* [135] Robert E. Roth “Interactive maps: What we know and what we need to know” In _Journal of Spatial Information Science_ 6, 2013, pp. 59–115 DOI: 10.5311/JOSIS.2013.6.105
  * [136] Robert E. Roth “Visual variables” In _International Encyclopedia of Geography: People, the Earth, Environment and Technology_ Oxford, UK: John Wiley & Sons, Ltd, 2017, pp. 1–11 DOI: 10.1002/9781118786352.wbieg0761
  * [137] Robert E. Roth, Cynthia A. Brewer and Michael S. Stryker “A typology of operators for maintaining legible map designs at multiple scales” In _Cartographic Perspectives_ , 2011, pp. 29–64 DOI: 10.14714/CP68.7
  * [138] Robert E. Roth et al. “User studies in cartography: opportunities for empirical research on interactive maps and visualizations” In _International Journal of Cartography_ 3.sup1, 2017, pp. 61–89 DOI: 10.1080/23729333.2017.1288534
  * [139] Igbal Safarov, Albert Meijer and Stephan Grimmelikhuijsen “Utilization of open government data: A systematic literature review of types, conditions, effects and users” In _Information Polity_ 22.1, 2017, pp. 1–24 DOI: 10.3233/IP-160012
  * [140] Bahador Saket, Alex Endert and Cagatay Demiralp “Task-based effectiveness of basic visualizations” In _IEEE Transactions on Visualization and Computer Graphics_ 25.7, 2019, pp. 2505–2512 DOI: 10.1109/TVCG.2018.2829750
  * [141] Arvind Satyanarayan and Jeffrey Heer “Lyra: An interactive visualization design environment” In _Computer Graphics Forum_ 33.3, 2014, pp. 351–360 DOI: 10.1111/cgf.12391
  * [142] Arvind Satyanarayan et al. “Critical reflections on visualization authoring systems” In _IEEE Transactions on Visualization and Computer Graphics_ , 2020, pp. 1–1 DOI: 10.1109/TVCG.2019.2934281
  * [143] Arvind Satyanarayan, Dominik Moritz, Kanit Wongsuphasawat and Jeffrey Heer “Vega-Lite: A grammar of interactive graphics” In _IEEE Transactions on Visualization and Computer Graphics_ 23.1, 2017, pp. 341–350 DOI: 10.1109/TVCG.2016.2599030
  * [144] Simon Scheider, Auriol Degbelo, Werner Kuhn and Holger Przibytzin “Content and context description - How linked spatio-temporal data enables novel information services for libraries” In _gis.Science_ , 2014, pp. 138–149
  * [145] Simon Scheider, Jim Jones, Alber Sánchez and Carsten Keßler “Encoding and querying historic map content” In _The 17th AGILE International Conference on Geographic Information Science - Connecting a Digital Europe Through Location and Place_ Castellón, Spain: Springer International Publishing, 2014, pp. 251–273 DOI: 10.1007/978-3-319-03611-3˙15
  * [146] Douglas C Schmidt “Why software reuse has failed and how to make it work for you” In _C++ Report_ 11.1, 1999
  * [147] Olaf Schnabel “Map Symbol Brewer - a new approach for a cartographic map symbol generator” In _Proceedings of the 22nd International Cartographic Conference_ , 2005
  * [148] Susan Schnur, Kenan Bektaş and Arzu Çöltekin “Measured and perceived visual complexity: a comparative study among three online map providers” In _Cartography and Geographic Information Science_ 45.3, 2018, pp. 238–254 DOI: 10.1080/15230406.2017.1323676
  * [149] Raimund Schnürer, Martin Ritzi, Arzu Çöltekin and René Sieber “An empirical evaluation of three-dimensional pie charts with individually extruded sectors in a geovisualization context” In _Information Visualization_ 19.3, 2020, pp. 183–206 DOI: 10.1177/1473871619896103
  * [150] Daeil Seo and Byounghyun Yoo “Interoperable information model for geovisualization and interaction in XR environments” In _International Journal of Geographical Information Science_ 34.7, 2020, pp. 1323–1352 DOI: 10.1080/13658816.2019.1706739
  * [151] Ben Shneiderman “Universal usability” In _Communications of the ACM_ 43.5, 2000, pp. 84–91 DOI: 10.1145/332833.332843
  * [152] C. Stolte, D. Tang and P. Hanrahan “Polaris: a system for query, analysis, and visualization of multidimensional relational databases” In _IEEE Transactions on Visualization and Computer Graphics_ 8.1, 2002, pp. 52–65 DOI: 10.1109/2945.981851
  * [153] Olivier Swienty, Tumasch Reichenbacher, Simone Reppermund and Joseph Zihl “The role of relevance and cognition in attention-guiding geovisualisation” In _The Cartographic Journal_ 45.3, 2008, pp. 227–238 DOI: 10.1179/000870408X311422
  * [154] Alex Tait “Visual hierarchy and layout” In _Geographic Information Science & Technology Body of Knowledge_, 2018 DOI: 10.22224/gistbok/2018.2.4
  * [155] David Thevenin and Joëlle Coutaz “Plasticity of user interfaces: framework and research agenda” In _Proceedings of INTERACT ’99_ Edinburgh, Scotland, UK: IOS Press, 1999, pp. 110–117
  * [156] Tyler Thrash et al. “The future of geographic information displays from GIScience, cartographic, and cognitive science perspectives (Vision Paper)” In _14th International Conference on Spatial Information Theory (COSIT 2019)_ Regensburg, Germany: Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, 2019, pp. 19:1–19:11 DOI: 10.4230/LIPIcs.COSIT.2019.19
  * [157] Sergio Trilles, Carlos Granell, Auriol Degbelo and Devanjan Bhattacharya “Interactive guidelines: public communication of data-based research in cities” In _PLOS ONE_ 15.1, 2020, pp. e0228008 DOI: 10.1371/journal.pone.0228008
  * [158] Q. Van Ho, P. Lundblad, T. Astrom and M. Jern “A web-enabled visualization toolkit for geovisual analytics” In _Information Visualization_ 11.1, 2012, pp. 22–42 DOI: 10.1177/1473871611425870
  * [159] Pierre Vanhulst, Florian Évéquoz, Raphaël Tuor and Denis Lalanne “Designing a classification for user-authored annotations in data visualization” In _Proceedings of the 13th International Joint Conference on Computer Vision, Imaging and Computer Graphics Theory and Applications (VISIGRAPP 2018) - Volume 3: IVAPP_ Funchal, Portugal: SCITEPRESS - ScienceTechnology Publications, 2018, pp. 85–96 DOI: 10.5220/0006613700850096
  * [160] Dalia Varanka and Lynn Usery “The map as knowledge base” In _International Journal of Cartography_ 4.2, 2018, pp. 201–223 DOI: 10.1080/23729333.2017.1421004
  * [161] Fernanda B. Viegas et al. “ManyEyes: a site for visualization at Internet scale” In _IEEE Transactions on Visualization and Computer Graphics_ 13.6, 2007, pp. 1121–1128 DOI: 10.1109/TVCG.2007.70577
  * [162] Luis M. Vilches‐Blázquez and Jhonny Saavedra “A framework for connecting two interoperability universes: OGC Web Feature Services and Linked Data” In _Transactions in GIS_ 23.1, 2019, pp. 22–47 DOI: 10.1111/tgis.12496
  * [163] Julius Volz, Christian Bizer, Martin Gaedke and Georgi Kobilarov “Discovering and maintaining links on the web of data” In _8th International Semantic Web Conference (ISWC 2009)_ Chantilly, Virginia, USA: Springer, Berlin, 2009, pp. 650–665 DOI: 10.1007/978-3-642-04930-9˙41
  * [164] Johan Wagemans et al. “A century of Gestalt psychology in visual perception II. Conceptual and theoretical foundations.” In _Psychological Bulletin_ 138.6, 2012, pp. 1218–1252 DOI: 10.1037/a0029334
  * [165] Gerhard Weikum, Johannes Hoffart and Fabian Suchanek “Knowledge harvesting: achievements and challenges” In _Computing and Software Science: State of the Art and Perspectives_ , 2019, pp. 217–235 DOI: 10.1007/978-3-319-91908-9˙13
  * [166] Hadley Wickham, Dianne Cook, Heike Hofmann and Andreas Buja “Graphical inference for infovis” In _IEEE Transactions on Visualization and Computer Graphics_ 16.6, 2010, pp. 973–979 DOI: 10.1109/TVCG.2010.161
  * [167] Mark D. Wilkinson et al. “The FAIR guiding principles for scientific data management and stewardship” In _Scientific Data_ 3.160018, 2016 DOI: 10.1038/sdata.2016.18
  * [168] Mark D. Wilkinson et al. “A design framework and exemplar metrics for FAIRness” In _Scientific Data_ 5.1, 2018, pp. 180118 DOI: 10.1038/sdata.2018.118
  * [169] Graham Wills and Leland Wilkinson “AutoVis: automatic visualization” In _Information Visualization_ 9.1, 2010, pp. 47–69 DOI: 10.1057/ivs.2008.27
  * [170] Kanit Wongsuphasawat et al. “Voyager: exploratory analysis via faceted browsing of visualization recommendations” In _IEEE Transactions on Visualization and Computer Graphics_ 22.1, 2016, pp. 649–658 DOI: 10.1109/TVCG.2015.2467191
  * [171] Lindsay Bertram Yeates “Thought experimentation: A cognitive Approach” In _Graduate Diploma in Arts (By Research) Dissertation_ , 2004
  * [172] Yun Zhang et al. “A decentralized model for spatial data digital rights management” In _ISPRS International Journal of Geo-Information_ 9.2, 2020 DOI: 10.3390/ijgi9020084
  * [173] Tian Zhao, Chuanrong Zhang and Weidong Li “Accessing distributed WFS data through a RDF query interface” In _International Conference on GIScience (GISCience 2016) - Short Paper Proceedings_ 1.1, 2016


# cuTN-QSVM: cuTensorNet-accelerated Quantum Support Vector Machine with
cuQuantum SDK ††thanks: **The first three authors contributed equally to this
work. The order of these names is based on the alphabetical arrangement of the
characters in their names.

Kuan-Cheng Chen  QuEST
Imperial College London
London, United Kingdom
<EMAIL_ADDRESS>Tai-Yue Li  National Synchrotron Radiation
Research Center
Hsinchu, Taiwan
<EMAIL_ADDRESS>Yun-Yuan Wang  NVIDIA AI Technology Center
NVIDIA Corp.
Taipei, Taiwan
<EMAIL_ADDRESS>Simon See  NVIDIA AI Technology Center
NVIDIA Corp.
Santa Clara, CA, USA
<EMAIL_ADDRESS>Chun-Chieh Wang  National Synchrotron Radiation
Research Center
Hsinchu, Taiwan
<EMAIL_ADDRESS>Robert Wille  Chair of Design Automation
Technical University of Munich
Munich, Germany
<EMAIL_ADDRESS>Nan-Yow Chen  National Center for HPC
Narlabs
Hsinchu, Taiwan
<EMAIL_ADDRESS>An-Cheng Yang  National Center for HPC
Narlabs
Hsinchu, Taiwan
<EMAIL_ADDRESS>Chun-Yu Lin  National Center for HPC
Narlabs
Hsinchu, Taiwan
<EMAIL_ADDRESS>
###### Abstract

This paper investigates the application of Quantum Support Vector Machines
(QSVMs) with an emphasis on the computational advancements enabled by NVIDIA’s
cuQuantum SDK, especially leveraging the cuTensorNet library. We present a
simulation workflow that substantially diminishes computational overhead, as
evidenced by our experiments, from exponential to quadratic cost. While state
vector simulations become infeasible for qubit counts over 50, our evaluation
demonstrates that cuTensorNet speeds up simulations to be completed within
seconds on the NVIDIA A100 GPU, even for qubit counts approaching 784. By
employing multi-GPU processing with Message Passing Interface (MPI), we
document a marked decrease in computation times, effectively demonstrating the
strong linear speedup of our approach for increasing data sizes. This enables
QSVMs to operate efficiently on High-Performance Computing (HPC) systems,
thereby opening a new window for researchers to explore complex quantum
algorithms that have not yet been investigated. In accuracy assessments, our
QSVM achieves up to 95% on challenging classifications within the MNIST
dataset for training sets larger than 100 instances, surpassing the
capabilities of classical SVMs. These advancements position cuTensorNet within
the cuQuantum SDK as a pivotal tool for scaling quantum machine learning
simulations and potentially signpost the seamless integration of such
computational strategies as pivotal within the Quantum-HPC ecosystem.

###### Index Terms:

Quantum Machine Learning, Quantum Simulation, Tensor Network, cuQuantum SDK,
cuTensorNet

##  I Introduction

In the rapidly evolving landscape of artificial intelligence (AI), machine
learning algorithms stand out as pivotal components driving advancements
across a multitude of domains[1]. These algorithms, distinguished into
supervised and unsupervised learning paradigms, harness the power of data to
uncover patterns or make predictions[2]. Supervised learning, in particular,
leverages pre-labeled data to train models, with the Support Vector Machine
(SVM) being a cornerstone technique in this category[3]. SVMs excel in
classifying data into distinct categories by finding an optimal hyperplane in
either the original or a higher-dimensional feature space[4]. However, the
computational demands of SVMs, especially in the context of large-scale “big
data” applications[5], pose significant challenges in terms of both
computational resources and execution time.

Enter the realm of quantum computing, a burgeoning field offering profound
computational speedups over classical approaches for certain problem types.
Among these, Quantum Support Vector Machines (QSVMs) emerge as a promising
quantum-enhanced technique for machine learning[6, 7], capable of drastically
reducing the computational resources required for SVMs. Leveraging quantum
algorithms, QSVMs achieve exponential speedups in both training and
classification tasks by performing calculations in parallel and employing
quantum-specific optimizations[6, 8, 9, 10].

However, in the current Noisy Intermediate-Scale Quantum (NISQ) era[11], the
practical utility of quantum computers is significantly constrained by their
availability and imperfect technological state. Challenges such as the
fidelity of qubits, the error rates of two-qubit gates, and the limited number
of available qubits present substantial hurdles[12, 13, 14]. Despite the
advent of several methodologies aimed at enhancing qubit fidelity—such as
Quantum Error Mitigation (QEM)[15, 16] and Dynamical Decoupling (DD)[17,
18]—these limitations persist, impeding the realization of quantum advantage
on quantum computing platforms in the current NISQ era[19, 20]. Consequently,
the design and validation of quantum-inspired algorithms, or hybrid classical-
quantum algorithms, are predominantly conducted through high-performance
classical simulations[10, 21]. Furthermore, quantum simulators have shown
considerable success in the near-term verification of quantum algorithms on
small qubit systems[22].

Within the scope of our research, we have engineered an advanced tensor-
network simulation framework, purpose-built to expedite the development of
QSVMs through the integration of the cuTensorNet library underlying cuQuantum
SDK[23]. This library is meticulously optimized for NVIDIA GPUs and can
facilitate QSVM algorithms, requiring noiseless simulations for quantum kernel
estimation as depicted in Fig. 1. A pre-computation mechanism is embedded
within this workflow, allowing for the reuse of quantum kernel values in the
QSVM’s complex learning stages, thereby bolstering the efficacy of both the
training and classification phases.

Our simulation of the QSVM algorithm, devised for parallel execution using the
Message Passing Interface (MPI), harnesses the formidable power of GPU
acceleration, equipping our QSVM simulator to efficiently manage vast data
sets with only a modest increment in memory requirements. Its flexibility
ensures its utility across various quantum machine learning paradigms.
Performance benchmarks of our simulator demonstrate that it achieves
significant speedups, often exceeding tens of times, which are exponentially
better than those achieved by state-of-the-art methods using Qiskit on
CPUs[24]. Furthermore, it boasts scalability to multi-GPUs—a testament to the
proficiency achieved by the NVIDIA Quantum Team—thereby affirming our
simulator’s role as a potent and scalable asset in the quantum machine
learning arena and the Quantum-HPC ecosystem[23, 21].

Figure 1: QSVM Simulator: Optimizes quantum kernel estimation and learning,
enhancing phase operation and objective evaluation, leading to swift and
precise classification outcomes.

This approach’s adeptness at handling up to 784 qubits for QSVMs allows for a
comprehensive scaling analysis, shedding light on the quantum acceleration
achieved in comparison to traditional classical solvers. These strides in QSVM
development signal a major progression towards practical deployment, charting
a path for the application of quantum-enhanced methodologies to complex, real-
world data classification challenges within the Quantum-HPC Ecosystem[10, 21,
25, 26, 27]. This paves the way for breakthroughs in quantum computing’s
applicability and marks a significant contribution to the quantum information
sciences.

##  II Background

In the research work by Rebentrost et. al.[6], a quantum algorithm is
introduced that revolutionizes the computational efficiency of SVMs for big
data classification. This QSVM leverages quantum mechanics to achieve
logarithmic complexity in terms of both the dimension of the vectors $N$
(qubit number) and the number of training examples $M$ (data size).

Traditionally, SVM is formulated to identify a hyperplane that maximizes the
margin between two classes, described in the primal form as:

$\min_{\mathbf{w},b}\frac{1}{2}||\mathbf{w}||^{2}$ (1)

subject to

$y_{j}(\mathbf{w}\cdot\mathbf{x}_{j}+b)\geq 1,\;\forall j,$ (2)

where $\mathbf{w}$ is the normal to the hyperplane, $b$ is the bias,
$\mathbf{x}_{j}$ are the feature vectors, and $y_{j}$ are the class labels.

The quantum version utilizes the kernel method, mapping the feature space to a
higher-dimensional space to allow for nonlinear classification. The kernel
matrix $K$, with elements $K_{jk}=k(\mathbf{x}_{j},\mathbf{x}_{k})$,
represents inner products in this space. The quantum algorithm efficiently
inverts this kernel matrix, a task that is computationally intensive in
classical SVM, especially as $M$ and $N$ grow.

The quantum algorithm for matrix inversion, central to QSVM, significantly
reduces the runtime complexity to $O(\log(NM))$[6]. This quantum routine for
non-sparse matrix exponentiation underpins the SVM’s optimization, facilitated
by the QML algorithm for linear equations, known as the HHL algorithm, which
solves $A|x\rangle=|b\rangle$ for Hermitian matrix $A$.

The SVM’s dual optimization problem is given by:

$L(\alpha)=\sum_{j=1}^{M}y_{j}\alpha_{j}-\frac{1}{2}\sum_{j,k=1}^{M}\alpha_{j}K_{jk}\alpha_{k},$
(3)

with constraints:

$\displaystyle\sum_{j=1}^{M}\alpha_{j}y_{j}$ $\displaystyle=0,$ (4)
$\displaystyle\alpha_{j}$ $\displaystyle\geq 0,\;\forall j.$ (5)

The QSVM algorithm employs quantum routines to approximate the solution vector
$\alpha$, defining the separating hyperplane in the feature space, thus
classifying new data points with reduced computational complexity. This
approach highlights the potential of quantum computing to tackle large-scale
machine learning problems, characteristic of big data challenges, by utilizing
quantum parallelism and efficient manipulation of high-dimensional data.

##  III Simulating QSVM using cuQuantum SDK and cuTensorNet

###  III-A Introduction of cuQuantum SDK and cuTensorNet

As the fields of quantum computing and advanced numerical simulations rapidly
expand, NVIDIA has introduced cuQuantumK[23], a comprehensive software
development kit (SDK), to accelerate quantum circuit simulations with NVIDIA
GPUs. The cuQuantum SDK consists of optimized libraries such as cuStateVec and
cuTensorNet. cuStateVec is dedicated to state-vector-based simulation methods,
enabling substantial acceleration and efficient memory utilization.
cuTensorNet, on the other hand, deals with tensor-network-based simulation,
where a quantum circuit is represented as a tensor network (Fig.2). In this
formulation, the sequence of pairwise contractions plays a role in
computational cost as illustrated in Fig.3. The cuTensorNet library provides
advanced features for tensor network contractions, including path
optimization, approximate tensor network simulation techniques, and multi-GPU
multi-node execution. These functionalities facilitate simulations at an
unprecedented scale with significant acceleration, empowering researchers to
delve into unexplored tensor network theories and complex quantum algorithms.
As the demand for more sophisticated computational methods grows within the
scientific community, cuStateVec and cuTensorNet stand out for their potential
to accelerate research domains across quantum physics, quantum chemistry, and
quantum machine learning.

Figure 2: Quantum Circuit Formulation of cuTensorNet Figure 3: Consider two
contraction paths: The upper path results in a higher computational cost

To boost the efficiency of tensor network computation, cuTensorNet delivers
modular and finely adjustable APIs, as shown in Fig. 4, tailored for
optimizing the pairwise contraction path on the CPU and improving contraction
performance on the GPU. This optimization is essential for minimizing both
computation cost and memory footprint. The pathfinder workflow is primarily
structured in the following manner:

1) Simplification: This initial stage focuses on reducing the complexity of
the whole tensor network and eliminating redundancies within the network. The
implementation involves rank simplification to minimize the number of tensors
by removing trivial tensor contractions from the network, resulting in a
smaller network for subsequent processing.

2) Division: After simplification, the tensor network undergoes a recursive
graph partitioning. This approach segments the network into multiple sub-
networks and forms a contraction tree. The binary tree defines the contraction
path and can be further optimized at the following stage.

3) Slicing and Reconfiguration: The slicing process selects a subset of edges
from a tensor network for explicit summation. This technique results in lower
memory requirements and allows parallel execution for each sliced contraction.
Reconfiguration considers several small subtrees within the full contraction
tree and attempts to reduce the contraction cost of the subtrees. cuTensorNet
implements dynamic slicing, which interleaves slicing with reconfiguration.

cuTensorNet can efficiently find a contraction path with high quality
quickly[23]. This capability accelerates the exploration of quantum mechanics
underlying complex systems and quantum machine learning models, particularly
in the processing and analysis of high-dimensional data.

Figure 4: Building blocks of cuTensorNet

Consequently, cuQuantum itself represents a comprehensive suite aimed at
bolstering quantum computing simulations. As part of NVIDIA’s broader
initiative to support the quantum computing ecosystem, cuQuantum provides a
range of tools and libraries engineered to optimize various facets of quantum
computing simulations, including the programming model CUDA Quantum[23, 28]
(CUDA-Q) and frameworks such as Qiskit[29], Pennylane[30], and Cirq.[31]. The
introduction of cuQuantum marks a significant stride towards bridging the
theoretical potential of quantum computing with the practical realities of
today’s computational resources. By offering a scalable and high-performance
platform for quantum simulations, cuQuantum not only democratizes access to
quantum computing research but also propels the field towards achieving real-
world quantum computing applications.

###  III-B Pipeline of simulating QSVM algorithm

In Fig. 5(a), the depicted pipeline of a QSVM commences with the initial
quantum state preparation in a canonical basis state $|0\rangle$. The number
of qubits depends on input data features, which can be adjusted using
principal components analysis (PCA) to evaluate QSVM with varying qubit
counts. A parameterized quantum circuit (QC) follows, designed to map
classical data onto a quantum feature space via tunable gates, with the gate
parameters refined throughout the training process to facilitate effective
data classification. This step, shown in Fig. 5(b), is counterpoised by the
introduction of the adjoint of the parameterized quantum circuit (QC†), which
ensures the reversion of ancillary qubits to their initial state, maintaining
the quantum state’s coherence for measurement. Central to the QSVM’s operation
is the quantum entanglement and interference within the high-dimensional
feature space, as visualized in the intertwined loops, where the data
classification is a result of the constructive and destructive interference
patterns that are encoded in the state’s amplitude probabilities. The
measurement stage yields classical outputs indicative of the data’s class
membership probabilities, signifying the culmination of the QSVM process—from
quantum state preparation and transformation through to interference-based
processing and eventual classical information retrieval—for enhanced machine
learning efficacy. In this paper, we use a parameterized quantum circuit based
on Block-Encoded State (BPS) wavefunctions[32, 33]. This enables QSVM to
maintain high classification accuracy even with a greater number of qubits.
Worthy of note, our quantum circuit does not decompose into small blocks;
instead, we entangle each qubit by adding CNOT gates with linear topology.
Such nearest-neighbor connectivity renders the QSVM circuit compatible with
near-term quantum hardware.

Figure 5: (a) The QSVM pipeline showcasing the quantum circuit transformation
of input data into feature space quantum states. (b) A schematic of the QSVM
circuit.

###  III-C Complexity of Quantum circuit simulation for QSVM

When executed on classical hardware such as CPUs and GPUs, the simulation of
the QSVM algorithm poses significant computational challenges. Figure 6
elucidates these challenges, indicating that the complexity scales
exponentially with the number of qubits as $O(2^{n})$ and quadratically with
data size as $O(n^{2})$. Additionally, the complexity scales exponentially
with the number of qubits $q$, which encode the feature space within the
quantum circuit. This aspect underscores the inherent computational intensity
of simulating quantum systems on classical infrastructure.

Figure 6: Computational complexity of QSVM simulation. The left graph
demonstrates that simulation time scales exponentially with the number of
qubits, as $O(2^{n})$, while the right graph shows that the number of quantum
circuits required scales quadratically with data size, as $O(n^{2})$.

This scenario highlights the computational complexity advantages that QSVM
offers in the realm of quantum machine learning. The simulation demands, in
terms of computation time and memory size, grow exponentially with larger
datasets and a greater number of qubits, a limitation not encountered when
QSVM is run on quantum computers. As demonstrated by Rebentrost et al. [6],
the complexity advantage of QSVM can exhibit logarithmic scaling with respect
to the product of the number of features and the size of the training set,
denoted as $O(\log(NM))$. However, in the NISQ era, the verification of
algorithms using traditional CPUs is inevitable. Therefore, this section
focuses on leveraging GPU acceleration to address the computational
bottlenecks encountered when simulating QSVM with large-scale qubit sizes and
processing large datasets.

###  III-D Simulating QSVM’s Kernel Matrix

In this section, we will discuss three methods for simulating a QSVM algorithm
as follows. One method involves using a CPU with opt-einsum to optimize the
simulation of quantum circuits on CPU devices. The other methods utilize
cuStateVector and cuTensorNet for simulating quantum circuits on GPU devices,
with acceleration provided by the cuQuantum SDK.

####  III-D1 Conventional Simulation of QSVM using CPU/opt-einsum

Optimized Einsum (opt-einsum) significantly enhances the execution efficiency
of einsum-like expressions across various libraries by optimizing contraction
orders and employing specialized routines such as BLAS and cuBLAS. Its
compatibility with multiple backends, like NumPy, Dask, PyTorch, and
TensorFlow, makes it a versatile tool for tensor operations on CPUs.

Input : Number of data1 $datasize1$, Number of data2 $datasize2$, Circuit
einstein summation expression $exp$, List of operands $operands$, Index of
data1 and data2 combinations $indices$, Contraction path $path$

  1. 1.

Initialize $kernel\\_matrix\in\mathbb{C}^{datasize1\times datasize2}$ with all
elements set to zero.

  2. 2.

Set the current operand index $i$ to $-1$.

  3. 3.

for  _$i_{1},i_{2}\in\\{1,\ldots,indices\\}$_ do

  1. (a)

Update the operands index $i\leftarrow i+1$.

  2. (b)

Compute amplitude
$amp\leftarrow\text{opt\\_einsum.contract}(exp,operands[i],path)$.

  3. (c)

Calculate and store
$kernel\\_matrix[i_{1}-1][i_{2}-1]\leftarrow\sqrt{\text{amp.real}^{2}+\text{amp.imag}^{2}}$.

end for

* 4.

Symmetrize $kernel\\_matrix$ by adding its transpose and an identity matrix:
$kernel\\_matrix\leftarrow
kernel\\_matrix+kernel\\_matrix^{T}+\text{diag}(\mathbb{1}_{datasize1})$.

return _$kernel\\_matrix$_

Algorithm 1 Get Kernel Matrix using opt-einsum

To ensure a fair comparison between CPU and GPU performance, we utilize the
opt-einsum package, which provides optimized tensor computation on CPUs
similar to the cuQuantum SDK available for NVIDIA GPUs. The detailed algorithm
for simulating the QSVM on CPUs, aimed at equalizing the computational
environment to the extent possible, is described in Algorithm 1.

####  III-D2 Simulation of QSVM using cuStateVector

Input : Number of data1 $datasize1$, Number of data2 $datasize2$, List of
quantum circuits $circuits$, Index of data1 and data2 combinations $indices$,
statevector simulator $simulator$

  1. 1.

Initialize $kernel\\_matrix\in\mathbb{C}^{datasize1\times datasize2}$ with all
elements set to zero.

  2. 2.

Set the current operand index $i$ to $-1$.

  3. 3.

for  _$i_{1},i_{2}\in\\{1,\ldots,indices\\}$_ do

  1. (a)

Update the circuits index $i\leftarrow i+1$.

  2. (b)

Save circuits[i] statevector.

  3. (c)

Set transpile(circuits[i], simulator).

  4. (d)

Run simulator and save result $result$.

  5. (e)

Compute amplitude $amp\leftarrow\text{result.get\\_statevector()}$.

  6. (f)

Calculate and store
$kernel\\_matrix[i_{1}-1][i_{2}-1]\leftarrow(\sqrt{\text{amp.real}^{2}+\text{amp.imag}^{2}})$.

end for

* 4.

Symmetrize $kernel\\_matrix$ by adding its transpose and an identity matrix:
$kernel\\_matrix\leftarrow
kernel\\_matrix+kernel\\_matrix^{T}+\text{diag}(\mathbb{1}_{datasize1})$.

return _$kernel\\_matrix$_

Algorithm 2 Get Kernel Matrix using statevector simulator

Furthermore, we implement the simulation using the NVIDIA cuQuantum framework,
replacing Algorithm 1 with calls to the cuStateVec library to accelerate state
vector simulation on GPUs, described in Algorithm 2. The advantage of using
cuStateVec includes a speedup of the simulation time by leveraging GPU
capabilities and enabling multi-GPU processing with MPI for distributed
computing. The effectiveness of cuStateVec in enhancing quantum-circuit-
simulation efficiency is evidenced in Lykov et al.’s research work using
cuStateVec and the cuQuantum SDK[22].

Input : Number of data1 $datasize1$, Number of data2 $datasize2$, Circuit
einstein summation expression $exp$, List of operands $operands$, Index of
data1 and data2 combinations $indices$, network options $options$

  1. 1.

Initialize $kernel\\_matrix\in\mathbb{C}^{datasize1\times datasize2}$ with all
elements set to zero.

  2. 2.

Set the current operand index $i$ to $-1$.

  3. 3.

Initialize the network with given $options$ to prepare for contraction
operations.

  4. 4.

for  _$i_{1},i_{2}\in\\{1,\ldots,indices\\}$_ do

  1. (a)

Update the operand index $i\leftarrow i+1$.

  2. (b)

Reset the network to its initial state before each contraction.

  3. (c)

Prepare the operands for contraction based on $i$.

  4. (d)

Compute amplitude $amp\leftarrow\text{Contract within the
network}(exp,operands[i],options)$.

  5. (e)

Calculate and store
$kernel\\_matrix[i_{1}-1][i_{2}-1]\leftarrow\sqrt{\text{amp.real}^{2}+\text{amp.imag}^{2}}$.

end for

* 5.

Symmetrize $kernel\\_matrix$ by adding its transpose and an identity matrix:
$kernel\\_matrix\leftarrow
kernel\\_matrix+kernel\\_matrix^{T}+\text{diag}(\mathbb{1}_{datasize1})$.

return _$kernel\\_matrix$_

Algorithm 3 Get Kernel Matrix using cuTensorNet with Network Context Figure 7:
Comparative visualization of quantum kernel matrices and their computation
speedups. (a), (b), and (c) illustrate the quantum kernel matrices generated
from state vector simulation, tensor network simulation, and tensor network
simulation with path reuse strategies, respectively. (d) and (e) feature the
parity plots for quantum kernel assessments comparing the outputs of state
vector simulations with tensor network and tensor network with path reuse
algorithms, demonstrating high concordance. (f) quantifies the performance
enhancement attributable to path reuse in tensor network simulations,
showcasing significant temporal reductions across an array of dataset sizes.

####  III-D3 Simulation of QSVM using cuTensorNet

However, even with opt-einsum facilitating CPU acceleration and cuStateVector
enabling GPU acceleration, challenges persist due to the complexity of
encoding the number of qubits $O(2^{n})$ and the size of the data $O(n^{2})$.
To surmount these challenges, we present an innovative approach using the
cuTensorNet library for QSVM simulation. In the creation of the tensor network
representation, we seamlessly integrate Qiskit and cuQuantum’s built-in
CircuitToEinsum object. Initially, Qiskit is used to construct a QuantumKernel
circuit, which is then transformed into ’expression’ and ’operand’ components
by CircuitToEinsum. Due to the identical topological structure of the quantum
circuit, the same ’expression’ component can be reused for subsequent pairs of
data. Meanwhile, the ’operand’ is updated with parameters from the previously
created operand. This approach rapidly transitions data pairs into tensor
networks and preserves computational efficiency. The derivation of the kernel
matrix—a pivotal component of the SVM—exploits a consistent ’path’ to greatly
minimize the repetition of contraction order calculations. The detailed
algorithm is described in Algorithm 3. This technique not only leverages the
computational strength of GPUs but also ensures path reusability, resulting in
a considerable acceleration of the simulation process and a dramatic reduction
in computational complexity. We will demonstrate those improvements in the
next section.

##  IV Performance and Benchmarking of QSVM with cuTensorNet

###  IV-A QSVM Simulation and cuTensorNet-Accelerated QSVM (cuTN-QSVM)

In the outlined simulation workflow, Fig. 1 and 5 illustrate the sequence from
the initial input of data to the generation of a quantum circuit for the
purpose of encoding. Subsequent steps involve the use of optimized compilation
to compute and simulate the quantum circuits, leading to the extraction of a
quantum kernel matrix. This matrix is then applied to develop a support vector
classifier (SVC).

However, in typical CPU-based workflows, bottlenecks arise in the progression
from the construction of quantum circuits to the calculation of the quantum
kernel matrix, where the complexity of simulating the QSVM algorithm scales
exponentially with the number of qubits, $O(2^{n})$, and quadratically with
data size, $O(n^{2})$. To alleviate these bottlenecks, we incorporate the
cuQuantum SDK into QSVM workflow, employing a method of assigned parameters
for the formulation of QSVM’s quantum circuits. We then maintain a consistent
’expression’ for the simulation of these circuits. Ultimately, we apply a
’path reuse’ strategy within the tensor network to compute the quantum kernel
matrix. This suite of strategies significantly mitigates the computational
complexity associated with processing large datasets, reducing it from
$O(n^{2})$ to $O(1)$ for pathfinding and substantially enhancing throughput
over traditional CPU computations. Importantly, as depicted in Fig. 7, the
expressions and paths used in the cuTensorNet during the QSVM simulation
process remain unchanged compared to those in CPU and cuStateVector, ensuring
that no accuracy is compromised for the sake of expedience. In addition to the
path reuse strategy, cuTensorNet offers concurrent execution for tensor
network contraction. This technique allows the continued contractions on the
GPU asynchronously when tensors are already on the device, thus enhancing
computational efficiency by continuing operations without delay. The
pronounced speedup achieved through the implementation of path reuse within
the cuTensorNet framework is detailed in Fig.7(g), where we report a fiftyfold
increase in speed compared to conditions without path reuse.

Figure 8: Workflow optimization for QSVM simulation through architectural
enhancements, integrating Qiskit/Cirq with cuQuantum SDK. This transition from
circuit building to tensor network conversion and kernel matrix computation
reduces computational time complexity, leveraging GPU acceleration and multi-
GPU strategies.

In the comprehensive workflow outlined in Fig. 8, the input data initiates the
construction of a quantum circuit, seamlessly integrated with Qiskit/Cirq. The
process advances by converting quantum circuits into tensor networks
represented as CuPy arrays, enabling the utilization of in-place operations to
efficiently update content for the same operands. Key to enhancing
computational efficiency within this framework is the strategic deployment of
direct conversion from data to operand, alongside expression reuse for
optimizing computational pathways. This step is crucial in minimizing
redundancy and ensuring the streamlined execution of the workflow. As the
process proceeds, CuPy’s capabilities are harnessed to accelerate the
computation of the kernel matrix, culminating in the application of the SVC.
Moreover, cuTensorNet, as part of the cuQuantum SDK, incorporates advanced
strategies such as path reuse and non-blocking operations across multi-GPU
configurations.

These approaches significantly reduce the computational overhead from a
conventional complexity of $O(2^{n})$ to a more scalable $O(n^{2})$, thereby
enhancing the practicality of executing extensive QSVM simulations with
improved processing times and efficiency in resource utilization. Fig. 9
illustrates that quantum simulation on the NVIDIA A100 GPU using cuStateVector
becomes practically infeasible for more than 50 qubits. However, by employing
cuTensorNet, single-contraction simulations can be completed within 0.2
seconds, even with up to 784 qubits. Additionally, Fig. 9 shows that the path
reuse strategy can further enhance the speed, offering more than tenfold
acceleration when increasing the number of qubits in the QSVM algorithm.

In the GPU-accelerated workflow utilizing cuTensorNet, as delineated in Fig.
8, we are able to expand the feature size (number of qubits) and scale up the
data volume for our QSVM algorithm. The evaluation of accuracy resulting from
these augmentations will be discussed in the following part, while an in-depth
assessment of resource management will be presented in the subsequent section.

Figure 9: Simulation time comparison for quantum statevector simulation and
two cuTensorNet approaches—path plus contraction (no-path reuse), and
contraction only (path reuse)—across a range of qubit numbers, up to the
equivalent of a 28x28 pixel grid (784 qubits).

###  IV-B Accuracy Benchmarking

Figure 10: Train and test accuracy of quantum and classical models as a
function of the number of qubits. The plot compares quantum model performance
with varying training set sizes (50 and 100 data points) against the optimal
classical model.

In our study’s accuracy benchmarking, we observed that the trained QSVM
demonstrates an improvement in accuracy with increasing dataset sizes shown in
Fig. 10. Additionally, there tends to be a positive correlation between the
number of qubits and accuracy; a larger quantity of qubits typically yields
higher accuracy. In the specific case under consideration, the optimized
configuration entailed an algorithm using 128 qubits on a sample set of 100.
This experiment also indicates that the accuracy sharply declines beyond
roughly 200 qubits, attributed to the off-diagonal kernel matrix elements
approaching zero. Under optimized parameters, the QSVM achieved test and
training accuracies of 94% and 100%, respectively, outperforming the
traditional SVM, which achieved accuracies of 90% in testing and 95% in
training with a comparable amount of data. These empirical results underscore
the enhanced capability of QSVM over traditional SVMs as data volume
increases, especially in big data analytics applications, a finding that is
corroborated by the work of Chen et al. [10].

Furthermore, in Fig. 11, we extend our evaluation to larger data sizes,
focusing on the MNIST dataset, particularly on digits 0 and 9, which are
inherently challenging to distinguish. Our analysis reveals that with a
limited training dataset size (i.e., 10 instances), the accuracy hovers near
50%. However, as the volume of training data surpasses 100 instances, the
accuracy significantly increases to approximately 95%, surpassing the
performance of the optimal classical SVM.

Figure 11: The relationship between dataset size and accuracy for a 128-qubit
QSVM algorithm, compared to the optimal classical model.

###  IV-C Simulation with Single CPU and GPU

In this section, we compare the performance of a CPU and a GPU, as illustrated
in Fig.12. To ensure a fair comparison, we employed Opt-Einsum for the
contraction process on a single AMD EPYC 7J13 CPU, contrasting this with a
single NVIDIA A100 GPU using cuTensorNet for the contraction process, with
path reuse implemented. The detailed pseudocodes are discussed in Section
III-D. Moreover, it was necessary to synchronize the contraction paths in Opt-
Einsum with those of cuTensorNet to ensure consistency. As depicted in Fig.12,
the speedup provided by the GPU relative to the CPU becomes more pronounced as
the number of simulated qubits increases. Consequently, for large-scale qubit
simulations, GPUs demonstrate enhanced scalability and promise substantial
benefits for future advanced qubit algorithms in simulation and emulation.

Figure 12: Benchmark QSVM circuit simulation time using a single CPU and a
single GPU.

##  V Distributed simulation and Resource Estimation in HPC

In the final section of our study, a multi-GPU instance was utilized to expand
the QSVM model to accommodate a dataset comprising 1,000 data points of 28x28
MNIST images. The implementation of multi-GPU resources to enhance quantum
circuit simulation via cuStateVector is thoroughly detailed in the research
conducted by Shaydulin et al. [34]. Our emphasis lies on leveraging the data
from these experiments to rigorously assess both the computational costs and
the temporal demands inherent in simulating the QSVM algorithm within a multi-
GPU processing framework.

In our computational environment, each GPU within a node is interconnected
using the high-bandwidth NVLink network, optimizing intra-node communication.
Inter-node data transfer mandates that information from the GPU be first
relayed to the CPU, then onward to the target node, a process that requires
careful selection of a communication protocol sensitive to the GPU’s
positional context. The MPI offers built-in capabilities for such operations,
which are enabled by setting the $MPI\\_GPU\\_SUPPORT\\_ENABLED$ environment
variable. By harnessing these integrated technological benefits, we have
successfully actualized the accelerated computational outcomes for managing
large-scale qubit systems and extensive datasets, as illustrated in Fig.13.
Comparative analysis indicates that our performance metrics are on par with
distributed simulation results documented in the existing scientific corpus,
as cited in Bayraktar et. al.’s and Lykov et. al.’s work[23, 22].

###  V-A Benchmarking cuQuantum Multi-GPU with MPI

Fig. 13 illustrates the execution time required for quantum simulations in
relation to the number of qubits. The data compares the performance of a
single A100 GPU to systems utilizing 2, 4 and 8 GPUs in conjunction with MPI
and within a single NVIDIA DGX node. It is evident from the results that the
incorporation of multi-GPUs significantly decreases computation time,
highlighting the strong linear speedup of cuTenserNet with MPI. The trend
indicates a substantial reduction in execution time as the number of GPUs is
increased, affirming the efficacy of multi-GPU setups in handling large
datasets.

Figure 13: Strong scaling of the QSVM simulation is observed for 1,000 data
points across 1, 2, 4, and 8 GPUs, demonstrating linear speedup.

###  V-B Large Dataset Processing with Multi-GPU instance

Figure 14 presents a comparative analysis of computational time across
different configurations, ranging from a single GPU (A100, 80GB) to 2, 4, and
8 multi-GPU arrangements using MPI for processing datasets of various sizes.
The results distinctly highlight the superior efficiency and scalability of
multi-GPU systems, especially when managing large-scale datasets. A notable
reduction in processing time is observed with the integration of an 8-GPU
setup, underscoring the considerable advantages of parallel computing for
large-scale data analysis. In Figure 14, experimental data (solid line) from
40 to 1,000 data points is extrapolated to estimate the processing time for
10,000 data points, corresponding to nearly 50 million circuits (dashed line).
The projection indicates that an eight-GPU system could achieve linear
acceleration, reducing a week-long processing task using the simulated QSVM to
approximately one day (blue line).

Figure 14: Execution time for quantum simulations against qubit count for a
single A100 GPU and MPI-based 2, 4 and 8 multi-GPU setups. The performance
enhancement with additional GPUs is evident, underscoring the benefits of
parallelized computation.

##  VI Conclusion

This paper has presented a comprehensive study on the application and efficacy
of cuTN-QSVMs in conjunction with NVIDIA’s cuQuantum SDK. Our findings have
consistently shown that by integrating cuTensorNet within our computational
workflow, we have been able to significantly decrease the computational
complexity involved in simulating QSVM algorithms, especially when working
with large qubit counts. Through rigorous performance benchmarking, our
approach has demonstrated not only enhanced simulation speeds and efficiency
but also scalability across multi-GPU platforms. Additionally, we have
observed substantial improvements in accuracy with increasing data set sizes,
underlining the potential of quantum computing methods in tackling complex,
large-scale data challenges. The integration of cuTensorNet and MPI-based
multi-GPU systems reveals a path forward for quantum computing that offers
practical advantages for both current research and future applications,
bridging the gap between quantum hardware and classical computing resources.
This points towards a promising direction for high-performance quantum
simulations within the Quantum-HPC ecosystem. Furthermore, we utilized cuTN-
QSVM to classify bone cell images captured by synchrotron transmission X-ray
microscopy (TXM) at the BL01B1 beamline of the Taiwan Light Source (TLS) in
Hsinchu, Taiwan. Bone cell image data possess a high-dimensional feature space
but limited data quantity. Traditional machine learning struggles with
classifying such data, but QSVM overcomes this, showcasing its advantages. Due
to page limit, the classification results of bone cells will be elaborated in
the conference.

## Acknowledgment

The authors extend their gratitude to W.C. Qian for invaluable assistance and
insights, which have been pivotal to the success of this research work. K.C.
is grateful for the financial support from both the Turing Scheme for the
Imperial Global Fellows Fund and the Taiwanese Government Scholarship to Study
Abroad. This work was supported by the National Science and Technology Council
(NSTC), Taiwan, under Grant NSTC 112-2119-M-007-008- and NSTC
113-2119-M-007-013-. The authors would like to thank the National Center for
High-performance Computing of Taiwan for providing computational and storage
resources. Additionally, the authors would like to extend special thanks to
the technical support provided by the NVAITC and NVIDIA Quantum team.

## Code Visability

The source code for generating the dataset and figures presented in this
research work is openly accessible at https://github.com/Tim-Li/cuTN-QSVM.git.

## References

  * [1] M. I. Jordan and T. M. Mitchell, Machine learning: Trends, perspectives, and prospects, Science 349 (2015), pp. 255–260.
  * [2] D. J. MacKay, Information theory, inference and learning algorithms, Cambridge university press, 2003.
  * [3] M. A. Hearst, S. T. Dumais, E. Osuna, J. Platt and B. Scholkopf, Support vector machines, IEEE Intelligent Systems and their applications 13 (1998), pp. 18–28.
  * [4] B. Ghaddar and J. Naoum-Sawaya, High dimensional data classification and feature selection using support vector machines, European Journal of Operational Research 265 (2018), pp. 993–1004.
  * [5] B. Gaye, D. Zhang and A. Wulamu, Improvement of support vector machine algorithm in big data background, Mathematical Problems in Engineering 2021 (2021), pp. 1–9.
  * [6] P. Rebentrost, M. Mohseni and S. Lloyd, Quantum support vector machine for big data classification, Physical review letters 113 (2014), p. 130503.
  * [7] X. Wang, Y. Du, Y. Luo and D. Tao, Towards understanding the power of quantum kernels in the nisq era, Quantum 5 (2021), p. 531.
  * [8] Z. Li, X. Liu, N. Xu and J. Du, Experimental realization of a quantum support vector machine, Physical review letters 114 (2015), p. 140504.
  * [9] C. Ding, T.-Y. Bao and H.-L. Huang, Quantum-inspired support vector machine, IEEE Transactions on Neural Networks and Learning Systems 33 (2021), pp. 7210–7222.
  * [10] K.-C. Chen, X. Xu, H. Makhanov, H.-H. Chung and C.-Y. Liu, Quantum-enhanced support vector machine for large-scale stellar classification with gpu acceleration, arXiv preprint arXiv:2311.12328 (2023).
  * [11] J. Preskill, Quantum computing in the nisq era and beyond, Quantum 2 (2018), p. 79.
  * [12] M. Kjaergaard et al., Superconducting qubits: Current state of play, Annual Review of Condensed Matter Physics 11 (2020), pp. 369–395.
  * [13] C. D. Bruzewicz, J. Chiaverini, R. McConnell and J. M. Sage, Trapped-ion quantum computing: Progress and challenges, Applied Physics Reviews 6 (2019).
  * [14] S. J. Evered et al., High-fidelity parallel entangling gates on a neutral-atom quantum computer, Nature 622 (2023), pp. 268–272.
  * [15] Z. Cai et al., Quantum error mitigation, Reviews of Modern Physics 95 (2023), p. 045005.
  * [16] K.-C. Chen, Short-depth circuits and error mitigation for large-scale ghz-state preparation, and benchmarking on ibm’s 127-qubit system, in 2023 IEEE International Conference on Quantum Computing and Engineering (QCE), vol. 2, IEEE, 2023, pp. 207–210.
  * [17] A. M. Souza, G. A. Alvarez and D. Suter, Robust dynamical decoupling for quantum computing and quantum memory, Physical review letters 106 (2011), p. 240501.
  * [18] M. Prest and K.-C. Chen, Quantum-error-mitigated detectable byzantine agreement with dynamical decoupling for distributed quantum computing, arXiv preprint arXiv:2311.03097 (2023).
  * [19] J. W. Z. Lau, K. H. Lim, H. Shrotriya and L. C. Kwek, Nisq computing: where are we and where do we go?, AAPPS bulletin 32 (2022), p. 27.
  * [20] S. Chen, J. Cotler, H.-Y. Huang and J. Li, The complexity of nisq, Nature Communications 14 (2023), p. 6001.
  * [21] K.-C. Chen, X. Li, X. Xu, Y.-Y. Wang and C.-Y. Liu, Quantum-hpc framework with multi-gpu-enabled hybrid quantum-classical workflow: Applications in quantum simulations, arXiv preprint arXiv:2403.05828 (2024).
  * [22] D. Lykov, R. Shaydulin, Y. Sun, Y. Alexeev and M. Pistoia, Fast simulation of high-depth qaoa circuits, in Proceedings of the SC’23 Workshops of The International Conference on High Performance Computing, Network, Storage, and Analysis, 2023, pp. 1443–1451.
  * [23] H. Bayraktar et al., cuquantum sdk: A high-performance library for accelerating quantum science, in 2023 IEEE International Conference on Quantum Computing and Engineering (QCE), vol. 1, IEEE, 2023, pp. 1050–1061.
  * [24] D. Fortunato, J. CAMPOS and R. ABREU, Mutation testing of quantum programs: A case study with qiskit, IEEE Transactions on Quantum Engineering 3 (2022), pp. 1–17.
  * [25] X. Vasques, H. Paik and L. Cif, Application of quantum machine learning using quantum kernel algorithms on multiclass neuron m-type classification, Scientific Reports 13 (2023), p. 11541.
  * [26] C.-H. H. Yang et al., A quantum kernel learning approach to acoustic modeling for spoken command recognition, in ICASSP 2023-2023 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), IEEE, 2023, pp. 1–5.
  * [27] S. L. Wu et al., Application of quantum machine learning using the quantum kernel algorithm on high energy physics analysis at the lhc, Physical Review Research 3 (2021), p. 033221.
  * [28] J.-S. Kim, A. McCaskey, B. Heim, M. Modani, S. Stanwyck and T. Costa, Cuda quantum: The platform for integrated quantum-classical computing, in 2023 60th ACM/IEEE Design Automation Conference (DAC), IEEE, 2023, pp. 1–4.
  * [29] R. Wille, R. Van Meter and Y. Naveh, Ibm’s qiskit tool chain: Working with and developing for real quantum computers, in 2019 Design, Automation & Test in Europe Conference & Exhibition (DATE), IEEE, 2019, pp. 1234–1240.
  * [30] V. Bergholm et al., Pennylane: Automatic differentiation of hybrid quantum-classical computations, arXiv preprint arXiv:1811.04968 (2018).
  * [31] S. V. Isakov et al., Simulations of quantum circuits with approximate noise using qsim and cirq, arXiv preprint arXiv:2111.02396 (2021).
  * [32] J. Martyn, G. Vidal, C. Roberts and S. Leichenauer, Entanglement and tensor networks for supervised image classification, arXiv preprint arXiv:2007.06082 (2020).
  * [33] T. Suzuki, T. Miyazaki, T. Inaritai and T. Otsuka, Quantum ai simulator using a hybrid cpu–fpga approach, Scientific Reports 13 (2023), p. 7735.
  * [34] R. Shaydulin et al., Evidence of scaling advantage for the quantum approximate optimization algorithm on a classically intractable problem, arXiv preprint arXiv:2308.02342 (2023).


# Focus the electromagnetic field to $10^{-6}\lambda$ for ultra-high
enhancement of field-matter interaction

Xiang-Dong Chen  En-Hui Wang  Long-Kun Shan  Ce Feng  Yu Zheng  Yang Dong
Guang-Can Guo  Fang-Wen Sun<EMAIL_ADDRESS>CAS Key Laboratory of Quantum
Information, School of Physical Sciences, University of Science and Technology
of China, Hefei, 230026, People’s Republic of China  CAS Center For Excellence
in Quantum Information and Quantum Physics, University of Science and
Technology of China, Hefei, 230026, People’s Republic of China

###### Abstract

Focusing electromagnetic field to enhance the interaction with matter has been
promoting researches and applications of nano electronics and photonics.
Usually, the evanescent-wave coupling is adopted in various nano structures
and materials to confine the electromagnetic field into a subwavelength space.
Here, based on the direct coupling with confined electron oscillations in a
nanowire, we demonstrate an extreme localization of microwave field down to
10${}^{-6}\lambda$. A hybrid nanowire-bowtie antenna is further designed to
focus the free-space microwave to this deep-subwavelength space. Detected by
the nitrogen vacancy center in diamond, the field intensity and microwave-spin
interaction strength are enhanced by 2.0$\times$108 and 1.4$\times$104 times,
respectively. Such an extreme concentration of microwave field will further
promote integrated quantum information processing, sensing and microwave
photonics in a nanoscale system.

Electromagnetic field can usually be focused at the scale of its wavelength.
However, in pursuit of a strong interaction with matter, the manipulation of
the electromagnetic field in a subwavelength space is one of the most
important tasks in nanoscience researches and applications, ranging from
integrated optics to biological sensingMarpaung  _et al._ (2019); Saha  _et
al._ (2012); Cheben  _et al._ (2018); Kauranen and Zayats (2012).
Nanostructures of dielectricHu  _et al._ (2018); Choi  _et al._ (2017); Fröch
_et al._ (2020), metallicKim  _et al._ (2015); Chikkaraddy  _et al._ (2016);
Chen  _et al._ (2018) and two-dimensional materialsAlcaraz Iranzo  _et al._
(2018); Xia  _et al._ (2014) have been developed to tightly confine the
electromagnetic field mainly based on the evanescent-wave coupling. For
example, the plasmonic nanostructure has been used for the light field
confinement at a scale smaller than $10^{-2}\lambda$Kim  _et al._ (2015);
Chikkaraddy  _et al._ (2016); Chen  _et al._ (2018). These confinements can
dramatically reduce the mode volume to highly increase the density of states
and enhance the light-matter interaction at the nanoscale, which has harnessed
the researches of single molecule spectroscopyZhang  _et al._ (2013), nano
laserAzzam  _et al._ (2020), nonlinear opticsSmirnova and Kivshar (2016), and
solar energyAtwater and Polman (2010).

Especially, the interaction between microwave field and matter at the
nanoscale strongly drives the development of quantum information processing,
sensing and microwave photonics. The deep-subwavelength confinement of the
microwave field will benefit the individual manipulation of multi-qubitWarring
_et al._ (2013); Harty  _et al._ (2016). Meanwhile, the enhancement of local
microwave field is of central importance to microwave-to-optics
conversionSalamin  _et al._ (2018); Wang  _et al._ (2018), fast spin qubit
manipulationFuchs  _et al._ (2009) and hybrid quantum system couplingClerk
_et al._ (2020); Kubo  _et al._ (2010). It indicates that, the efficient
localization and detection of microwave field at the nanoscale is highly
required for develop a practical quantum information device. Further more, the
wireless qubit manipulation with a compact and scalable system will decrease
the power consumption and reduce the heat load in a cryostatBlais  _et al._
(2020); Pauka  _et al._ (2021).

Here, we study the field confinement based on the direct coupling between
electromagnetic field and confined electrons in a low dimensional nano
material. An extreme-confinement of a microwave field with an ultra-strong
intensity is realized by utilizing the near field radiation of the electron
oscillation in an Ag nanowire. Using the NV center in diamond as a noninvasive
probe, we show that the microwave field can be localized down to 291 nm,
corresponding to a scale of $10^{-6}\lambda$. For far-field spin manipulation,
we design a hybrid nanowire-bowtie structure to focus the microwave field
directly from the free space to a deep-subwavelength volume. As a result, the
microwave-spin interaction strength is highly enhanced by observing a
1.4$\times$104 times enhancement of the Rabi oscillation frequency,
corresponding to increasing the field intensity by 2.0$\times$108 times.
Further considering the light guiding effect of the Ag nanowire, this antenna
can be used for delivering and concentrating both light and microwave field.
Subsequently, a wireless platform can be developed for the integrated quantum
information processing and quantum sensing.

Figure 1: The principle of localizing and detecting the microwave field. (a)
Sketch of the nanowire-bowtie antenna. A single crystal diamond plate is
placed under the nanowire. The spin state transition of NV center in diamond
is pumped by the localized microwave. (b) The image of the microwave
distribution is obtained by recording the spin state transition of NV center.
The insert is the integrated cross-section profile. The solid line is the fit
with equation (1). The power of the microwave that is radiated by the horn
antenna is 14 $\mu$W.

##  I Results

###  I.1 The design of experiments

As shown in Fig. 1(a), the nanowire-bowtie hybrid antenna consists of an Ag
nanowire with a diameter of 120 nm and a metallic bowtie structure. The gap
between the two arms of the bowtie structure is $W_{gap}=$ 8 $\mu$m. The
length of the bowtie structure is 6.5 cm, while the widths at the end and at
the gap are 1 cm and 160 $\mu$m, respectively (details in Supplementary
information). A double-ridged horn antenna radiates the microwave signal into
the free space. The nanowire-bowtie structure then receives the far-field
microwave (with a distance of approximate 20 cm).

The NV center in a single crystal diamond plate is generated by nitrogen ion
implanting. The depth is approximate 20 nm. The ground state of NV center is a
spin triplet. The transition between the $m_{s}=0$ and $m_{s}=\pm 1$ can be
pumped by a resonant microwave. It subsequently changes the fluorescence
intensity of NV center. To detect the microwave, we record the optically
detected magnetic resonance (ODMR) of NV center under a continuous-wave
microwave pumping. The contrast of the ODMR signal increases with the
amplitude of the microwave field (Supplementary information). To non-
invasively map the localized microwave with a high spatial resolution, the
charge state depletion (CSD) nanoscopyChen  _et al._ (2015) is applied for the
diffraction-unlimited ODMR measurement. It is based on the charge state
manipulation and detection of NV center. The resolution of CSD nanoscopy is
approximate 100 nm here, in comparison with the 500 nm resolution of the
confocal microscopy (Supplementary information).

###  I.2 Microwave localization and detection

In Fig. 1 (b), we show the magnetic component of the microwave field near the
Ag nanowire. Here, without an external magnetic field, the resonant microwave
frequency for NV center spin transition is 2.87 GHz. The result shows that the
near-field microwave is confined near the Ag nanowire. The width of the cross-
section profile is 291$\pm 10$ nm, corresponding to 2.8$\times
10^{-6}\lambda$, where $\lambda$ is the microwave wavelength in vacuum. The
distribution of the magnetic component can be well fitted by a reciprocal
function:

$|B_{MW}|\propto\frac{1}{\sqrt{r^{2}+r_{0}^{2}}},$ (1)

where $r$ is the distance from the nanowire in the xy plane and
$r_{0}=$84$\pm$3 nm, which is determined by the radius of the nanowire and the
depth of NV center. In contrast, the evanescent-field coupling shows an
exponential decay as a function of distanceKauranen and Zayats (2012).

Figure 2: Detect the microwave field vector at the nanoscale. (a) The
illustration of the direction of nanowire and NV center axes. (b) Frequency-
scanning ODMR results of NV centers with four symmetry axes. (c) The
fluorescence intensity of NV centers is enhanced due to the interaction with
the Ag nanowire. The cross-section profile in the bottom is used to locate the
relative position of nanowire (red points in the image). (d) The image of
$|B_{MW2}|$. The cross-section profile is the integrated signal of the whole
image. The solid line is the simulation with a straight line current by
setting the x axis position of the nanowire to 0. (e)(f) The microwave vector
is revealed by comparing different projections. The scale bars in all the
images are 400 nm in length. And the power of the microwave source is set to
174 $\mu$W. Figure 3: Spin manipulation with the localized microwave field.
The wide field imaging illustrates the microwave field enhancement with
different structures: (a) nanowire-bowtie antenna; (b) bowtie antenna without
nanowire; (c) no structure on diamond surface. The scale bars are 20 $\mu$m in
length. $B_{sat}$ is the saturation microwave amplitude defined in equation
(S2) in Supplementary information. (d)-(f) are Rabi oscillations of NV center
with different structures in (a)-(c), respectively. The inset in (d) shows
pulse sequences for the Rabi oscillation measurement. The Rabi oscillation in
(d) is measured with NV center ensemble under the nanowire. And Rabi
oscillations in (e) and (f) are measured with a single NV center. $P_{MW}$ is
the power of the microwave source.

To reveal the mechanism of the microwave confinement, we separate the ODMR
signal from the four categories of NV centers with different symmetry axes and
obtain the vector information of the localized microwave field, as shown in
Fig. 2(a). Here, the x, y, and z axes are defined as the edges of the diamond
plate. The symmetry axes of the four categories of NV centers are shown as NV1
(-$\sqrt{2}$, 0, 1), NV2 ($\sqrt{2}$, 0, 1), NV3 (0, $\sqrt{2}$, 1), NV4 (0,
-$\sqrt{2}$, 1). An external static magnetic field $\mathbf{B_{0}}$ is applied
to split the four categories (NVi), which is shown in Fig. 2(b). We assume
that the localized microwave field distribution with different frequencies is
the same. By recording the ODMR signal of NVi centers at different positions,
we obtain the distribution of the magnetic component $B_{MWi}$, which is
perpendicular to the NVi centers’ symmetry axis.

To precisely map the microwave vector distribution, the position of the
nanowire is firstly located according to the fluorescence enhancement of NV
center, as shown in Fig. 2(c). We find that the position of the magnetic
component’s maximum does not always match the position of the nanowire. As
shown in Fig. 2(d), the maximum of $B_{MW2}$ is approximate 100 nm away from
the nanowire. In Figs. 2(e) and 2(f), by simultaneously detecting the signals
from different NV categories, we further highlight the difference of the
distributions with $|B_{MW2}|/|B_{MW1}|$ and $|B_{MW4}|/|B_{MW3}|$,
respectively. With the structure in Fig. 2, the amplitude of $B_{MW3}$ and
$B_{MW4}$ are almost same, while the amplitude of $B_{MW1}$ is mirroring
$B_{MW2}$ with respect to the yz plane. The results show that the vector of
the magnetic component is in the xz plane. Comparing with the simulation
(Supplementary information), it suggests that the localized microwave field is
mainly determined by a straight line current that is transmitting through the
Ag nanowire. Such a field confinement is not from the evanescent-wave
coupling. The small mismatch between the simulation with a straight line
current and the experiments might be caused by the error of nanowire’s
location, the distribution of NV center in z axis, and the accuracy of CSD
microscopy resolution’s estimation.

###  I.3 Far-field spin manipulation

The extreme confinement leads to a significant enhancement of the localized
microwave field’s intensity. It subsequently enhances the interaction with a
spin qubit. In Fig. 3, we compares the localized microwave field with three
different structures: the nanowire-bowtie hybrid antenna, the bowtie antenna
without Ag nanowire, and no antenna. The results show that, with a nanowire-
bowtie antenna, the microwave is significantly enhanced near the Ag nanowire,
while a bowtie antenna without Ag nanowire slightly increases the localized
microwave field in the gap between the two arms.

Figure 4: The localized microwave field distribution is changed with the
polarization of the free-space microwave. (a)(b) The electrical polarization
of the free-space microwave is parallel to the direction of nanowire-bowtie
structure. (c)(d) The electrical polarization is perpendicular to the
nanowire-bowtie structure. The scale bars are 20 $\mu$m in length. $E_{free}$
denotes the electrical component of the free-space microwave. $P_{MW}$ and
$B_{sat}$ are defined as same as that in Fig. 3.

The extreme microwave field enhancement can be used for fast and high spatial
resolution spin qubit manipulation. Here, we use it to pump the Rabi
oscillation of NV center. As shown in Fig. 3(d), with a nanowire-bowtie
antenna, the Rabi frequency of the NV center under the Ag nanowire is
approximate 1.6 $\mu$s-1 with a 14 $\mu$W microwave excitation. Without Ag
nanowire, the Rabi oscillation frequency of the NV center in the gap of the
bowtie antenna is approximate 0.89 $\mu$s-1 with a 21 W microwave excitation.
In contrast, without any nanostructure, the Rabi oscillation frequency is only
0.14 $\mu$s-1 under a 21 W microwave excitation. The results indicate that, by
utilizing the nanowire-bowtie antenna for spin manipulation, the Rabi
frequency can be improved by at least 1.4$\times 10^{4}$ times, corresponding
to increasing the local microwave intensity by 2.0$\times 10^{8}$ times. The
observation of Rabi oscillation also indicates that the coherence of both the
spin and the localized microwave is preserved, which is crucial for quantum
applications.

The individual addressing of multi qubits from far field can be further
explored by utilizing the polarization dependence of the microwave
localization. Here, we rotate the horn antenna to change the polarization of
the free-space microwave. The results in Fig. 4 show that, with an electrical
polarization parallel to the bowtie-nanowire antenna, a stronger localized
microwave field is observed near the nanowire-bowtie antenna. The polarization
isolation of the localized microwave intensity with the nanowire-bowtie
antenna is higher than 20 dB, but lower than 40 dB. Therefore, encoding the
microwave for different spin manipulation into the polarization, the nanowire-
bowtie antennas with different directions can be used to selectively
manipulate the qubits at different positions from the far-field.

##  II discussion

Integrating and miniaturizing the electrical and optical device is essential
for the practical quantum processing and sensing applicationsKim  _et al._
(2019); Mehta  _et al._ (2020). Various electrical conductive and
ferromagnetic structures have been studied to transmit the microwave signal
for spin manipulation at the nanoscaleStaacke  _et al._ (2019); Bertelli  _et
al._ (2020). The enhancement of the nanoscale microwave field has also been
demonstrated with an in-plane slotted patch antennaSalamin  _et al._ (2018).
However, the Johnson noise of a large metal film will cause the spin
relaxation Kolkowitz  _et al._ (2015); Ariyaratne  _et al._ (2018), while the
Johnson noise from an Ag nanowire is negligible (Supplementary information).
Our method provides a solution to efficient spin manipulation from the far-
field. It simplifies the quantum processing device, avoids the thermal leakage
in a cryostat. In addition, the light guiding effect of an Ag nanowire can be
utilized to optically pump and collect the fluorescence of individual
qubitHuck  _et al._ (2011); Kumar  _et al._ (2013); Wei  _et al._ (2018). The
simultaneous integration of electrical and optical components certainly will
help to develop a compact and integrated quantum processing device.

Focusing and detecting the nanoscale electromagnetic field can be further used
to enhance the sensitivity of the spin-based metrology. Recently, the magnetic
concentrator of ferrite material has been applied for the detection of static
magnetic field with NV centerFescenko  _et al._ (2020); Zhang  _et al._
(2021). The key is how small the electromagnetic field can be focused and
detected. Utilizing the nanowire-bowtie antenna, we can improve the
sensitivity by 1.4$\times 10^{4}$ times. Combining with the coherent spin
manipulationBarry  _et al._ (2020), it will help to realize the ultra-weak
microwave signal sensing, such as for a quantum radar.

In conclusion, we have demonstrated the extreme concentration of a microwave
field by utilizing the confined electron’s oscillation in a low dimensional
material. The results can be used for integrated quantum information
processing and high sensitivity quantum sensing.

## Methods

Sample preparation. The electrical grade diamond plates with
$\left\\{100\right\\}$ surface and $<$110$>$ edges are purchased from Element
6. The size of the plate is $2\times 2\times 0.5$ mm3. The NV center ensemble
is produced through nitrogen ion implanting with an energy of 15 keV and a
dosage of 1013/cm2. The diamond is annealed at 850∘C for 2 hours to improve
the production efficiency of NV centers. The density of NV centers is
estimated to be approximate 5000/$\mu$m2. After the NV center is produced, Ag
nanowires are dropped on the surface of diamond with a spin processor. Then, a
small metallic bowtie structure of chromium/gold (5/200 nm thickness) film is
produced on the diamond surface through lift-off. Finally, a large in-plane
bowtie antenna is made with a copper foil tap. The Au film on the diamond
plate is ohm connected to the copper tape with silver glue.

Figure 5: The schematic diagram of experimental setup for the CSD nanoscopy.
DM1-3, long-pass dichroic mirror with edge wavelengths of 658.8, 536.8 and 605
nm, respectively; AOM, acousto-optic modulator; SPCM, single-photon-counting
modulator; PBS, polarizing beam splitter.

Experimental setup. The CSD nanoscopy setup for ODMR measurement is based on a
home-built confocal microscope, as shown in Fig. 5. Diamond plate is mounted
on a piezo-stage (P-733.3DD, PI). CW lasers with wavelengths of 532
(Coherent), 589 (MGL-III-589nm, New Industries Optoelectronics) and 637 nm
(MRL-III-637nm, New Industries Optoelectronics) are modulated by acousto-optic
modulators (AOMs, MT200-0.5-VIS, AA). A vortex phase plate (VPP-1a, RPC
photonics) is used to produce a doughnut-shaped 637 nm laser beam. The lasers
pump the NV center in diamond from the backside through an objective (Leica)
with 0.7 NA. The collected fluorescence is time gated by another AOM. Then, it
is detected by a single-photon-counting-module (SPCM-AQRH-15-FC, Excelitas)
after passing through a long-pass filter (edge wavelength 668.9 nm, Semrock).
In the wide field microscope, a 532 nm CW laser (MLL-III-532nm, New Industries
Optoelectronics) is used to pump the NV center. The fluorescence is detected
by a CCD camera (iXon897, Andor).

Two microwave generators (SMB100A and SMA 100A, Rhode&Schwartz) are used to
produce microwave signal with different frequencies. The microwave pulse is
controlled by microwave switches (ZASWA-2-50DR, MiniCircuits). Then, the two
channels are combined by a combiner (ZFRSC-42-S, MiniCircuits) and amplified
by a microwave amplifier (60S1G4A, Amplifier Research). The microwave is
radiated into free space by a horn antenna (LB-2080-SF, Chengdu AINFO Inc.).

Acknowledgment

This work was supported by National Key Research and Development Program of
China (No. 2017YFA0304504); Anhui Initiative in Quantum Information
Technologies (AHY130100); National Natural Science Foundation of China (Nos.
91536219, 91850102).

## References

  * Marpaung  _et al._ (2019) David Marpaung, Jianping Yao, and José Capmany, “Integrated microwave photonics,” Nat. Photon. 13, 80–90 (2019).
  * Saha  _et al._ (2012) Krishnendu Saha, Sarit S. Agasti, Chaekyu Kim, Xiaoning Li, and Vincent M. Rotello, “Gold nanoparticles in chemical and biological sensing,” Chem. Rev. 112, 2739–2779 (2012).
  * Cheben  _et al._ (2018) Pavel Cheben, Robert Halir, Jens H Schmid, Harry A Atwater, and David R Smith, “Subwavelength integrated photonics,” Nature 560, 565–572 (2018).
  * Kauranen and Zayats (2012) Martti Kauranen and Anatoly V. Zayats, “Nonlinear plasmonics,” Nat. Photon. 6, 737–748 (2012).
  * Hu  _et al._ (2018) Shuren Hu, Marwan Khater, Rafael Salas-Montiel, Ernst Kratschmer, Sebastian Engelmann, William M. J. Green, and Sharon M. Weiss, “Experimental realization of deep-subwavelength confinement in dielectric optical resonators,” Sci. Adv. 4 (2018).
  * Choi  _et al._ (2017) Hyeongrak Choi, Mikkel Heuck, and Dirk Englund, “Self-similar nanocavity design with ultrasmall mode volume for single-photon nonlinearities,” Phys. Rev. Lett. 118, 223605 (2017).
  * Fröch  _et al._ (2020) Johannes E. Fröch, Sejeong Kim, Connor Stewart, Xiaoxue Xu, Ziqing Du, Mark Lockrey, Milos Toth, and Igor Aharonovich, “Photonic nanobeam cavities with nanopockets for efficient integration of fluorescent nanoparticles,” Nano Lett. 20, 2784–2790 (2020).
  * Kim  _et al._ (2015) Myung-Ki Kim, Hongchul Sim, Seung Ju Yoon, Su-Hyun Gong, Chi Won Ahn, Yong-Hoon Cho, and Yong-Hee Lee, “Squeezing photons into a point-like space,” Nano Lett. 15, 4102–4107 (2015).
  * Chikkaraddy  _et al._ (2016) Rohit Chikkaraddy, Bart de Nijs, Felix Benz, Steven J. Barrow, Oren A. Scherman, Edina Rosta, Angela Demetriadou, Peter Fox, Ortwin Hess, and Jeremy J. Baumberg, “Single-molecule strong coupling at room temperature in plasmonic nanocavities,” Nature 535, 127–130 (2016).
  * Chen  _et al._ (2018) Wen Chen, Shunping Zhang, Qian Deng, and Hongxing Xu, “Probing of sub-picometer vertical differential resolutions using cavity plasmons,” Nat. Commun. 9, 801 (2018).
  * Alcaraz Iranzo  _et al._ (2018) David Alcaraz Iranzo, Sébastien Nanot, Eduardo J. C. Dias, Itai Epstein, Cheng Peng, Dmitri K. Efetov, Mark B. Lundeberg, Romain Parret, Johann Osmond, Jin-Yong Hong, Jing Kong, Dirk R. Englund, Nuno M. R. Peres, and Frank H. L. Koppens, “Probing the ultimate plasmon confinement limits with a van der waals heterostructure,” Science 360, 291–295 (2018).
  * Xia  _et al._ (2014) Fengnian Xia, Han Wang, Di Xiao, Madan Dubey, and Ashwin Ramasubramaniam, “Two-dimensional material nanophotonics,” Nat. Photon. 8, 899–907 (2014).
  * Zhang  _et al._ (2013) R. Zhang, Y. Zhang, Z.C. Dong, S. Jiang, C. Zhang, L.G. Chen, L. Zhang, Y. Liao, J. Aizpurua, Y. Luo, J. L. Yang, and J. G. Hou, “Chemical mapping of a single molecule by plasmon-enhanced raman scattering,” Nature 498, 82–86 (2013).
  * Azzam  _et al._ (2020) Shaimaa I Azzam, Alexander V Kildishev, Ren-Min Ma, Cun-Zheng Ning, Rupert Oulton, Vladimir M Shalaev, Mark I Stockman, Jia-Lu Xu, and Xiang Zhang, “Ten years of spasers and plasmonic nanolasers,” Light Sci. Appl. 9, 90 (2020).
  * Smirnova and Kivshar (2016) Daria Smirnova and Yuri S. Kivshar, “Multipolar nonlinear nanophotonics,” Optica 3, 1241–1255 (2016).
  * Atwater and Polman (2010) Harry A Atwater and Albert Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9, 205–213 (2010).
  * Warring  _et al._ (2013) U. Warring, C. Ospelkaus, Y. Colombe, R. Jördens, D. Leibfried, and D. J. Wineland, “Individual-ion addressing with microwave field gradients,” Phys. Rev. Lett. 110, 173002 (2013).
  * Harty  _et al._ (2016) T. P. Harty, M. A. Sepiol, D. T. C. Allcock, C. J. Ballance, J. E. Tarlton, and D. M. Lucas, “High-fidelity trapped-ion quantum logic using near-field microwaves,” Phys. Rev. Lett. 117, 140501 (2016).
  * Salamin  _et al._ (2018) Y. Salamin, B. Baeuerle, W. Heni, F. C. Abrecht, A. Josten, Y. Fedoryshyn, C. Haffner, R. Bonjour, T. Watanabe, M. Burla, D. L. Elder, L. R. Dalton, and J. Leuthold, “Microwave plasmonic mixer in a transparent fibre-wireless link,” Nat. Photon. 12, 749–753 (2018).
  * Wang  _et al._ (2018) Cheng Wang, Mian Zhang, Xi Chen, Maxime Bertrand, Amirhassan Shams-Ansari, Sethumadhavan Chandrasekhar, Peter Winzer, and Marko Lončar, “Integrated lithium niobate electro-optic modulators operating at cmos-compatible voltages,” Nature 562, 101–104 (2018).
  * Fuchs  _et al._ (2009) G. D. Fuchs, V. V. Dobrovitski, D. M. Toyli, F. J. Heremans, and D. D. Awschalom, “Gigahertz dynamics of a strongly driven single quantum spin,” Science 326, 1520–1522 (2009).
  * Clerk  _et al._ (2020) A. A. Clerk, K. W. Lehnert, P. Bertet, J. R. Petta, and Y. Nakamura, “Hybrid quantum systems with circuit quantum electrodynamics,” Nat. Phys. 16, 257–267 (2020).
  * Kubo  _et al._ (2010) Y. Kubo, F. R. Ong, P. Bertet, D. Vion, V. Jacques, D. Zheng, A. Dréau, J. F. Roch, A. Auffeves, F. Jelezko, J. Wrachtrup, M. F. Barthe, P. Bergonzo, and D. Esteve, “Strong coupling of a spin ensemble to a superconducting resonator,” Phys. Rev. Lett. 105, 140502 (2010).
  * Blais  _et al._ (2020) Alexandre Blais, Steven M Girvin, and William D Oliver, “Quantum information processing and quantum optics with circuit quantum electrodynamics,” Nat. Phys. , 247–256 (2020).
  * Pauka  _et al._ (2021) S. J. Pauka, K. Das, R. Kalra, A. Moini, Y. Yang, M. Trainer, A. Bousquet, C. Cantaloube, N. Dick, G. C. Gardner, M. J. Manfa, and D. J. Reilly, “A cryogenic cmos chip for generating control signals for multiple qubits,” Nat. Electron. 4, 64–70 (2021).
  * Chen  _et al._ (2015) Xiangdong Chen, Changling Zou, Zhaojun Gong, Chunhua Dong, Guangcan Guo, and Fangwen Sun, “Subdiffraction optical manipulation of the charge state of nitrogen vacancy center in diamond,” Light Sci. Appl. 4, e230 (2015).
  * Kim  _et al._ (2019) Donggyu Kim, Mohamed I Ibrahim, Christopher Foy, Matthew E Trusheim, Ruonan Han, and Dirk R Englund, “A cmos-integrated quantum sensor based on nitrogen-vacancy centres,” Nat. Electron. 2, 284–289 (2019).
  * Mehta  _et al._ (2020) Karan K Mehta, Chi Zhang, Maciej Malinowski, Thanh-Long Nguyen, Martin Stadler, and Jonathan P Home, “Integrated optical multi-ion quantum logic,” Nature 586, 533–537 (2020).
  * Staacke  _et al._ (2019) Robert Staacke, Roger John, Max Kneiß, Marius Grundmann, and Jan Meijer, “Highly transparent conductors for optical and microwave access to spin-based quantum systems,” npj Quantum Inf. 5, 98 (2019).
  * Bertelli  _et al._ (2020) Iacopo Bertelli, Joris J. Carmiggelt, Tao Yu, Brecht G. Simon, Coosje C. Pothoven, Gerrit E. W. Bauer, Yaroslav M. Blanter, Jan Aarts, and Toeno van der Sar, “Magnetic resonance imaging of spin-wave transport and interference in a magnetic insulator,” Sci. Adv. 6, eabd3556 (2020).
  * Kolkowitz  _et al._ (2015) S. Kolkowitz, A. Safira, A. A. High, R. C. Devlin, S. Choi, Q. P. Unterreithmeier, D. Patterson, A. S. Zibrov, V. E. Manucharyan, H. Park, and M. D. Lukin, “Probing johnson noise and ballistic transport in normal metals with a single-spin qubit,” Science 347, 1129–1132 (2015).
  * Ariyaratne  _et al._ (2018) Amila Ariyaratne, Dolev Bluvstein, Bryan A Myers, and Ania C Bleszynski Jayich, “Nanoscale electrical conductivity imaging using a nitrogen-vacancy center in diamond,” Nat. Commun. 9, 2406 (2018).
  * Huck  _et al._ (2011) Alexander Huck, Shailesh Kumar, Abdul Shakoor, and Ulrik L. Andersen, “Controlled coupling of a single nitrogen-vacancy center to a silver nanowire,” Phys. Rev. Lett. 106, 096801 (2011).
  * Kumar  _et al._ (2013) Shailesh Kumar, Alexander Huck, and Ulrik L. Andersen, “Efficient coupling of a single diamond color center to propagating plasmonic gap modes,” Nano Lett. 13, 1221–1225 (2013).
  * Wei  _et al._ (2018) Hong Wei, Deng Pan, Shunping Zhang, Zhipeng Li, Qiang Li, Ning Liu, Wenhui Wang, and Hongxing Xu, “Plasmon waveguiding in nanowires,” Chem. Rev. 118, 2882–2926 (2018).
  * Fescenko  _et al._ (2020) Ilja Fescenko, Andrey Jarmola, Igor Savukov, Pauli Kehayias, Janis Smits, Joshua Damron, Nathaniel Ristoff, Nazanin Mosavian, and Victor M. Acosta, “Diamond magnetometer enhanced by ferrite flux concentrators,” Phys. Rev. Research 2, 023394 (2020).
  * Zhang  _et al._ (2021) Chen Zhang, Farida Shagieva, Matthias Widmann, Michael Kuebler, Vadim Vorobyov, Polina Kapitanova, Junich Isoya, and Jörg Wrachtrup, “Diamond magnetometry and gradiometry with subpicotesla dc sensitivity,” Preprint at https://arxiv.org/abs/2012.15706 (2021).
  * Barry  _et al._ (2020) John F. Barry, Jennifer M. Schloss, Erik Bauch, Matthew J. Turner, Connor A. Hart, Linh M. Pham, and Ronald L. Walsworth, “Sensitivity optimization for nv-diamond magnetometry,” Rev. Mod. Phys. 92, 015004 (2020).