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# Parameterized process characterization with reduced resource requirements

Vicente Leyton-Ortega<EMAIL_ADDRESS>Computational Sciences and
Engineering Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA
Tyler Kharazi  Computational Sciences and Engineering Division, Oak Ridge
National Laboratory, Oak Ridge, TN 37831, USA  Raphael C. Pooser
Computational Sciences and Engineering Division, Oak Ridge National
Laboratory, Oak Ridge, TN 37831, USA

###### Abstract

Quantum Process Tomography (QPT) is a powerful tool to characterize quantum
operations, but it requires considerable resources making it impractical for
more than 2-qubit systems. This work proposes an alternative approach that
requires significantly fewer resources for unitary processes characterization
without prior knowledge of the process and provides a built-in method for
state preparation and measurement (SPAM) error mitigation. By measuring the
quantum process as rotated through the $X$ and $Y$ axes on the Bloch Sphere,
we can acquire enough information to reconstruct the quantum process matrix
$\chi$ and measure its fidelity. We test the algorithm’s performance against
standard QPT using simulated and physical experiments on several IBM quantum
processors and compare the resulting process matrices. We demonstrate in
numerical experiments that the method can improve gate fidelity via a noise
reduction in the imaginary part of the process matrix, along with a stark
decrease in the number of experiments needed to perform the characterization.

Quantum computation, gate characterization

##  I Introduction

Modern quantum computers are marred by noise that limits the computational
reach of these devices. The sources of this noise are myriad, including
initial state preparation errors, noise introduced during the computation via
decoherence and gate noise, and imprecise state readout at measurement
Magesan2012 . There has been extensive work to improve quantum processor units
(QPUs) at the hardware level with methods that are generally not accessible at
the end-user level. Given the current proliferation of noisy intermediate-
scale quantum resources, we seek to develop strategies that end-users can use
to calibrate a set of qubits on a physical QPU in a cost-effective (less
resource-intensive) manner. Furthermore, new algorithms to isolate and
characterize noise are critical for quantifying where QPUs need improvement
and benchmarking algorithm performance Eisert2020 .

One such characterization method, quantum tomography, provides a set of tools
to characterize the behavior of quantum dynamical processes through a series
of measurements on a complete basis, typically the Pauli basis. Standard
quantum process tomography (QPT) reconstructs the underlying quantum process
$\mathcal{E}$ by performing state tomography on a set of identical quantum
states after applying certain quantum operations, i.e., a quantum circuit
NielsenBook ; Poyatos1997 ; Chuang1997 . Through this tomographic
reconstruction in state space, one can infer the region where each generated
state lies by applying maximum likelihood estimation Baumgratz2013 , Bayesian
credibility BlumeKohout2010 ; Ferrie2014 , or confidence regions
BlumeKohout2012 ; Christandl2012 ; Wang2019 .

Process tomography is also a key component in noise characterization and noise
mitigation for quantum algorithms Zhang2020 . Through a large number of
circuit evaluations, or shots, one can deploy statistical and numerical
methods to recover the underlying process matrix representation of
$\mathcal{E}$ Alexander2020 ; Korotkov2013 ; Mohseni2008 . However, the
resource requirements inhibit the scalability of QPT-based methods on NISQ
hardware. For a complete determination of an n-qubit quantum process, one
needs to prepare $4^{n}\times 3^{n}$ independent circuit executions to specify
a quantum process completely (see Appendix A for more details). This resource
overhead makes QPT impractical for characterizing processes involving more
than a few qubits. For example, the complete characterization of a 3-qubit
quantum process requires $12^{3}=1728$ independent experiments, with each
experiment repeated many times to gather sufficient statistics. On the
publicly available IBM QPUs, i.e., IBMQ Bogota, a user can send at most 900
independent experiments, falling far short of the 1728 required to
characterize a 3-qubit process fully. One may send the complete set of
experiments in two separate batches of circuits, but this leaves open the
possibility that the device may have changed significantly between
experimental runs. Without dedicated access to a QPU, process tomography is
practically challenging for $n>3$ qubit systems. To dodge this problem,
ancilla-Altepeter2003 ; Leung2003 ; Ariano2003 ; Ariano2001  and error-
correction-basedOmkar2015a ; Omkar2015b ; Mohseni2006 ; Mohseni2007  QPT
schemes were introduced. These methods require sophisticated state and
measurement preparations. The number of experiments can be reduced even more
if some prior information about the process is known using compressed sensing
techniquesFlammia2010 ; Flammia2012 . However, the success of compressed
sensing depends on the accuracy of the rank knowledge given for the quantum
processRodionov2014 ; Shabani2011 . An adaptive measurement technique
introduces a way to characterize any unitary process that does not require any
prior assumption about the processKim2020 , but it requires an optimization
routine to adapt the initial states and measurement operators. On the other
hand, the resource requirements can be reduced if one does not require a
complete characterization of the quantum process; for instance, randomized
benchmarking is commonly use to compute gate fidelities on superconducting
QPUs Emerson2005 ; Knill2008 ; Magesan2011 . In summary, these methods assume
a specific structure: low-rank restrictions Shabani2011 , two-qubit processes
Govia_2020 , and a unitary structure that only requires measurements of the
diagonal elements of the rotated process matrix.

We consider the results from standard QPT as a reference to measure the
performance of our method. In addition to the experimental overhead, QPT
assumes perfect readout measurement, yet it is highly sensitive to state
preparation and measurement (SPAM) errors Eisert2020 ; Korotkov2013 . This
assumption can lead to underestimating process fidelity by rolling SPAM errors
into the same process as the gates one is trying to characterize. Here, we
reduce the complexity of QPT without sparsity assumptions while still offering
an exponential improvement in resource cost over standard QPT by assuming a
very simple noise model consisting of SPAM error and rotation error (see Fig.
2). Characterization uses a series of rotations and measurements tailored to
limited access quantum chips such as cloud-based IBM quantum devices, which we
dub parametrized process characterization (PPC). We further provide a method
to unravel SPAM errors from process characterization by fitting the projective
measurement of key quantum states generated by the quantum process to a
statistical model influenced by SPAM-type errors. The resulting fit parameters
then allow us to reconstruct the underlying quantum process.

##  II Methods

To illustrate the general idea of PPC, we first give a one-qubit example that
can be extended to a more general case. We wish to characterize some quantum
process ${\cal U}:\rho\rightarrow U\rho\,U^{\dagger}$, with $\rho$ and $U$ as
a one-qubit quantum state and unitary operator, respectively. Without loss of
generality, we shall consider the rotation $Y_{\theta}=\hbox{\rm
exp}[-i\theta\sigma_{y}/2]$ and assume that rotation is a noiseless unitary
operation in the experimental setup. The projective measurement, along the
$z$-axis in the Bloch sphere, of the state $Y_{\theta}U|s\rangle$, for
$|s\rangle\in\\{|0\rangle,|1\rangle\\}$, reads

$P^{\cal
U}_{s}(\theta)=\left(\begin{array}[]{c}s_{\theta/2}^{2}+|{U}_{s,0}|^{2}c_{\theta}-({U}_{s,0}{U}_{s,1}^{*}+{U}_{s,1}{U}_{s,0}^{*})s_{\theta}\\\\[5.69054pt]
s_{\theta/2}^{2}+|{U}_{s,1}|^{2}c_{\theta}+({U}_{s,0}{U}_{s,1}^{*}+{U}_{s,1}{U}_{s,0}^{*})s_{\theta}\end{array}\right)\
,$ (1)

with ${U}_{kl}=\langle l|{U}|k\rangle$, $c_{\theta}=\cos\theta$, and
$s_{\theta}=\sin\theta$.

To account for readout error, we introduce a classical assignment error
modeled by the transition matrix:

$T=\left(\begin{array}[]{cc}t_{00}&1-t_{11}\\\\[5.69054pt]
1-t_{00}&t_{11}\end{array}\right),$ (2)

with $t_{00}$ and $t_{11}$ as the probabilities of measuring correctly the
states $|0\rangle$ and $|1\rangle$, respectively. $T$ is obtained
experimentally via calibration measurements. This matrix represents a binary
asymmetric channel, i.e., $t_{00}\neq t_{11}$, that maps the original
probability distribution to the experimental observation $Q^{\cal
U}_{s}(\theta)=T\cdot P^{\cal U}_{s}(\theta)$, which can be rewritten as:

$Q^{\cal U}_{s}(\theta)=\left(\begin{array}[]{c}1-t_{11}\\\
1-t_{00}\end{array}\right)+|T|P^{\cal U}_{s}(\theta-\theta_{0}).$ (3)

This establishes a way to determine the quantum process by fitting $U_{s,0}$
and $U_{s,1}$ with the experimental data $T$ and $Q^{\cal U}_{s}(\theta)$.
Notably, with the assumption a transition matrix can be describe readout
error, and that errors occur only along the direction of rotation, a single
parameter in Eq. 3 can be used to fit the data. Here $\theta_{0}$ is an
initial phase representing either a state preparation error or a compilation
error in the rotation operator $Y_{\theta}$. In summary, we evaluate the
action of the unitary operator $U$ on a set of rotated states
$Y_{\theta}|0\rangle$, and by fitting the model for the measurement output
$T\cdot P_{s}^{\cal U}(\theta)$ to the experimental output $Q_{s}^{\cal
U}(\theta)$ in (see Eq. 3), we can estimate the components for $U$ in the
computational basis.

This procedure can be extended to an $n$-qubit system, considering
$Y^{s}_{\theta}$ as the main rotation; the superscript $s$ stands for the
physical qubit where the rotation is applied. As before, it is enough to
consider a subset of the $n$-qubit computational basis where the $s$-th qubit
is in the ground state, i.e., the set
$\\{|\varphi_{k,s}\rangle=|k_{0},\cdots,k_{2^{n}}\rangle\in\\{|0\rangle,|1\rangle\\}^{n}:k_{s}=0\\}$,
and determine the action of the unitary operator $U$ on different rotations
applied on this set,

$|\psi_{k,s}(\theta)\rangle=U\cdot Y^{s}_{\theta}|\varphi_{k,s}\rangle$ (4)

for different values of $\theta$ in $[-\pi,\pi]$. The projective measurement
from these states, $T\cdot P_{k,s}^{\cal U}(\theta)$, can be seen as a
function of $\theta$ with parameters given by the components of $U$, i.e., a
function $F_{k,s}(\theta;U_{00},\cdots,U_{(2^{n},2^{n})})$. Thus, the fit
parameters of this function to the experimental measurement data give the
estimate for $U$. Before applying the algorithm, we use a calibration
procedure to determine the transition matrix $T$ and the phase correction
$\theta_{0}$ that uses a similar angular sweep (see Section V).

Figure 1: Diagrammatic sketch of PPC in the Bloch sphere for one-qubit
systems. With the information obtained from the rotation on the states ${\cal
U}|0\rangle$ and ${\cal U}|1\rangle$ we get information about the quantum
process represented by $\cal U$. The dotted lines represent the rotations of
the states ${\cal U}|0\rangle$ and ${\cal U}|1\rangle$ around the $y$-axis.

##  III Single Qubit Quantum Process Characterization

For an $n$-qubit quantum process characterization ${\cal U}$, it is necessary
to execute $N_{PPC}=2^{n-1}(n+2)N_{\theta}$ quantum circuits with $N_{\theta}$
as the number of rotations the interval $[-\pi,\pi]$ is divided into;
$2^{n}N_{\theta}$ quantum circuits for calibration; and $2^{n-1}nN_{\theta}$
quantum circuits to obtain data for fitting an estimate of $U$. Both the
calibration and the estimation quality depends on $N_{\theta}$ -a large enough
value ensures slight deviations $(\sigma\sim N_{\theta}^{-1/2})$ of the model
from the experimental data. Note that $N_{\theta}$ does not depend on the
number of qubits. Since QPT scales differently with the number of qubits,
$N_{QPT}=12^{n}$, PPC with a moderate number of angles becomes a favorable
method in cases where $n>2$, since $\log(N_{QPT}/N_{{PPC}})\sim(n-1)\log
6-\log(N_{\theta}/2)$. Figure 2 shows resource scaling for each protocol.

Figure 2: _Number of quantum circuit executions for quantum process
characterization:_ in this figure, we compare the resources used by QPT
against the resources used by PPC. For different numbers of rotations
($N_{\theta}=31,51,$and $71$ lines), QPT surpasses PPC in the number of
resources required for its execution for $n>2$.

We performed simulations and on-hardware experiments for one and two-qubit
systems using native gates as the target operations to characterize. For these
experiments, we set $N_{\theta}=51$ rotations and $N=5000$ experimental
repetitions of PPC circuits and employ the Python Symfit symfit  library to
fit the model to the experimental data. Symfit is a Python module that uses
symbolic methods to determine the Jacobian in the fitting process
analytically. We test our procedure characterizing the X- and H-gate for one-
qubit systems and the CX-gate for two-qubit systems applied on different
plaquettes on several IBM QPUs. We use $Y_{\theta}$ for the calibration and
characterization since this rotation has lower fidelity than $X_{\theta}$ (see
appendix B), and thus constitutes a good benchmark for the method. For the QPT
experiments, we use the tomography module of Qiskit Ignis Qiskit .

A common way to describe the quantum process $\cal U$ is through the process
matrix ${\chi}$ defined as

${\cal U}(\rho)=\sum_{kl}\chi_{kl}P_{k}\rho P_{l},$ (5)

with $\rho$ as a one qubit state, and
$P_{k}\in\\{\mathbb{I},\sigma_{x},\sigma_{y},\sigma_{z}\\}$ (Pauli basis). It
is simple to find the process matrix ${\chi}$, we need to find the
representation of $U$ in the Pauli basis and compare $U\rho_{0}U^{\dagger}$
with Eq. (5), which gives $\chi_{kl}=u_{k}u^{*}_{l}$ with $u_{k}={\rm
tr}\\{{\cal U}P_{k}\\}/2$.

When comparing QPT and PPC-generated process matrices, we observe
qualitatively similar results for the real part, $\chi_{re}$, while observing
differences in the imaginary part, $\chi_{im}$. In Figures 3 and 4, we present
the heat plots of the process matrix from the numerical and _on-hardware_
experiments comparing both methods using the gates H and CX as targets,
respectively. We further observe that the $\chi^{PPC}_{im}$ differs from
$\chi^{QPT}_{im}$, even in the noiseless numerical experiments. The imaginary
part, which gives information about the quantum error Korotkov2013 , in both
procedures is slightly different, owing to the assignment error mitigation
implemented in PPC. This isolation of SPAM errors is impossible to do under
standard QPT. A common way to compare the empirical and the expected operation
is through the process fidelity $F_{\chi}={\rm tr}[\chi\chi_{0}]/4^{n}$, with
$\chi_{0}$ as the noiseless process matrix. In table 1, we show the process
fidelity values for every experiment. The PPC results show closer values to
$1$ than the QPT results in the numerical experiments, where we used a
noiseless simulator as the initial test of the method. In the physical
experiments, we still observe that $\chi_{PPC}$ has a more minor contribution
from the imaginary part than the result from $\chi_{QPT}$. The numerical
results for $\chi^{PPC}_{im}$ and $\chi^{QPT}_{im}$ are irrelevant since the
statistical error $N^{-1/2}/2\sim 7\times 10^{-3}$Knips2015  in the projective
measurement. For the physical experiment, we observed similar values for one-
and two-qubit in the standard deviation ($\sigma_{PPA}\sim 2\times 10^{-3}$
and $\sigma_{QPT}\sim 6\times 10^{-3}$) after bootstrapping results from 21
experiments using each method in $10^{4}$ resamples. These deviations
determine what values are statistically relevant in the experiment. Therefore,
the results for one-qubit experiments are closer to an ideal behavior than the
results from two-qubit gates. Additionally, we determine the difference
between the process matrices $\chi^{PPC}$ and $\chi^{QPT}$ applying the
distance $d_{\infty}(\cdot,\cdot)$

Figure 3:  _Comparison between tomography results for the one-qubit system_. Here we present the $\chi$-matrix representation in the Pauli matrices for the Hadamard gate $\rm H$ using QPT and PPC. In a) and b), we depict the $\chi$-matrix using QPT and PPC, respectively, by running the experiment in a noiseless simulator. In c) and d), we show the resulting $\chi$ matrices, by QPT and PPC, respectively, by running the experiment on the 0th qubit on the IBMQ-Bogota backend. In both cases, we considered $N=5000$ shots and least square estimation for QPT, and $N=5000$ shots and $N_{\theta}=51$ for PPC. There is a good agreement in the $\chi$-matrix’s real part, but a difference in the imaginary part, even when using the noiseless simulator. In the noiseless simulator, the imaginary part is statistically irrelevant since the statistical error is in the order of $N^{-1/2}/2\sim 7\times 10^{-3}$Knips2015 . On the other hand, in the QPT experiments, the imaginary part is statistically relevant due to SPAM errors. At the bottom, we present the $d_{\infty}(\cdot)$ distance between the resulting process matrices. Table 1: Process fidelity $F_{\chi}$ of local and non-local gates.  configuration | gate |  $F_{\chi}$ PPC |  $F_{\chi}$ QPT
---|---|---|---
Numerical |  | 1.0 | 0.99
IBMQ-Bogota, qubit 0 |  | 0.99 | 0.92
IBMQ-Bogota, qubit 2 |  | 0.99 | 0.95
IBMQ-Santiago, qubit 1 | X | 0.99 | 0.97
IBMQ-Santiago, qubit 3 |  | 0.99 | 0.99
IBMQ-Quito, qubit 0 |  | 0.92 | 0.93
IBMQ-Quito, qubit 1 |  | 0.99 | 0.99
IBMQ-Boeblingen, qubit 0 |  | 0.99 | 0.96
IBMQ-Boeblingen, qubit 4 |  | 0.99 | 0.92
Numerical |  | 1.0 | 0.99
IBMQ-Bogota, qubit 0 |  | 0.96 | 0.90
IBMQ-Bogota, qubit 2 |  | 0.99 | 0.95
IBMQ-Santiago, qubit 1 | H | 0.99 | 0.97
IBMQ-Santiago, qubit 3 |  | 0.99 | 0.99
IBMQ-Quito, qubit 0 |  | 0.93 | 0.94
IBMQ-Quito, qubit 1 |  | 0.99 | 0.99
IBMQ-Boeblingen, qubit 0 |  | 0.99 | 0.95
IBMQ-Boeblingen, qubit 4 |  | 0.96 | 0.90
Numerical |  | 1.0 | 0.98
IBMQ-Bogota, qubits [1,2] |  | 0.97 | 0.75
IBMQ-Bogota, qubits [0,1] | CX | 0.96 | 0.73
IBMQ-manhattan, qubits [0,1] |  | 0.99 | 0.88
IBMQ-manhattan, qubits [11,17] |  | 0.99 | 0.98
IBMQ-Boeblingen, qubit 0 |  | 0.99 | 0.86
IBMQ-Boeblingen, qubit 4 |  | 0.99 | 0.81

##  IV multi-qubit quantum process characterization

Consider the action of a $n$-qubit quantum operator ${\cal U}$ on the state

$\displaystyle|\psi_{k,s}(\theta)\rangle$ $\displaystyle=$ $\displaystyle
Y_{\theta}^{s}|k_{0},\cdots,k_{s-1},0,k_{s+1},\cdots,k_{n}\rangle$ (6)
$\displaystyle=$ $\displaystyle Y_{\theta}^{s}|k,0_{s}\rangle.$

Above, we used a short notation to identify where the rotation is being
applied, in this case on the ground state of the $s$-th qubit while the rest
are at $|k_{0},\cdots,k_{s-1},k_{s+1},\cdots,k_{n}\rangle$. The probability
distribution for ${\cal U}|\psi_{k,s}(\theta)\rangle$ reads as

$P^{\,\cal
U}_{k,s}(\theta)=\frac{1}{2}\left(A_{k,s}+B_{k,s}\,c_{\theta}+C_{k,s}\,s_{\theta}\right),$
(7)

where $A_{k,s}$, $B_{k,s}$, and $C_{k,s}$ are vectors in $\mathbb{R}^{2^{n}}$,
with components

$\displaystyle[A_{k,s}]_{j}$ $\displaystyle=$ $\displaystyle|\langle j|{\cal
U}|k,0_{s}\rangle|^{2}+|\langle j|{\cal U}|k,1_{s}\rangle|^{2}\ ,$ (8)
$\displaystyle\,[B_{k,s}]_{j}$ $\displaystyle=$ $\displaystyle|\langle j|{\cal
U}|k,0_{s}\rangle|^{2}-|\langle j|{\cal U}|k,1_{s}\rangle|^{2}\ ,$ (9)
$\displaystyle\,[C_{k,s}]_{j}$ $\displaystyle=$ $\displaystyle\langle j|{\cal
U}|k,0_{s}\rangle\langle j|{\cal U}|k,1_{s}\rangle^{*}+$
$\displaystyle\quad\quad\langle j|{\cal U}|k,0_{s}\rangle^{*}\langle j|{\cal
U}|k,1_{s}\rangle,\ j\in\\{0,1\\}^{n}\ .$

By measuring $N$ systems identically prepared in the state ${\cal
U}|\psi_{k,s}(\theta)\rangle$ for every $\theta$ in ${\cal
S}_{N_{\theta}}=\\{(j/N_{\theta}-1)\pi:j\in\mathbb{N},0\leq j\leq
N_{\theta}\\}$; we estimate the probability distributions $Q_{k,s}^{j}=({\rm
C}^{k,j}_{0}/N,\cdots,{\rm C}^{k,j}_{2^{n}}/N)$ for $j=0,\cdots,N_{\theta}$,
with ${\rm C}^{k,j}_{m}$ as the number of outcomes ‘$m$’ $\in\\{0,1\\}^{n}$
for the $j$-th angle. The original distributions $P^{j}_{k,s}$, after
mitigating the assignment error (see next section for details), are obtained
by minimizing the functions $f_{j}(P)=||Q^{j}_{k,s}-T\cdot P||_{F}$, where
$||\cdot||_{F}$ is the Frobenius norm. With that information, i.e.
$\\{P^{0}_{k,s},\cdots,P^{N_{\theta}}_{k,s}\\}$ and the model (7), we can
compute the matrix elements for $\cal U$. Therefore, we can determine the
process matrix $\chi$.

Figure 4: _$\chi$ -matrix for the CX-gate:_ Here we depict the $\chi$-matrix
representation in Pauli matrices of the CNOT-gate, obtained by the QPT and PPC
methods. In a) and b) we used a noiseless simulator to run QPT and PPC,
respectively. In c) and d), we used the qubits 0 and 1 on IBMQ-Bogota backend
to run QPT and PPC respectively. In the experiments we used $N=5000$ shots and
least squared estimation for the QPT configuration, and $N=5000$ shots and
$N_{\theta}=51$ for the PPC configuration. We obtained similar results to the
one-qubit case, there is good agreement with the real part, but a slight
difference in the imaginary ones. Again, part of the imaginary values stem
from an inherent error introduced by the estimation procedure in the QPT
method. In this case, the imaginary part produced by PPC and QPT in the
numerical experiments are still statistical no relevant since we can consider
the same level of statistical error used in the one-qubit case Knips2015 ,
i.e. $\sim 7\times 10^{-3}$. For the physical experiments, the imaginary part
overpass the standard deviations ($\sigma_{PPA}\sim 2\times 10^{-3}$ and
$\sigma_{QPT}\sim 6\times 10^{-3}$) reveling imperfections in the quantum
gate. In this case, the distance $d_{\infty}(\cdot,\cdot)$ in the simulation
is similar with the experimental result.

##  V calibration

We assume the assignment error effects are represented by a transition matrix
$T_{kl}=P(l|k)$, where $P(l|k)$ is the conditional probability of observing
$|j\rangle$ when the system has been prepared in the state $|i\rangle$. This
is typically determined by measuring the number of outcomes ‘$j$’, $c_{j}$, of
$N$ identically-prepared states of $|i\rangle$; $T_{ij}=c_{j}/N$. However,
this procedure is limited in its explanatory power, as we cannot determine
which part of the readout error comes from the state preparation.
Additionally, this method scales exponentially with the number of qubits
considered since we must evaluate the conditional probabilities for all
$2^{n}$ states of the system. We follow a modified calibration routine that
tracks assignment error as a function of the rotation about the $y$-axis to
discriminate the assignment error from every possible state preparation error
in each experiment.

We rotate the states $|0,k_{0},\cdots,k_{n-1}\rangle$ around the $y$-axis of
the first qubit, for $k\in\\{0,1\\}^{n-1}$, yielding the output
$c_{\theta/2}|0,k_{0},\cdots,k_{n-1}\rangle+s_{\theta/2}|1,k_{0},\cdots,k_{n-1}\rangle$.
Thus, the expected probability distribution will have two components,
$c_{\theta/2}^{2}$ and $s_{\theta/2}^{2}$, for instance

$\displaystyle P_{k=0\cdots 0}(\theta)$ $\displaystyle=$
$\displaystyle(c_{\theta/2}^{2},\underbrace{0,\cdots,0}_{2^{n-1}-1},s_{\theta/2}^{2},0,\cdots,0)^{T},$
$\displaystyle P_{k=10\cdots 0}(\theta)$ $\displaystyle=$
$\displaystyle(0,c_{\theta/2}^{2},\underbrace{0,\cdots,0}_{2^{n-1}-2},0,s_{\theta/2}^{2},0,\cdots,0)^{T},$
$\displaystyle\vdots$ $\displaystyle P_{k=0\cdots 01}(\theta)$
$\displaystyle=$
$\displaystyle(\underbrace{0,\cdots,0}_{2^{n-1}-1},c_{\theta/2}^{2},0,\cdots,0,s_{\theta/2}^{2})^{T}.$

Under the bit-flip noise model, an empirical probability distribution
$Q_{k}(\theta)$ is related to $P_{k}(\theta)$ by $Q_{k}(\theta)=T\cdot
P_{k}(\theta)$, with

$T=\left(\begin{array}[]{ccc}t_{00}&\cdots&t_{0,2^{n}}\\\
\vdots&\ddots&\vdots\\\ t_{2^{n},0}&\cdots&t_{2^{n},2^{n}}\end{array}\right)\
.$ (12)

Taking into account the structure of $P_{k}(\theta)$, we can determine two
columns of $T$ per state $Y_{\theta}|0,k_{0},\cdots,k_{n-1}\rangle$. We
measure the states $Y_{\theta_{j}}|0,k_{0},\cdots,k_{n-1}\rangle$,
$\theta_{j}\in{\cal S}_{\theta}$, and estimate the probability distribution by
the sampling distribution of the outcomes, i.e. we obtain the quantities
$Q_{k}^{j}=({\rm C}^{k,j}_{0}/N,\cdots,{\rm C}^{k,j}_{2^{n}}/N)^{T}$. We fit
the acquired data with the model $T\cdot P_{k}(\theta-\theta_{0})$. In the
model, we have considered an initial phase $\theta_{0}$ to describe
preparation error.

###  V.1 Two-qubit transition matrix and initial preparation

For the two-qubit system, we can consider the following model for the expected
distributions of the states $Y^{0}_{\theta}|00\rangle$ and
$Y^{0}_{\theta}|01\rangle$,

$\displaystyle P_{0}(\theta)$ $\displaystyle=$
$\displaystyle(c_{\theta/2}^{2},0,s_{\theta/2}^{2},0)^{T},$ $\displaystyle
P_{1}(\theta)$ $\displaystyle=$
$\displaystyle(0,c_{\theta/2}^{2},0,s_{\theta/2}^{2})^{T},$ (13)

and the emperical distributions $Q_{k}(\theta)=T\cdot P_{k}(\theta)$,

$\displaystyle Q_{0}(\theta)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}t_{00}\\\ t_{10}\\\ t_{20}\\\
t_{30}\end{array}\right)c_{\theta/2}^{2}+\left(\begin{array}[]{c}t_{02}\\\
t_{12}\\\ t_{22}\\\ t_{32}\end{array}\right)s_{\theta/2}^{2},$ (22)
$\displaystyle Q_{1}(\theta)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}t_{01}\\\ t_{11}\\\ t_{21}\\\
t_{31}\end{array}\right)c_{\theta/2}^{2}+\left(\begin{array}[]{c}t_{03}\\\
t_{13}\\\ t_{23}\\\ t_{33}\end{array}\right)s_{\theta/2}^{2}.$ (31)

Now, we execute the quantum circuits

(33) (35)

and for $\theta_{j}\in{\cal S}_{\theta}$. We determine the transition matrix
elements $t_{ij}$ by fitting the model (22) to the acquired data
$Q_{k}^{j}=({\rm C}_{00}^{k,j}/N,{\rm C}_{01}^{k,j}/N,{\rm
C}_{10}^{k,j}/N,{\rm C}_{11}^{k,j}/N)^{T}$ from the above quantum circuit
executions.

##  VI Discussion

To summarize, we introduced an alternative method to characterize a unitary
quantum process based on a rotational sweeping procedure without prior
information about the process. Instead of using a complete set of measurement
operators, we measure the unknown quantum state rotated around different
angles. Additionally, we proposed a pre-characterization process to define the
assignment errors enclosed in the transition matrix and then mitigated those
effects in the quantum process characterization. In the scheme presented here,
we perform $2^{n-1}(n+2)N_{\theta}$ experiments to characterize an $n$-qubit
process, in which the number of angles $N_{\theta}$ does not scale with the
number of qubits. Therefore, we can fix this parameter to get enough data to
fit the model with minor deviations. We considered $N_{\theta}=51$ for a
suitable fitting with relatively slight deviations ($\sim 10^{-4}$) from ideal
behavior; this value can be used for any $n$-qubit system. Additionally, QPT
is affected by errors in preparing the initial states and tomography
measurements. By contrast, in our method, we control the initial preparation
error by considering an initial phase in the model and the readout error by
mitigating the assignment error.

The PPC algorithm is affected by the fitting error that depends on the number
of rotational divisions, $N_{\theta}$. This procedure assumes a high-fidelity
rotation and trusted quantum states $|0\rangle$ and $|1\rangle$ in the same
way that QPT relies on high fidelity measurement. One way to mitigate a
possible imperfection in the rotation process is by using a different
compilation for the rotations $Y_{\theta}$ and $X_{\theta}$. Instead of
decomposing any rotation in terms of $X_{\pm\pi/2}$ and $Z_{\theta}$, we can
use the compilation procedure introduced in Gokhale2020 ; a decomposition in
terms of $X_{\theta}$. The alternative decomposition brings a shorter pulse
structure and commensurately higher fidelity.

To validate the performance of the PPC procedure, we compared the process
matrix obtained by our approach with the outcome from the QPT. We compared the
process fidelity $F_{\chi}$ of the results, from PPC and QPT, of different
one- and two-qubits gates on-hardware and in-silico. As expected, we found
minor differences between ${\rm Re}\\{\chi_{PPC}\\}$ and ${\rm
Re}\\{\chi_{QPT}\\}$; there was, however, a remarkable difference in their
imaginary parts, even in the numerical experiments. The matrix elements
$(\chi^{QPT/PPA}_{im})_{ij}$ can be attributed to numerical errors. They are
in the range of the statistical error present in the simulation and therefore
do not contribute to the characterization analysis. We observed statistically
relevant values for $\chi^{QPT}_{im}$ in the physical experiments and
neglishible ones in $\chi^{PPA}_{im}$, other consequence of the SPAM
mitigation in the PPA process in one-qubit quantum processes. The PPA
implements the mitigation of preparation errors in local rotations, therefore
the imaginary part $\chi^{QPT}_{im}$ contents relevant data. We establish the
difference of the process matrices via the distance $d_{\infty}(\cdot,\cdot)$,
and the process fidelity as performance metric that gives a perfect score in
the noiseless simulation for the PPA (see Table 1).

Additionally, the PPC protocol allows the calculation of the error process
matrix directly from the quantum gate’s characterization without appealing to
the QPT process introduced by Korotkov Korotkov2013 . The imaginary part of
the error process matrix provides information about the process fidelity and
imperfections. A natural extension of this work is studying the error process
matrix for low-depth quantum circuits. The method may also prove useful in
providing tighter measures of crosstalk effects in quantum processes. For
example, tomography on one qubit, while its neighbors undergo local unitaries,
can reveal correlated noise idle1 ; idle2 . The improved scalability of PPC
over QPT allows us to extend this tomographic method to more significant
numbers of qubits, a valuable feature for crosstalk identification and
characterization.

##  Appendix A Quantum process tomography implementation

The QPT algorithm finds the process matrix $\chi_{\mathcal{E}}$ of a quantum
map $\mathcal{E}:|\rho\rangle\rangle\rightarrow|E\rho
E^{\dagger}\rangle\rangle$, by measuring the resulting state
$\mathcal{E}|\rho_{k}\rangle\rangle$, from a basis of initial states
$\mathcal{P}:\\{|\rho_{1}\rangle\rangle,\cdots,|\rho_{4^{n}}\rangle\rangle\\}$,
onto different directions
$\mathcal{M}:\\{|E_{1}\rangle\rangle,\cdots,|E_{M}\rangle\rangle\\}$ in the
Hilbert space, with $n$ and $M$ as the number of qubits and number of
measurement operators respectively. Here, we have introduced the superoperator
notation for the statistical operator, where operators become superkets and
quantum maps becomes superoperators, i.e,
$\mathcal{E}(\rho)\rightarrow\mathcal{E}|\rho\rangle\rangle$ (more details in
Greenbaum2015 ). In the superoperator notation, the goal is to determine the
matrix representation $[\chi_{\mathcal{E}}]_{ij}=\langle\langle
j||\mathcal{E}||i\rangle\rangle$ in the Pauli basis $\\{|i\rangle\rangle\\}$,
by the measurements

$\lambda_{ij}=\langle\langle E_{j}|\mathcal{E}|\rho_{i}\rangle\rangle.$ (37)

We can establish the relation between $\lambda$ and $\chi_{\mathcal{E}}$ by
inserting the completeness identity $\sum_{i}|i\rangle\rangle\langle\langle
i|=\mathbb{I}$ in Eq. 37,

$\lambda_{ij}=\sum_{k,l=1}^{4^{n}}[\chi_{\mathcal{E}}]_{lk}\ \langle\langle
E_{j}|k\rangle\rangle\langle\langle l|\rho_{i}\rangle\rangle.$ (38)

We can arrange the terms as

$\displaystyle y_{j+(i-1)\times M}$ $\displaystyle=$
$\displaystyle\lambda_{ij},$ (39) $\displaystyle x_{k+(l-1)\times 4^{n}}$
$\displaystyle=$ $\displaystyle[\chi_{\mathcal{E}}]_{ij}$ (40) $\displaystyle
B_{j+(i-1)\times M,\,k+(l-1)\times 4^{n}}$ $\displaystyle=$
$\displaystyle\langle\langle E_{j}|k\rangle\rangle\langle\langle
l|\rho_{i}\rangle\rangle,$ (41)

and transform 38 into a more convenient expression, $B\,\vec{x}-\vec{y}=0$,
for a numerical solution.

B.1 One-qubit process matrix: Without loss of generality, consider the
characterization of a one-qubit quantum map $\mathcal{E}$ using the following
intitial states and measurement operators

$\begin{aligned}
\mathcal{P}:\left\\{\right.\,&|Z_{p}\rangle\rangle,|Z_{m}\rangle\rangle,\\\
&|X_{p}\rangle\rangle,|Y_{p}\rangle\rangle\,\left.\right\\}\\\
&\end{aligned}\quad,\quad\begin{aligned}
\mathcal{M}:\left\\{\right.\,&|Z_{p}\rangle\rangle,|Z_{m}\rangle\rangle,\\\
&|X_{p}\rangle\rangle,|X_{m}\rangle\rangle,\\\
&|Y_{p}\rangle\rangle,|Y_{m}\rangle\rangle\,\left.\right\\}\ \ ,\end{aligned}$
(42)

where we have introduced the projectors $Z_{p}=(\mathbb{I}+\sigma_{z})/2$,
$Z_{m}=(\mathbb{I}-\sigma_{z})/2$, $X_{p/m}=Y_{-\pi/2}Z_{p/m}Y_{\pi/2}$, and
$Y_{p/m}=X_{-\pi/2}Z_{p/m}X_{\pi/2}$. Since $Z_{p}=|0\rangle\langle 0|$,
$Z_{m}=|1\rangle\langle 1|$, $X_{p}=|+\rangle\langle+|$, and
$Y_{p}=|i\rangle\langle i|$, the set $\mathcal{P}$ is generated by the
preparation of the states $\\{|0\rangle,|1\rangle,|+\rangle,|i\rangle\\}$. The
quantum chip detector measures $\sigma_{z}$ by default. The possible outcomes
of a measurement on an arbitrary state $\rho$ are $i=0,1$, where $i=0$
corresponds to $\langle\sigma_{z}\rangle=+1$ and $i=1$ to
$\langle\sigma_{z}\rangle=-1$, with probabilities

$p_{0}={\rm tr}\\{\rho Z_{p}\\},\quad p_{1}={\rm tr}\\{\rho Z_{m}\\}.$ (43)

Now, to measure $\sigma_{x}$ and $\sigma_{y}$ we need to consider the
projectors $X_{p}$, $X_{m}$, and $Y_{p}$, $Y_{m}$, respectively, by
transforming the $x$\- and $y$-axis into the $z$-axis in the Bloch sphere and
measuring $\sigma_{z}$ (see the definitions below Eq. 44). Therefore, the set
$\mathcal{M}$ is generated by the gates
$\\{\mathbb{I},Y_{-\pi/2},X_{-\pi/2}\\}$. In Figure 5 there is a sketch of the
required circuits to gate characterization. Thus, the number of quantum
circuits for a complete characterization is less than the number of
independent terms in the matrix process $\chi_{\mathcal{E}}$. On the other
hand, one can observe that the number of measurement operators, size of
$\mathcal{M}$, is enough for the solution of Eq. 37, since ${\rm
dim}\\{\mathcal{M}\\}\times{\rm dim}\\{\mathcal{P}\\}>{\rm dim}\\{\vec{y}\\}$.

B.2 $n$-qubit process matrix: The natural extension of the initial states and
the measurement operators in the $n$-qubit quantum process characterization
follows:

$\begin{aligned}
\mathcal{P}_{n}:\left\\{\right.\,&|Z_{p}\rangle\rangle,|Z_{m}\rangle\rangle,\\\
&|X_{p}\rangle\rangle,|Y_{p}\rangle\rangle\,\left.\right\\}^{\otimes n}\\\
&\end{aligned}\quad,\quad\begin{aligned}
\mathcal{M}_{n}:\left\\{\right.\,&|Z_{p}\rangle\rangle,|Z_{m}\rangle\rangle,\\\
&|X_{p}\rangle\rangle,|X_{m}\rangle\rangle,\\\
&|Y_{p}\rangle\rangle,|Y_{m}\rangle\rangle\,\left.\right\\}^{\otimes n}\ \
,\end{aligned}$ (44)

where $\mathcal{P}_{n}$ is generated by the preparation preparation of the
states $\\{|0\rangle,|1\rangle,|+\rangle,|i\rangle\\}^{\otimes n}$. Now, the
possible outcomes of the $\sigma_{z}^{\otimes n}$ measurement on an arbitrary
state are $s=\\{0,1\\}^{n}$, with probabilities

$\displaystyle p_{0\cdots 0}$ $\displaystyle=$ $\displaystyle{\rm
tr}\\{Z_{p}\otimes\cdots\otimes Z_{p}\\}$ $\displaystyle p_{010\cdots 0}$
$\displaystyle=$ $\displaystyle{\rm tr}\\{Z_{p}\otimes Z_{m}\otimes
Z_{p}\otimes\cdots\otimes Z_{p}\\}$ $\displaystyle\vdots$ $\displaystyle
p_{1\cdots 1}$ $\displaystyle=$ $\displaystyle{\rm
tr}\\{Z_{m}\otimes\cdots\otimes Z_{m}\\}.$ (45)

For the $\\{\sigma_{z},\sigma_{y},\sigma_{z}\\}^{\otimes n}$ measurements we
need to apply the projectors $\\{X_{p},X_{n}\\}^{\otimes n}$ and
$\\{Y_{p},Y_{n}\\}^{\otimes n}$, respectively. Thus, the set $\mathcal{M}_{n}$
is generated by $\\{\mathbb{I},Y_{-\pi/2},X_{-\pi/2}\\}^{\otimes n}$. Again,
as in the one-qubit case, we get a redundant amount of measurements, ${\rm
dim}\\{{\cal M}_{n}\\}\times{\rm dim}\\{{\cal P}_{n}\\}=4^{n}\times 6^{n}$ for
the number $16^{n}$ of independent elements in $\chi$.

Figure 5: Preparation and Measurement set of quantum circuits for one-qubit
quantum process tomography.

##  Appendix B Fitting parameters

In this section we shall discuss the fitting details used in the post-
processing step in the PPC protocol.

The number of rotations and shots plays an essential role in the quantum
process characterization. We consider a one-qubit experiment to benchmark the
calibration process, which follows the same principle as the characterization.
In this experiment, we consider the qubit 0 in the IBMQ-Bogota quantum chip.
For the calibration we use rotations about the $x$\- and $y$-axis, with a
different number of rotations and shots. Figure 6 shows the conditional
probabilities and the initial phase for each experimental setup as a function
of the number of rotations and shots, with error bars indicating the standard
deviation in each measurement. We choose a suitable setup where the
conditional probabilities do not vary significantly concerning the result
using the highest values $N_{\theta}=71$ and $N=5000$. For $Y_{\theta}$ and
$X_{\theta}$ we found the optimal points $N_{\theta}=51$ and $N_{\theta}=41$,
respectively (see shadow regions in Figure 6). One important feature is the
dependence of the parameters’ standard deviation on $N_{\theta}$ and $N$,
which slightly improves the experimental setup’s refinement.

Figure 6:  _Fitting parameters:_ Conditional probabilities and initial phase
using $Y_{\theta}$ rotation in a), b), and c), and using the $X_{\theta}$
rotation in d), e), and f), respectively. The error bars represent the
standard deviation of each parameter in the fitting process. The shadow
regions indicate the optimal values for $N_{\theta}$ and $N$.

###### Acknowledgements.

We thank Ryan Bennink for valuable discussions. This work was supported as
part of the ASCR Quantum Testbed Pathfinder Program at Oak Ridge National
Laboratory under FWP # ERKJ332. This research used resources of the Oak Ridge
Leadership Computing Facility, which is a DOE Office of Science User Facility
supported under Contract DE-AC05-00OR22725.

## Author contributions

V.L.O. introduced the PPC method. V.L.O. and T.K. developed the theoretical
formalism, performed the analytic calculations and performed the experiments.
R.C.P. analyzed and interpreted experimental data and suggested experiments to
run. V.L.O, T.K., and R.C.P. cowrote the manuscript.

## Competing interests

The authors declare that there are no competing interests.

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