$. The quantities $\chi_{mn}$ are the
elements of $\chi$ matrix which characterize the given quantum process
$\Lambda$. The $\chi$ matrix is also referred as process matrix. The $\chi$
matrix satisfies the following three properties:

  1. 1.

$\chi$ should be hermitian. i.e. , $\chi=\chi^{\dagger}$

  2. 2.

Trace($\chi$) = 1

  3. 3.

$\chi$ should be positive. i.e. , $\chi\geq 0$

The standard procedure of estimating full $\chi$ matrix involves solving
linear system of equations of the form[54]:

$\mathcal{\beta}\chi=\mathcal{\lambda}$ (1.42)

where matrix $\beta$ is the coefficient matrix $\mathcal{A}$, column matrix
$\chi$ contains the elements $\chi_{mn}$ which are to be determined and column
matrix $\lambda$ is actual experimental data.

####  1.5.1 Dealing with invalid density and process matrices via convex
optimization methods

Utilizing the standard linear inversion technique for reconstructing quantum
states and processes often leads to unphysical density and process matrices
respectively. This issue is evident in scenarios like QPT, where non-
completely positive maps emerge as result of experimental reconstruction of
processes using linear inversion method. This signifies that inversion is not
the optimal fit for raw tomography outcomes. Incorrect or unphysical
reconstructions could mislead assessments of quantum state and channel.
Approaches such as convex optimization, a fundamental concept in machine
learning, strive to identify parameters within a model that align best with
prior information. Convex optimization yields a global optimum that provides
the most accurate fit to raw data. By employing convex optimization, one can
extract comprehensive and accurate information from measurement results
ensuring the positivity of quantum states and processes. Variety of convex
optimization techniques are used in the past to perform valid QST and QPT.
Most of the convex optimization methods are based on least square (LS)
optimization problem (${\|\mathcal{A}x-\mathcal{B}\|}_{2}$) subject to
positivity condition ($\rho\geq 0$)[53, 55, 56, 57]. However, computation
complexity grow exponentially as the size of the system which make these
methods computationally expensive in terms of time required to reach global
optimal solution. To address this issue, these methods are further refined
using more advance algorithm like hyperplane intersection projection algorithm
which results in projected least-squares QPT protocol[58] and projected
gradient descent algorithms for QST[59]. These convex optimization methods
offer several distinct advantages, including its independence from prior
knowledge about the system and its ability to operate without the need for
additional ancillary qubits. Specifically, in the context of QPT, the convex
optimization method allows for the computation of a complete set of Kraus
operators.

Moreover, compressed sensing (CS) optimization method has also been
extensively used to perform QST as well as QPT in which the $\|.\|_{l_{1}}$
($l_{1}$ norm of variable vector) is the main objective function with multiple
constraints[60, 61, 62]. It tunes out that CS optimization method not only
deals with unphysicality issue but also with complexity issue. That is, CS
method allows to reconstruct valid density and processes matrix using heavily
reduced data set and in some cases outperforms LS method [60, 63]. However,
the applicability of CS method is limited and required prior knowledge about
the sparsity and noise present in the measurement data. The CS optimization
method turns out to be promising while characterizing high dimensional quantum
gates.

####  1.5.2 Data driven based QST and QPT via machine learning methods

Machine learning methods primarily address the issue of complexity and allow
to perform full QST as well as QPT using incomplete measurement set. Majority
of such protocols are primarily based on data driven approach which includes
protocols like self guided tomography in which quantum state is iteratively
learned by optimizing a projection measurement without any data storage or
post-processing[64, 65], adaptive quantum tomography method using Bayesian
approach[66] and via linear regression estimation[57, 67]. Though self guided
and adaptive protocols are efficient as compared to standard method, it
requires large number of projective measurements which are hard to implement
on ensemble quantum system.

##### Artificial Neural Networks (ANNs)

ANN is a sub-branch of machine learning method and the framework is inspired
from neurons in human brain. ANN consists of many artificial neurons which are
connected to each other. The neuron is activated when its value is greater
than threshold value called bias. The Feed-Forward-Neural network (FFNN) is
one the type of ANN model where the architecture consists of three layers:
input layer, hidden layer and output layer. Data is fed into input layer which
is passed on layer by layer till it arrives at the output. Data is divided
into two parts i.e. train data-set and test data-set. As name implies, train
data-set is used to train the model (update network parameters, weights and
biases) and test data-set is used to evaluate the network. Consider ’$m$’
training elements
$\\{(\vec{x}^{(1)},\hat{\vec{y}}^{(1)}),(\vec{x}^{(2)},\hat{\vec{y}}^{(2)}),\cdots,(\vec{x}^{(m)},\hat{\vec{y}}^{(m)})\\}$
where $\vec{x}^{(i)}$ is $i^{th}$ input and $\hat{\vec{y}}^{(i)}$ is
corresponding labelled output. Feeding these inputs to the network produces
outputs
$[\tilde{\vec{y}}^{(1)},\tilde{\vec{y}}^{(2)},...,\tilde{\vec{y}}^{(m)}]$.
Since network parameters are initialized randomly, predicted output is not
equal to expected output. Training of this network can be achieved by
minimizing the mean-squared-error cost function,
$J(w,b)=\frac{1}{m}\sum_{i=1}^{m}||\hat{\vec{y}}^{(i)}-\tilde{\vec{y}}^{(i)}||^{2}$
with respect to network parameters by stochastic gradient descent,

$\displaystyle w_{ij}\rightarrow w^{\prime}_{ij}=$ $\displaystyle
w_{ij}-\frac{\eta}{m^{\prime}}\sum_{i=1}^{m^{\prime}}\frac{\partial}{\partial
w_{ij}}\mathcal{L}(\vec{x}^{(i)})$ (1.43) $\displaystyle b_{i}\rightarrow
b^{\prime}_{i}=$ $\displaystyle
b_{i}-\frac{\eta}{m^{\prime}}\sum_{i=1}^{m^{\prime}}\frac{\partial}{\partial
b_{i}}\mathcal{L}(\vec{x}^{(i)})$ (1.44)

where $\mathcal{L}(x^{(i)})=||\hat{\vec{y}}^{(i)}-\tilde{\vec{y}}^{(i)}||^{2}$
is the cost function of randomly chosen $m^{\prime}$ training inputs $x^{(i)}$
and $\eta$ is the learning rate. $w^{\prime}_{ij}$ and $b^{\prime}_{i}$ are
updated weights and biases. Once the model is trained then it can be applied
to test data set which can be a experimentally obtained data set. In the
recent years, such ANN models are applied to perform efficient QST as well as
QPT[68, 69, 70, 71, 72, 73].

####  1.5.3 Selective and direct QST and QPT protocols

The selective and direct estimation of desired element of density and process
matrices is referred as direct QST (DQST) and QPT (DQPT) respectively. The
objective of DQST and DQPT is very well explored in the past where plenty of
algorithms and protocols are proposed to selectively compute element of
density and process matrix[74, 75, 76]. Although the traditional methods like
adaptive, self guided, maximum likelyhood estimation (MLE) and ancilla
assisted methods offer some advantages over standard QPT, they still are not
very useful when only certain elements of the density or process matrix need
to be estimated. The methods like selective and efficient quantum process
tomography (SEQPT) based on quantum 2-design states[77, 78, 79] turns out to
efficient while performing DQPT whereas weak measurement (WM) based techniques
have been used to perform efficient DQST as well as DQPT where in certain
special cases WM method outperform projective measurements[80, 81].

####  1.5.4 Simulation and characterization of open quantum dynamics

One of the main building blocks of a quantum computer is the underlying
physical system and its time evolution under a given Hamiltonian [18], while
the main obstacle in building such a quantum computer is its unwanted and
inevitable interaction with its environment, generally referred to as
decoherence. Efforts to mitigate decoherence led to studies of open quantum
dynamics, whereby the time evolution of a quantum system is studied using a
master equation approach [82, 83]. In real situations, the physical system
under consideration is continuously interacting with its environment, causing
its time evolution to be non-unitary, contributing significantly to errors in
the computational output and reducing the quality of the quantum device [84].
Hence the task of designing quantum algorithms to simulate open quantum
dynamics is important from a fundamental as well as a practical point of view.

The duality quantum algorithm for simulating evolution of an open quantum
system was proposed [85, 86] where the time evolution of the open quantum
system is realized by using Kraus operators. In duality algorithm the
evolution operator is a linear combination of unitary operators resulting in
desired non-unitary evolution. However, the method is experimentally expensive
and requires ancilla system of dimension equal to number of Kraus operators
characterizing given open quantum system. One the other hand, a class of
promising quantum algorithms to simulate arbitrary non-unitary evolutions on
quantum devices have been reported, which are primarily based on the dilation
technique namely, the Stinespring dilation algorithm [87] and Sz.-Nagy’s
dilation algorithm [88]. The basic tenet of these algorithms is, to construct
a unitary operation in a higher-dimensional Hilbert space, which simulates the
desired non-unitary evolution in a lower-dimensional Hilbert space. The
Stinespring dilation algorithm requires a larger Hilbert space dimension,
which makes it computationally and experimentally expensive, as compared to
the Sz.-Nagy algorithm. The Sz.-Nagy algorithm has been used to experimentally
simulate a single-qubit amplitude damping channel on the IBM quantum processor
[89].

###  1.6 Organization of the thesis

Chapter 2 presents overview of standard QST and QPT methods based on the
linear inversion technique. A detailed description of the associated non-
physicality issue concerning states and processes is given. Additionally, the
CCO-based QST and QPT protocols are introduced to effectively address and
resolve the non-physicality problem.

Chapters 3 and 4 are dedicated to the consideration of the scalability concern
of QST and QPT protocols using the CS algorithm and techniques based on
artificial neural networks, respectively. The performance of these methods is
demonstrated on both NMR and IBM quantum processors. In Chapters 5 and 6, the
challenge of selectively and directly estimating desired elements of unknown
density and process matrices is addressed. The modified selective quantum
process tomography (MSQPT) protocol is proposed, and the weak measurement
method is applied on NMR and IBM processors, respectively.

Chapter 7 focuses on the simulation and characterization of arbitrary open
quantum dynamics through the utilization of Sz-Nagy’s dilation algorithm on an
NMR processor. Chapter 8 provides a summary of the main results obtained
throughout the study and outlines potential directions for future research.

##  Chapter 2 Achieving valid quantum states and processes on an NMR quantum
information processor through convex optimization

###  2.1 Introduction

In recent years, much research has been conducted to design high-fidelity
quantum devices based on quantum technology. This requires the
characterization of quantum states and processes, which are essential in
studying the behavior of quantum processors and validating quantum devices.
The characterization is usually done through quantum state tomography (QST)
and quantum process tomography (QPT). Both QST and QPT are statistical
processes that consist of two elements: a set of measurements and an estimator
that maps the measurement outcomes to an estimate of the unknown state or
process. Since the sample size is finite and systematic errors are inevitable,
there is always some uncertainty or error associated with the estimated state,
which can sometimes result in unphysical density matrices[90]. Therefore, it
is crucial to have an estimation protocol that produces valid quantum states
(processes).

To date, several tomography protocols have been proposed and successfully
applied to physical systems, such as the state of nuclear spin ensembles[91,
92, 93], photon polarization states[57, 94], and infinite-dimensional coherent
states of light. These protocols are typically based on least-square linear
inversion[91, 94, 95, 96, 90, 97]. However, there are also other strategies
for QST, such as maximum likelihood estimation[92], hedged likelihood function
estimation[98], model averaging[99], adaptive bayesian state estimation[67],
linear regression[57], gradient approach(self guided)[64], numerical
strategies[100], weak measurement[101], and controlled-SWAP quantum
network[102]. Similar protocols have also been proposed for QPT, including
ancilla-assisted QPT[103], simplified QPT[104], selective QPT using quantum
2-design states[77], self-consistent QPT[105], compressed sensing QPT[60],
adaptive measurement-based QPT[106], and selective QPT via sequential weak
value measurement of incompatible observables[107]. These protocols have been
implemented on various platforms, such as NMR[108, 79, 92, 91],
superconducting qubits[109, 110, 111], nitrogen vacancy centers[112], linear
optics[106, 113], and ion-trap quantum processors[114]. Despite the
availability of many tomography protocols, most of them do not produce valid
density (process) matrices. Protocols such as adaptive measurements and self-
guided tomography that do produce valid states/processes often involve a large
number of projective measurements, which can be challenging to implement on
ensemble quantum computers. Additionally, in the MLE protocol, knowledge of
the noise distribution in the system is crucial for constructing the
likelihood function. Care should be taken when estimating special states, such
as entangled states, using MLE[115, 116].

In this chapter, a method is presented to address the issue of unphysical
experimentally reconstructed density matrices and process matrices. The method
involves optimizing a least square objective function while taking into
consideration the positivity condition as a nonlinear constraint and the unit
trace condition as a linear constraint[53]. This transforms the standard
linear inversion-based tomography problem into a constrained convex
optimization (CCO) problem, which ensures the positivity of the reconstructed
state/process and produces valid quantum states and quantum processes. The CCO
tomography method does not require any prior knowledge about the system or any
additional ancillary qubits. The advantages of the CCO method over the
standard method are demonstrated by characterizing unknown quantum states and
quantum processes for two- and three-qubit quantum systems. Additionally, for
QPT, the complete set of valid Kraus operators for a given quantum process can
be efficiently computed via the unitary diagonalization of the experimentally
reconstructed positive process matrix, which was not previously possible using
other QPT techniques.

###  2.2 QST via standard and CCO method

####  2.2.1 Linear inversion based standard QST on NMR

The general density matrix corresponding to n-qubit system has $4^{n}-1$
independent real parameters and can be reconstructed experimentally by
determining all these $4^{n}-1$ numbers[92]. QST is usually done by performing
set of repeated projective measurements on multiple copies of identically
prepared states in different measurement bases. On an ensemble quantum
computer like NMR, it is very hard to perform projective measurements, in that
case QST is carried out by measuring expectation values of certain fixed set
of basis operators which are further related to single quantum coherence (off-
diagonal elements $\langle m|\rho|n\rangle$, such that total z-angular
momentum quantum numbers between $|m\rangle$ and $|n\rangle$ differ by one)
terms in the underlying density matrix. This is generally done by rotating the
state via different unitary transformations before performing the measurement
to collect information about different elements of the density matrix[91].

##### 2-qubit standard QST on NMR

For a system of two qubits, the general form of density matrix $\rho$ is given
as,

$\rho=\begin{bmatrix}x_{1}&x_{2}+ix_{3}&x_{4}+ix_{5}&x_{6}+ix_{7}\\\
x_{2}-ix_{3}&x_{8}&x_{9}+ix_{10}&x_{11}+ix_{12}\\\
x_{4}-ix_{5}&x_{9}-ix_{10}&x_{13}&x_{14}+ix_{15}\\\
x_{6}-ix_{7}&x_{11}-ix_{12}&x_{14}-ix_{15}&x_{16}\end{bmatrix}$ (2.1)

where all $x_{i}$’s are real numbers.

Figure 2.1: Schematic of standard QST based on linear inversion.

The NMR signal corresponding to 1st spin is associated with
$\rho_{24}=x_{11}+ix_{12}$ and $\rho_{13}=x_{4}+ix_{5}$ while signal
corresponding to 2nd spin is associated with $\rho_{34}=x_{14}+ix_{15}$ and
$\rho_{12}=x_{2}+ix_{3}$. These elements are generally referred to as readout
positions of density matrix and are also interpreted as transition amplitudes
between two energy eigenstates:

$\displaystyle\rho_{24}$
$\displaystyle:|01\rangle\longleftrightarrow|11\rangle$
$\displaystyle\rho_{13}$
$\displaystyle:|00\rangle\longleftrightarrow|10\rangle$
$\displaystyle\rho_{34}$
$\displaystyle:|10\rangle\longleftrightarrow|11\rangle$
$\displaystyle\rho_{12}$
$\displaystyle:|00\rangle\longleftrightarrow|01\rangle$

Note that these are the only four allowed transitions possible in two qubit
system as, $\Delta m=\pm 1$ (see Fig.1.5). All other transitions are forbidden
by selection rule. In the case of 2-qubit system, the complete NMR signal from
two spins consists of 4 spectrums (2 spectrum per nuclear spin). Each spectrum
corresponds to given transition. The absorption part of given spectrum is
associated with real part of corresponding density matrix element whereas
dispersion part corresponds to imaginary part of density matrix element. For
example, the intensity of absorption part of spectrum (integrated area)
corresponding to element $\rho_{24}$ is proportional to $x_{11}$ likewise
intensity of dispersion part of the same spectrum is proportional to $x_{12}$.
So the readout from two spins will give the four density matrix elements or in
more general sense it gives rise to 8 linear equations. To extract other
elements of density matrix we apply certain unitary gates on density matrix
and then perform measurement. One of the possible complete set of unitary
gates for 2-qubit QST is given as:
$\\{II,R_{x}^{2},R_{y}^{2},R_{x}^{1}R_{x}^{2}\\}$ where $I$ is single qubit
identity matrix and $R_{x(y)}^{i}$ denotes $90^{\circ}$ rotation about $x(y)$
axis on $i$th qubit. Unitary gate, $II=I\otimes I$ is identity operation on
both qubits and so on. The complete set of unitary gates followed by
measurements is referred as tomographically complete set of measurements. For
a given tomographically complete set of measurements a system of linear
equations can be formed. Each tomographic unitary gate yields 8 linear
equations, so in total we get 32 equations. An extra equation is added as unit
trace condition: $x_{1}+x_{8}+x_{13}+x_{16}=1$. For this system of 33 linear
equations, the coefficient matrix $\mathcal{A}$ is of dimension $33\times 16$
and column matrix $\mathcal{B}$ is of dimension $33\times 1$. The column
matrix $x$ containing the variables $x_{i}$s is of dimension $16\times 1$.
Solving linear inversion problem given below will yield the column matrix $x$
and so do given unknown density matrix $\rho$ as,

$\displaystyle\mathcal{A}x$ $\displaystyle=\mathcal{B}$
$\displaystyle\mathcal{A}^{T}\mathcal{A}x$
$\displaystyle=\mathcal{A}^{T}\mathcal{B}$ $\displaystyle x$
$\displaystyle=(\mathcal{A}^{T}\mathcal{A})^{-1}\mathcal{A}^{T}\mathcal{B}$
(2.2)

The elements of column matrix $\mathcal{B}$ are the intensities of NMR
spectrums obtained experimentally and matrix
$(\mathcal{A}^{T}\mathcal{A})^{-1}\mathcal{A}^{T}$ can be computed
analytically. Computing $x$ using Eq.2.2 and constructing $\rho$ is referred
as standard QST based on linear inversion method.

##### 3-qubit standard QST on NMR

The $8\times 8$ dimensional density matrix for 3-qubit system will have 64
real independent parameters (trace condition excluded), i.e. column matrix $x$
will be of dimension $64\times 1$. The NMR signal corresponding to given spin
is associated with density matrix elements as follows:

1st spin or 1st qubit $\displaystyle:\rho_{48},\rho_{26},\rho_{37},\rho_{15}$
2nd spin or 2nd qubit $\displaystyle:\rho_{57},\rho_{13},\rho_{68},\rho_{24}$
3rd spin or 3rd qubit $\displaystyle:\rho_{56},\rho_{12},\rho_{78},\rho_{34}$

These readout elements are associated with transition amplitudes between two
energy eigenstates:

$\displaystyle\rho_{48}$
$\displaystyle:|011\rangle\longleftrightarrow|111\rangle,\quad\rho_{26}:|001\rangle\longleftrightarrow|101\rangle,\quad\rho_{37}:|010\rangle\longleftrightarrow|110\rangle,\quad\rho_{15}:|000\rangle\longleftrightarrow|100\rangle$
$\displaystyle\rho_{57}$
$\displaystyle:|100\rangle\longleftrightarrow|110\rangle,\quad\rho_{13}:|000\rangle\longleftrightarrow|010\rangle,\quad\rho_{68}:|101\rangle\longleftrightarrow|111\rangle,\quad\rho_{24}:|001\rangle\longleftrightarrow|011\rangle$
$\displaystyle\rho_{56}$
$\displaystyle:|100\rangle\longleftrightarrow|101\rangle,\quad\rho_{12}:|000\rangle\longleftrightarrow|001\rangle,\quad\rho_{78}:|110\rangle\longleftrightarrow|111\rangle,\quad\rho_{34}:|010\rangle\longleftrightarrow|011\rangle$

The complete NMR signal from three spins will have 12 spectrums (4 spectrums
per spin). A given tomographic unitary operation will yield 24 linear
equations. The complete set of tomographic unitary gates for 3-qubit QST is
given as:

$\\{III,R_{y}^{3},R_{y}^{1},R_{y}^{2}R_{y}^{3},R_{x}^{1}R_{y}^{2}R_{x}^{3},R_{x}^{1}R_{x}^{2}R_{y}^{3},R_{x}^{1}R_{x}^{2}R_{x}^{3}\\}$

where $III=I\otimes I\otimes I$ and so on. For given 7 tomographic operations
and including unit trace condition we get total of 169 linear equations. So
for 3-qubit case, $\mathcal{A}$ will be $169\times 64$ dimensional matrix and
$\mathcal{B}$ will be $64\times 1$ dimensional column matrix. Using Eq.2.2,
column matrix $x$ can be determined and $\rho$ can be constructed.

##### n-qubit standard QST on NMR

For $n$-qubit system, one can efficiently construct tomographically complete
optimal set of unitary rotations using integer programming method[117]. For
example, in case of 4-qubit system the cardinality of such optimal set turns
out to be 15. That is, it is necessary to execute 15 tomographic rotation
operations to reconstruct the complete density matrix of a 4-qubit system,
which entails solving a total of 961 linear equations. In the case of 4-qubit
QST, one possible set is given below:

$\begin{gathered}\\{IIII,R_{x}^{4},R_{x}^{1}R_{x}^{4},R_{x}^{1}R_{y}^{4},R_{y}^{1}R_{y}^{2},R_{x}^{2}R_{y}^{3}R_{y}^{4},R_{y}^{2}R_{x}^{3}R_{y}^{4},\\\
R_{y}^{2}R_{y}^{3}R_{y}^{4},R_{x}^{1}R_{x}^{2}R_{x}^{3},R_{x}^{1}R_{y}^{2}R_{y}^{3},R_{y}^{1}R_{x}^{3}R_{x}^{4},R_{y}^{1}R_{y}^{2}R_{y}^{3}\\\
R_{x}^{1}R_{x}^{2}R_{x}^{3}R_{x}^{4},R_{y}^{1}R_{x}^{2}R_{x}^{3}R_{x}^{4},R_{y}^{1}R_{x}^{2}R_{y}^{3}R_{y}^{4}\\}\end{gathered}$

Similarly, for 5-qubit system tomographically complete optimal set of unitary
rotations of cardinality 33 is given below:

$\begin{gathered}\\{IIIII,R_{x}^{5},R_{x}^{4}R_{y}^{5},R_{y}^{4}R_{y}^{5},R_{x}^{3}R_{y}^{5},R_{y}^{3}R_{y}^{5},R_{x}^{2}R_{x}^{3}R_{x}^{4},\\\
R_{y}^{2}R_{y}^{3}R_{y}^{4},R_{x}^{1}R_{y}^{3}R_{x}^{4},R_{x}^{1}R_{x}^{2}R_{y}^{4},R_{x}^{1}R_{y}^{2}R_{y}^{5},R_{x}^{1}R_{y}^{2}R_{x}^{3}\text{,
}\\\
R_{y}^{1}R_{x}^{3}R_{y}^{4},R_{y}^{1}R_{y}^{3}R_{y}^{5},R_{y}^{1}R_{x}^{2}R_{y}^{5},R_{y}^{1}R_{x}^{2}R_{y}^{3},R_{y}^{1}R_{y}^{2}R_{y}^{5},\\\
R_{y}^{1}R_{y}^{2}R_{x}^{4},R_{x}^{2}R_{x}^{3}R_{x}^{4}R_{x}^{5},R_{y}^{2}R_{y}^{3}R_{x}^{4}R_{x}^{5},R_{y}^{2}R_{y}^{3}R_{y}^{4}R_{x}^{5},\\\
R_{x}^{1}R_{y}^{3}R_{x}^{4}R_{x}^{5},R_{x}^{1}R_{x}^{2}R_{y}^{4}R_{x}^{5},R_{x}^{1}R_{y}^{2}R_{x}^{3}R_{x}^{5},R_{y}^{1}R_{x}^{3}R_{y}^{4}R_{x}^{5}\text{,
}\\\
R_{y}^{1}R_{x}^{2}R_{y}^{3}R_{x}^{5},R_{y}^{1}R_{y}^{2}R_{x}^{4}R_{x}^{5},R_{x}^{1}R_{x}^{2}R_{x}^{3}R_{x}^{4}R_{y}^{5},\\\
R_{x}^{1}R_{x}^{2}R_{x}^{3}R_{y}^{4}R_{y}^{5},R_{x}^{1}R_{x}^{2}R_{y}^{3}R_{x}^{4}R_{y}^{5},R_{x}^{1}R_{x}^{2}R_{y}^{3}R_{y}^{4}R_{y}^{5},\\\
R_{y}^{1}R_{y}^{2}R_{x}^{3}R_{x}^{4}R_{y}^{5},R_{y}^{1}R_{y}^{2}R_{x}^{3}R_{y}^{4}R_{y}^{5}\\}\end{gathered}$

The QST of 5-qubit density matrix involves solving 5281 number of linear
equations.

One can see that, as the system size increases the complexity of QST task
increases exponentially and to address this scalability issue one need to
design more efficient algorithms. The schematic representation of standard QST
is given in Fig.2.1.

####  2.2.2 Fidelity measure for quantum states

Fidelity serves as a fundamental concept in the realm of quantum information,
offering a mathematical framework for quantifying the degree of similarity
between two quantum states. In practical applications, numerous scenarios
arise where such comparisons prove beneficial. For instance, due to inherent
imperfections and noise in experimental preparations of quantum states, there
arises a genuine interest in determining the closeness between the actual
state produced and the intended target state. This concern commonly arises in
quantum communications and quantum computing, where the goal is to either
generate or transmit precisely defined quantum states amidst the challenges
posed by noise and other sources of error. Consequently, fidelity for mixed
states can be regarded as a more practical measure, given its generic nature,
as opposed to pure state fidelity, which represents an idealized scenario.
Nevertheless, defining fidelity for mixed states lacks a unique, clear-cut
definition, leading to the existence of several different approaches[118].

The Uhlmann-Jozsa (UJ) fidelity and normalized trace distance are two of the
most widely used state fidelity measures in the literature. The UJ-fidelity
$\mathcal{F}_{UJ}(\rho,\sigma)$ between two density matrices $\rho$ and
$\sigma$ is defined as, the maximal transition probability between the
purification of a pair of density matrices $\rho$ and $\sigma$. The
mathematical formula for UJ-fidelity is given as[118, 119],

$\mathcal{F}_{UJ}(\rho,\sigma):=\max_{|\psi\rangle,|\varphi\rangle}|\langle\psi\mid\varphi\rangle|^{2}=(\operatorname{Tr}\sqrt{\sqrt{\rho}\sigma\sqrt{\rho}})^{2}$
(2.3)

where $|\psi\rangle$ and $|\phi\rangle$ are the purifications of $\rho$ and
$\sigma$ respectively. Similarly, the normalized trace norm
$\mathcal{F}_{tr}(\rho,\sigma)$ between two density matrices $\rho$ and
$\sigma$ is defined as[120],

${\mathcal{F}}_{tr}(\rho,\sigma)=\frac{|{\rm
Tr}(\rho\sigma^{\dagger})|}{\sqrt{{\rm Tr}(\rho^{\dagger}\rho){\rm
Tr}(\sigma^{\dagger}\sigma)}}$ (2.4)

Both the fidelity measures given in Eq.2.3 and 2.4 satisfy a list of fidelity
axioms (the most basic requirements to be satisfied by any generalization of
fidelity measure) proposed by Jozsa in order to be a suitable fidelity measure
between pair of mixed states[120]:

  * -

Normalization: $0\leq\mathcal{F}(\rho,\sigma)\leq 1$

  * -

Symmetric: $\mathcal{F}(\rho,\sigma)=\mathcal{F}(\sigma,\rho)$

  * -

Unitary invariance: $\mathcal{F}(\rho,\sigma)=\mathcal{F}(U\rho
U^{\dagger},U\sigma U^{\dagger})$

  * -

Orthogonality support: $\mathcal{F}(\rho,\sigma)=0$ iff $\rho\sigma=0$

  * -

$\mathcal{F}(\rho,\sigma)=\rm{Tr}(\rho,\sigma)$ if either $\rho$ or $\sigma$
is a pure state.

When employing the Uhlmann-Jozsa (U-J) fidelity measure, it is imperative that
the density matrices $\rho$ and $\sigma$ be positive semi definite - a
condition that may not hold true for experimentally reconstructed density
matrices. To circumvent this concern, in this thesis, we have opted to utilize
the normalized trace norm as a state and process fidelity measure, unless
explicitly stated otherwise. One can also find the application of normalized
trace norm as a valid fidelity measure in References [120, 121, 61].

(a) $|01\rangle$

(b) $|0+\rangle$

(c) $(|00\rangle+|11\rangle)/\sqrt{2}$

(d) $(|01\rangle+|10\rangle)/\sqrt{2}$

Figure 2.2: Real (left) and imaginary (right) parts of experimental tomographs
of two qubit states, constructed via standard (1st row) and CCO (2nd row)
method are shown.

####  2.2.3 Convex optimization based QST

The reconstructed $\rho$ by standard protocol is hermitian in nature and has
trace 1 but there is no guarantee that it will be positive since the
positivity constrain is not explicitly included in standard protocol. And in
general it is very hard to incorporate positivity condition in any general
state estimation protocol. To address this situation, the linear inversion-
based standard QST problem has been reformulated into a constrained convex
optimization (CCO) problem. The standard mathematical form of CCO problem in
context of QST is given as follows,

$\displaystyle\\!\min_{x}$ $\displaystyle{\|\mathcal{A}x-\mathcal{B}\|}_{2}$
(2.5a) $\displaystyle s.t.$ $\displaystyle\rho\geq 0,$ (2.5b)
$\displaystyle\mathrm{Tr}(\rho)=1.$ (2.5c)

where $\|v\|_{2}$ denotes the $l_{2}$ norm of vector $v$. The least square
objective function given in Eq.2.5a is defined in the paper[53]. The CCO
problem stated in Eq.2.5a is formulated using YALMIP MATLAB package[122] which
employed SeDuMi[123] as solver. Upon solving the aforementioned CCO problem,
the valid density matrix can be obtained. This matrix provides the least
square fit to the experimental data and reveals the actual quantum state.

In order to assess the effectiveness of QST using the CCO method, various two-
and three-qubit quantum states were experimentally prepared and subjected to
tomography. Subsequently, their fidelities and corresponding eigenvalues were
calculated using both the standard method and the CCO method. Some of the
experimentally constructed tomographs for 2-qubit states via standard and CCO
method are depicted in the Figs.2.2. The fidelity between the theoretically
expected ($\rho_{\rm theo}$) and the experimentally reconstructed ($\rho_{\rm
expt}$) quantum state has been computed using Eq.2.4. The computed fidelities
for several different quantum states were slightly improved for the CCO method
as compared to the standard method. However, the main advantage of the CCO
method is that the experimentally reconstructed density matrix is always
positive semi-definite and hence represents a valid quantum state.

Table 2.1: Eigenvalues for the two-qubit density matrix, obtained from experimentally reconstructed density matrices via standard and CCO QST. Quantum state |  Standard |  CCO
---|---|---
$|00\rangle$ |  -0.0488, -0.0171, 0.0499, 1.0160 |  0, 0, 0.0225, 0.9775
$|01\rangle$ |  -0.0429, -0.0222, 0.0364, 1.0287 |  0, 0, 0.0067, 0.9933
$|10\rangle$ |  -0.1486, -0.0911, 0.1915, 1.0482 |  0, 0, 0.0807, 0.9193
$|11\rangle$ |  -0.1457, -0.0955, 0.1933, 1.0480 |  0, 0, 0.0808, 0.9192
$(|00\rangle+|11\rangle)/\sqrt{2}$ |  -0.0822, -0.0456, 0.0508, 1.0778 |  0, 0, 0.0105, 0.9895
$(|01\rangle-|10\rangle)/\sqrt{2}$ |  -0.0950, -0.0370, 0.0624, 1.0696 |  0, 0, 0.0142, 0.9858
$(|00\rangle-|11\rangle)/\sqrt{2}$ |  -0.1315, -0.0455, 0.1180, 1.0591 |  0, 0, 0.0592, 0.9408
$(|01\rangle+|10\rangle)/\sqrt{2}$ |  -0.1175, -0.0278, 0.0910, 0.0543 |  0, 0, 0.0397, 0.9603
$(|01\rangle+|11\rangle)/\sqrt{2}$ |  -0.0892, -0.0493, 0.1060, 1.0326 |  0, 0, 0.0255, 0.9745
$(|00\rangle+|01\rangle)/\sqrt{2}$ |  -0.0587, -0.0166, 0.0683, 1.0070 |  0, 0, 0.0375, 0.9625
$(|10\rangle+|11\rangle)/\sqrt{2}$ |  -0.1017, -0.0730, 0.1209, 1.0538 |  0, 0, 0.0381, 0.9619
$(|00\rangle+|10\rangle)/\sqrt{2}$ |  -0.0884, -0.0469, 0.1093, 1.0260 |  0, 0, 0.0303, 0.9697
$(|01\rangle+i|11\rangle)/\sqrt{2}$ |  -0.0936, -0.0436, 0.0987, 1.0385 |  0, 0, 0.0267, 0.9733
$(|10\rangle+i|11\rangle)/\sqrt{2}$ |  -0.1122, -0.0962, 0.1549, 1.0536 |  0, 0, 0.0544, 0.9456
$(|00\rangle+i|10\rangle)/\sqrt{2}$ |  -0.0898, -0.0420, 0.1028, 1.0290 |  0, 0, 0.0304, 0.9696
$(|00\rangle+i|01\rangle)/\sqrt{2}$ |  -0.0862, -0.0379, 0.0837, 1.0405 |  0, 0, 0.0329, 0.9671
$(|00\rangle+|01\rangle+|10\rangle+|11\rangle)/2$ |  -0.0823, -0.0293, 0.0974, 1.0142 |  0, 0, 0.0293, 0.9707
$(|00\rangle+i|01\rangle+|10\rangle+i|11\rangle)/2$ |  -0.0917, -0.0619, 0.1120, 1.0416 |  0, 0, 0.0298, 0.9702
$(|00\rangle+|01\rangle+i|10\rangle+i|11\rangle)/2$ |  -0.0728, -0.0110, 0.0770, 1.0068 |  0, 0, 0.0298, 0.9702
$(|00\rangle+i|01\rangle+i|10\rangle-|11\rangle)/2$ |  -0.0828, -0.0347, 0.0904, 1.0271 |  0, 0, 0.0234, 0.9766
Table 2.2:  Two-qubit quantum state fidelity obtained from standard and CCO method. Quantum state | Standard | CCO
---|---|---
$|00\rangle$ | 0.9969 | 0.9993
$|01\rangle$ | 0.9926 | 0.9948
$|10\rangle$ | 0.9690 | 0.9947
$|11\rangle$ | 0.9665 | 0.9928
$(|00\rangle+|11\rangle)/\sqrt{2}$ | 0.9921 | 0.9968
$(|01\rangle-|10\rangle)/\sqrt{2}$ | 0.9887 | 0.9942
$(|00\rangle-|11\rangle)/\sqrt{2}$ | 0.9796 | 0.9911
$(|01\rangle+|10\rangle)/\sqrt{2}$ | 0.9786 | 0.9956
$(|01\rangle+|11\rangle)/\sqrt{2}$ | 0.9832 | 0.9946
$(|00\rangle+|01\rangle)/\sqrt{2}$ | 0.9929 | 0.9966
$(|10\rangle+|11\rangle)/\sqrt{2}$ | 0.9834 | 0.9966
$(|00\rangle+|10\rangle)/\sqrt{2}$ | 0.9883 | 0.9982
$(|01\rangle+i|11\rangle)/\sqrt{2}$ | 0.9853 | 0.9952
$(|10\rangle+i|11\rangle)/\sqrt{2}$ | 0.9771 | 0.9953
$(|00\rangle+i|10\rangle)/\sqrt{2}$ | 0.9895 | 0.9980
$(|00\rangle+i|01\rangle)/\sqrt{2}$ | 0.9920 | 0.9984
$(|00\rangle+|01\rangle+|10\rangle+|11\rangle)/2$ | 0.9884 | 0.9964
$(|00\rangle+i|01\rangle+|10\rangle+i|11\rangle)/2$ | 0.9856 | 0.9958
$(|00\rangle+|01\rangle+i|10\rangle+i|11\rangle)/2$ | 0.9919 | 0.9969
$(|00\rangle+i|01\rangle+i|10\rangle-|11\rangle)/2$ | 0.9898 | 0.9966

###  2.3 QPT via standard and CCO method

####  2.3.1 Standard QPT

Quantum process tomography (QPT) can quantitatively characterize an unknown
quantum process. Any quantum state $\rho$ undergoing a physically valid
process can described by a completely positive (CP) map, and an unknown
process $\Lambda$ can be described in the operator-sum representation of Kraus
operators[54]:

$\Lambda(\rho)=\sum_{i=1}^{d^{2}}A_{i}\rho A_{i}^{\dagger}$ (2.6)

where $A_{i}$’s are the Kraus operators satisfying
$\sum_{i}A_{i}A_{i}^{\dagger}=I$. The Kraus operators can be expanded using a
fixed complete set of basis operators $\\{E_{i}\\}$ as
$A_{i}=\sum_{m}a_{im}E_{m}$, then Eq.2.6 can be rewritten as,