∎

11institutetext: T. Ichikawa 22institutetext: Center for Quantum Information
and Quantum Biology, Osaka University, Osaka 560-0043, Japan
22email<EMAIL_ADDRESS>
# Bibliometric analysis of topic structure in quantum computation and quantum
algorithm research ††thanks: This work was supported by MEXT Quantum Leap
Flagship Program (MEXT Q-LEAP) Grant Number JPMXS0120319794.

Tsubasa Ichikawa

(Dated: )

###### Abstract

We present a bibliometric analysis of the research papers on quantum
computation and quantum algorithms published in 1985-2020. We identify three
distinct periods from the trend of the annual number of published papers, and
show the 20 top contributing countries in each period in terms of the number
of publications and the number of total citations. The bibliographic coupling
network of the publications in the latest period is characterized as a dense,
small-world network with a small diameter, which contains 14 large
communities, whose topics are the fabrication of qubits in various physical
systems, studies of the quantum algorithms, and other related topics. Quantum
machine learning, one of the emerging topics in quantum computation, is found
to be the fifth-largest independent community.

###### Keywords:

bibliometrics bibliographic coupling Louvain method citation analysis quantum
computation quantum algorithms

††journal: Scientometrics

##  1 Introduction

Quantum computation (QC) is an interdisciplinary research field, aiming at the
implementation of quantum algorithms (QA), which utilize the principles of
quantum mechanics NC2000 . QC and QA have attracted much attention from
researchers since the discovery of QA outperforming the corresponding
classical algorithms. For example, Shor’s factorization algorithm Shor1994
performs prime factoring exponentially faster than the classical algorithms
known so far.

Moreover, the last five years have witnessed the growing attention to QC and
QA: the U.S. national quantum initiative Monroe2019 ; Raymer2019  came into
force in 2018, and other countries also started governmental investigations
Riedel2019 ; Yamamoto2019 ; Sussman2019 ; Robertson2019  to make an ecosystem
of quantum information science and technology (QIST).

With the growing interest in the QIST, there are several works to perform
quantitative analyses on the state of the art of the QIST Tolcheev2018 ;
Olijnyk2018 ; Farinhold2019 ; Chen2019 ; Scheidsteger2021 ; Dhawan2021 ;
Seskir2021 ; Brito2021 . These works clarified that i) the number of
publications in the QIST is rapidly growing, ii) the leading countries of the
QIST research are the US and China, and iii) Institutions in Europe have more
collaborative works than those in the other countries. In addition to these,
more specific analyses have been conducted in quantum cryptography Olijnyk2018
and quantum machine learning Dhawan2021 . The collaboration network in quantum
information authors is found to be a small-world network Brito2021 .

This paper supports these findings and appends new insights from the viewpoint
of the complex network analysis Barabasi2016 . More precisely, we clarify the
long-term trends of the researches on QC and QA from 1985 to 2020 by
bibliometric analysis and show that the bibliographic coupling network
Kessler1963  of QC and QA papers published in 2014-2020 is a dense, small-
world network with a small diameter, which has a clearly distinct topic
structure with emerging topics. Thus, this work implies that QC and QA
researches are mutually related deeply, and growing rapidly with the new
topics emerging.

The rest of the paper is organized as follows: In Sec. 2, we give a brief
account of the method employed here. In Sec. 3, we show the results obtained
in our analyses. Basic statistics of the dataset and the community structure
of the topics are presented. Section 4 is devoted to our conclusion.

##  2 Method

#### Data collection.

We collected 14,450 papers from the Clarivate Web of Science (WoS) database on
26 March 2021. We applied a query to the following attributes of the WoS
database: abstracts, titles, author keywords, and keyword plus. By referring
to Farinhold2019 , we constructed the query such that

> (“quantum computer” OR “quantum computing” OR “quantum compute” OR “quantum
> computers” OR “quantum algorithm” OR “quantum algorithms” OR “quantum
> inspired”)
>
> NOT
>
> (“post-quantum” OR “quantum resistant” OR keywords:cryptography OR “quantum
> key” OR conscious* OR brain OR dream OR QTAIM),

which was designed to collect the papers with respect to QC and QA but not to
quantum cryptography and its related topics. The difference between our query
and that in Farinhold2019  is laid on the treatment of quantum-inspired
algorithms, which has been recently drawing the interest of the quantum
information community Arrazola2020 . Bearing this trend in mind, we used the
keyword “quantum inspired” to include the papers of this topic in the
resultant dataset, whereas Farinhold2019  used it to exclude such papers.

#### Preprocessing.

Out of the 14,450 papers, we extracted the 14,086 papers which were published
until the end of 2020 and whose authors are not anonymous. In addition, we
deleted one paper that caused an error in constructing the bibliographic
coupling network Kessler1963 , but no qualitative influence to the descriptive
statistics mentioned below. The above preprocessing resulted in the 14,085
papers to be analyzed in this paper.

#### Outline of Analyses.

We performed the following statistical analyses. Throughout our analyses, we
used R and its libraries Bibliometrix Aria2017  and wordcloud2.

  1. 1.

We analyzed the annual number of the publications and annual transition of the
average citation number per year and paper. From the behaviors of these, we
divided the period 1985-2020 into three mutually exclusive subperiods, that
is, 1985-2003, 2004-2013, and 2014-2020.

  2. 2.

For each subperiod, we obtained the descriptive statistics by country. The
indices analyzed were: the number of the publications, total number of
citations, and the average number of citations per paper.

  3. 3.

For the latest subperiod 2014-2020, we performed the community detection. We
constructed the bibliographic coupling network, where the coupling strength
between two papers is given by the Jaccard coefficient Jaccard1912  with
respect to the similarity of the reference lists. The clustering method we
have employed was the Louvain method Brondel2008 , which is of modularity
optimization algorithms and known to be fast and accurate. Size distribution
of the communities obtained was analyzed.

  4. 4.

For the large communities having more than one hundred papers, we identified
their topics through their WordClouds obtained from the word occurrence
distribution over the abstracts, titles, author keywords, and keyword plus in
the dataset. To make the WordClouds, 24 stopwords were chosen from the
frequently appearing words over all the papers in the total period 1985-2020.
We list the stopwords in Table 1.

Table 1: List of the stopwords in this paper. algorithm | algorithms | also | based | can
---|---|---|---|---
computer | computing | gate | gates | information
number | paper | problem | proposed | quantum
results | show | state | states | system
systems | time | two | using |

##  3 Results

###  3.1 Overview: 1985-2020

Figure 1: (Left) The number of publications of QC and QA researches. (Right)
The average citations of the papers per year.

The left panel of Fig. 1 shows the annual shift of the number of publications
in QC and QA researches. Whereas the recent publications, Fujii2015  for
example, set the starting point of QC and QA research at the publication of a
Feynman’s seminal paper Feynman1982  in 1982, it is not the case for the
present study, since Feynman’s paper has no abstract and contains no query
keywords in its title, and thereby out of our dataset. Instead, our dataset
starts with the three papers published in 1985: the paper on universal quantum
computation by Deutsch Deutsch1985 , that on the relation between reversible
computation and quantum computation by Peres Peres1985 , and a brief
explanation on Deutsch1985  by Maddox Maddox1985 , who was an editor of Nature
at that time.

The number of publications had been almost negligible until 1993, but started
to increase from 1994, which is the year that Shor’s factoring algorithm was
put forward Shor1994 . In 1985-2020, we can observe three distinct subperiods,
1985-2003, 2004-2013, and 2014-2020, from the behavior of the inclinations of
the line graph.

Two relevant papers lie on the boundaries of these subperiods. One is a
Kitaev’s paper published in 2003 Kitaev2003 , which proposed a quantum error-
correcting code called the toric code. The toric code is distinguished from
the other quantum error-correcting codes up until then, not only because its
descendants, called surface codes, achieve high threshold values of error rate
allowed to perform fault-tolerant quantum computation Fujii2015 , but also
because it led to much literature on the investigation of topological matter.
The other is a paper of the Martinis group in 2014 Barends2014 , which
provided an experimental demonstration of the quantum elementary gates working
with the error rate below the threshold.

The right panel of Fig. 1 shows the annual shift of the average article
citations per year. The singular value in 1985 is because our dataset contains
three papers in 1985, out of which Deutsch1985  and Peres1985  have been cited
frequently (2048 citations for Deutsch1985  and 397 for Peres1985 ). Putting
this singular behavior in 1985 aside, we observe that relatively many highly
cited papers had been published during 1995-1999, whence the average article
citations per year has been almost stable. This implies that on the basis of
the fundamental works performed in 1995-1999, QC and QA community has been
grown steadily and established a stable citation network. Indeed, we can find
the several papers in 1995-1999 which became the basic materials of the
standard textbook of QC and QA community NC2000 : Shor’s 9 qubit code Shor1995
, Steane’s code Steane1996 , and Grover’s search algorithm Grover1998 , to
name a few.

On the basis of the above observations, we may set three subperiods:
1985-2003, 2004-2013, and 2014-2020. The first subperiod 1985-2003 is the
founding period, which began with Deutsch’s paper, went through the rapid
increase of the publications, and ended up with the official acceptance of the
toric code. In contrast, the second subperiod 2004-2013 had been almost stable
both in the number of annual publications and average article citations per
year. This stability ended up with the experiments conducted by the Martinis
group, whence the number of published papers has rapidly increased with the
average article citations per year being stable.

###  3.2 Major countries in three subperiods

(a) Founding period: 1985-2003.

(b) Stable period: 2004-2013.

(c) Growth period: 2014-2020.

Figure 2: Top 20 countries of the number of the papers in each subperiod. (a)
1985-2003, (b) 2004-2013, and (c) 2014-2020.

Figure 2 shows the top twenty countries of the number of papers in each
subperiod. The US has the largest number of papers in each subperiod. On the
other hand, the country with the second-largest number of publications is
different in each subperiod: Japan in 1985-2003, and China in 2004-2013 and
2014-2020. In addition, the gap between the number of publications of the US
and that of the second-ranked country has been narrowing in the second and
third subperiods. The ratio of multiple country publications has been
increasing in many top countries. However, there are exceptional countries
where the ratio of multiple country publications has been stable, such as
Japan, Russia, and India.

Table 2: Top 20 countries of the total citations of the papers and their
average article scitations in each subperiod. (a) 1985-2003, (b) 2004-2013,
and (c) 2014-2020.

(a) Founding period: 1985-2003.
Country Total Citations Average Article Citations USA 68007 97.99 UNITED
KINGDOM 12456 84.16 GERMANY 7904 69.33 AUSTRALIA 5556 54.47 JAPAN 4929 32.86
SWITZERLAND 4927 153.97 CANADA 3214 44.64 NETHERLANDS 3156 101.81 AUSTRIA 2460
136.67 ITALY 2203 24.75 CHINA 1800 25.00 RUSSIA 1575 22.83 KOREA 1467 73.35
SPAIN 1272 97.85 ISRAEL 1028 57.11 BELGIUM 998 90.73 DENMARK 926 77.17 FRANCE
879 26.64 INDIA 528 22.00 MEXICO 387 64.50

(b) Stable period: 2004-2013.
Country Total Citations Average Article Citations USA 63924 47.32 UNITED
KINGDOM 16037 45.95 GERMANY 15063 49.07 CANADA 13008 49.46 CHINA 12488 15.99
AUSTRALIA 12041 50.59 JAPAN 6664 17.58 AUSTRIA 6313 92.84 NETHERLANDS 5568
154.67 ISRAEL 4598 82.11 FRANCE 4525 35.35 ITALY 2789 17.32 SPAIN 2594 46.32
BRAZIL 2156 20.73 DENMARK 1985 56.71 SWITZERLAND 1871 66.82 KOREA 1371 24.05
RUSSIA 1262 12.88 INDIA 1097 8.44 SINGAPORE 631 15.39

(c) Growth period: 2014-2020.
Country Total Citations Average Article Citations USA 26466 16.70 CHINA 11481
9.48 UNITED KINGDOM 6222 16.08 AUSTRALIA 4563 21.94 GERMANY 4015 15.75 JAPAN
3862 14.68 NETHERLANDS 3026 22.42 CANADA 3015 12.51 INDIA 1798 4.12 FRANCE
1733 13.75 AUSTRIA 1729 24.70 SWITZERLAND 1717 14.19 ITALY 1488 10.33 SPAIN
1357 13.05 DENMARK 1339 38.26 RUSSIA 905 4.95 SOUTH AFRICA 783 27.00 KOREA 580
5.23 SWEDEN 550 14.47 IRAN 497 4.92

Let us turn to Table 2, which shows the top twenty countries of the total
citations of the papers and their average article citations in each subperiod.
The ranking of total citations shows a similar trend to that of the total
number of papers. This is a natural behavior since the total number of
citations increases as the total number of papers increases. On the other
hand, small countries in the population such as Austria, Denmark, and
Switzerland, which are the countries of the founders of quantum mechanics, are
highly ranked in average article citations. In particular, it is notable that
South Africa has been ranked since 2014. This is because quantum machine
learning Wittek2014 ; Schuld2018  has been a promising research field in Noisy
Intermediate-Scale Quantum (NISQ) computer research Preskill2018  since around
2014, and a strong research group in quantum machine learning is in South
Africa.

The above observations show that although the US has been the leading country
in QC research, the gap between the US and other countries is narrowing, and
the internationalization of research is progressing. Besides, with the
development of the NISQ computer researches, the influence of emerging
countries is growing.

###  3.3 Network statistics and community structure in 2014-2020

Table 3: Descriptive statistics of the bibliographic coupling network of the papers published in 2014-2020. The size is the number of the vertices $|V|$, and density is defined by $|E|/|V|^{2}$, where $|E|$ being the number of the edges. The square root density, denoted by $d$ in Melancon2006 , is defined by $\sqrt{|E|/|V|^{2}}$. Measure | Value
---|---
Size | 6777
Average degree | 317
Density | 0.091
Square root density | 0.301
Average path length | 2.056
$\ln|V|/\ln k$ | 1.532
Diameter | 7
Figure 3: Citation number-rank distribution of the references of the papers
published in 2014-2020.

Let us hereafter focus on the latest subperiod 2014-2020. As stated above,
this term witnessed the emergence of new topics, which allow us to expect a
rich topic structure. To discuss this, we deal with the descriptive statistics
of the bibliographic network of the papers published in 2014-2020 and the
size-rank distribution of the community obtained by the Louvain method.

Table 3 shows the descriptive statistics of the basic measures of the
bibliographic coupling network. The square root of the network density is
0.301, which is higher than those of many real-world datasets analyzed in
Melancon2006 . This shows that the network considered here is relatively
dense, suggesting that the papers of QC and QA share the basic literature
often to be commonly cited. Indeed, the citation number distribution of the
references in the papers published in 2014-2020 obeys a power law (Fig. 3).

The average path length of the small-world networks is approximated by
$\ln|V|/\ln k$, where $k$ is the average degree Barabasi2016 . The
bibliographic network considered here has an average path length of 2.056,
which is close to the value expected for the small-world network $\ln|V|/\ln
k\approx 1.532$. This suggests that the bibliographic network is small-world
property. In addition to this, the bibliographic coupling network has a
diameter of $7$. This is larger than that expected for the random network
$\ln|V|/\ln k\approx 1.532$ Barabasi2016 , and smaller than that for the high-
energy physics citation network, whose diameter is 12 Gehrke2003 ;
Leskovec2005 . Summing up, the bibliographic network is a dense, small-world
network with a small diameter.

Applying the Louvain method to the bibliographic coupling network, we obtained
911 communities, whose size-rank distribution is shown in Fig. 4. Out of the
911 communities, the fourteen largest communities consist of more than a
hundred papers. Besides, there exists a gap in the community size between the
14th and 15th communities: whereas the 14th community consists of 114 papers,
the 15th consists of 14 papers. Such a gap is found in a community structure
of an SNS network and reproduced by a connected nearest-neighbor model with
random edge creation to non-nearest neighboring vertices Yuta2007 .

Figure 4: The size-rank distribution of the communities obtained by the
Louvain method from the bibliographic coupling network 2014-2020.

###  3.4 Characters of major communities

To find the subjects of the major communities mentioned above, we made the
WordClouds of the words appearing in the abstracts of the papers, and
identified the subjects. For example, Fig. 5 shows the WordCloud of the
largest community, suggesting that its issues are of superconducting qubits
and error tolerance.

Figure 5: The WordCloud of the largest community of the papers published in
2014-2020.

Table 4 summarizes the statistics and subjects of the largest 14 communities,
which contain 5767 papers (85.1% of 6777 papers). Three remarks are in order.
First, these 14 communities can be classified into three categories as
follows:

  1. 1.

Nine communities on QC and QA using quantum gates (No. 1, 2, 3, 4, 5, 6, 7,
11, 13), which contain 4670 papers.

  2. 2.

One community on the adiabatic quantum computation and quantum annealing (No.
9) of 270 papers.

  3. 3.

Four communities on the topics related to QC (No. 8, 10, 12, 14), which
contain 827 papers.

This shows that QC and QA using quantum gates attract much attention to the
researchers and dominate the research field of QC.

Table 4: Statistics of the major communities of the papers published in 2014-2020. The last row shows the statistics all the publications in this period. No. | Publications | Ratio (%) | Median | Subjects
---|---|---|---|---
|  |  | of citations |
1 | 895 | 13.2 | 4 | Fabrication and use of low-error qubits,
|  |  |  | in particular, superconducting qubits.
2 | 838 | 12.4 | 3 | Photon quantum computing, boson sampling.
3 | 760 | 11.2 | 2 | Quantum algorithms, in particular,
|  |  |  | application to image processing.
4 | 547 | 8.1 | 3 | Quantum simulations.
5 | 465 | 6.9 | 2 | Quantum machine learning.
6 | 430 | 6.3 | 5 | Topological matter and
|  |  |  | topological quantum computation.
7 | 388 | 5.7 | 4 | Silicon qubits.
8 | 346 | 5.1 | 3 | Quantum inspired algorithms.
9 | 270 | 4.0 | 3 | Adiabatic quantum computing
|  |  |  | and quantum annealing.
10 | 219 | 3.2 | 1 | Construction and implementation
|  |  |  | of reversible logic gates.
11 | 206 | 3.0 | 5 | NV center qubits.
12 | 148 | 2.2 | 1 | Post-quantum cryptography.
13 | 141 | 2.1 | 9 | Molecular spin qubits.
14 | 114 | 1.7 | 3 | Quantum walk.
Total | 6777 | 100.0 | 2 |

Second, the communities concerning the fabrication of qubits (No. 1, 6, 7, 11,
13) have medians of citations higher than that of all the papers published in
2014-2020. This suggests the relevance of creating the qubits in good quality.
Indeed, it was pointed out as DiVincenzo’s criteria DiVincenzo2000  that
building a quantum computer needs scalable systems with well-characterized
qubits on which we can perform qubit-wise initialization, gate operations, and
measurements within their coherent time.

Third, quantum machine learning, which is a relatively new research area, is
an independent community. This suggests the rapid growth of quantum machine
learning research. Indeed, whereas the composite annual growth rate (CAGR) of
the publication numbers from 2014 to 2020 is 14.2%, the CAGR of quantum
machine learning publications is 55.4%.

##  4 Conclusion

In this paper, we investigated the trends and topics in the research of QC and
QA. We have found that the number of publications is rapidly growing since
2014, and since then the emerging countries participated in QC research.
Besides, we identified the major 14 topics of QC research in the period
2014-2020. We observed not only the longstanding topics such as the
fabrication of the qubits in good quality, but also the emerging topic such as
quantum machine learning. In addition, we have clarified that the papers of QC
and QA comprise a dense, small-world network with a small diameter. Notably
enough, the size-rank distribution of the communities (sets of the papers
sharing the same topics) has a gap between the sizes of the 14 largest
communities and the others. These findings suggest that in QC and QA research,
the topics are mutually related deeply and rapidly growing, with new topics
emerging.

In closing, we mention that it could be of interest that one applies the
method of the present analysis to other QIST research fields such as quantum
communication or quantum metrology, which have different origins from QC and
QA, and thereby their topic structures would be expected to differ from that
of QC and QA bibliographic coupling network. It is also of importance to
investigate a generation mechanism of the gap in the community size
distribution. These remaining issues could uncover characteristic features of
the QIST research communities.

## References

  * (1) M. Nielsen, and I. Chuang, Quantum computation and quantum information (1st edition), 676 pages. Cambridge university press, Cambridge (2000).
  * (2) P. W. Shor, Algorithms for quantum computation: discrete logarithms and factoring. Proceedings 35th Annual Symposium on Foundations of Computer Science, 124-134 (1994).
  * (3) Ch. Monroe, M. G. Raymer, and J. Taylor, The U.S. national quantum initiative: from act to action. Science, 364 440-442 (2019).
  * (4) M. G. Raymer, and Ch. Monroe, The US national quantum initiative. Quantum Science and Technology, 4 020504 (2019).
  * (5) M. Riedel, M. Kovacs, P. Zoller, J. Mlynek, and T. Calarco, Europe’s quantum flagship initiative. Quantum Science and Technology, 4 020501 (2019).
  * (6) Y. Yamamoto, M. Sasaki, and H. Takesue, Quantum information and science in Japan. Quantum Science and Technology, 4 020502 (2019).
  * (7) B. Sussman, P. Corkum, A. Blais, D. Cory, and A. Damascelli, Quantum Canada. Quantum Science and Technology, 4 020503 (2019).
  * (8) T. M. Roberson, and A. G. White, Charting the Australian quantum landscape. Quantum Science and Technology, 4 020505 (2019).
  * (9) V. O. Tolcheev, Scientometric Analysis of the Current State and Prospects of the Development of Quantum Technologies. Automatic Documentation and Mathematical Linguistics, 52, 121-133 (2018).
  * (10) N. V. Olijnyk, Examination of China’s performance and thematic evolution in quantum cryptography research using quantitative and computational techniques. PloS one 13, e0190646 (2018).
  * (11) J. Farinhold, Trend in quantum computing. Technical report (2019).
  * (12) Y. Chen, Z. Zhang, C. Tao, J. Xu, Q. Tian, J. Gulín-González, and Q. Liu, Scientometric method for comparing on the performance of research units in the field of quantum information. Proceedings of the 17th International Conference on Scientometrics and Informetrics-ISSI2019-with a Special STI Indicators Conference Track, 399-410 (2019).
  * (13) T. Scheidsteger, R. Haunschild, L. Bornmann, C. Ettl, Bibliometric Analysis in the Field of Quantum Technology, Quantum Reports. 3, 549-575 (2021).
  * (14) S. M. Dhawan, B. M. Gupta, and G. M. N. Mamdapur, Quantum Machine Learning: A Scientometric Assessment of Global Publications during 1999-2020, International Journal of Knowledge Content Development and Technology, 11, 29-44 (2021).
  * (15) Z. C. Seskir, and A. U. Aydinoglu, The landscape of academic literature in quantum technologies. International Journal of Quantum Information, 19 2150012 (2021).
  * (16) S. Brito, R. Oliveira, R. Oliveira, and R. Chaves, A portrait of the collaboration network in quantum information. arXiv:physics/2112.03403.
  * (17) A.-L. Barabasi, Network science (1st edition), 475 pages. Cambridge university press, Cambridge (2016).
  * (18) M. M. Kessler, Bibliographic Coupling Between Scientific Papers. American Documentation, 14, 10-25 (1963).
  * (19) J. M. Arrazola, A. Delgado, B. R. Bardhan, and S. Lloyd, Quantum-inspired algorithms in practice. Quantum, 4, 307 (2020).
  * (20) M. Aria, C. Cuccurullo, Bibliometrix: An R-tool for comprehensive science mapping analysis. Journal of Informetrics, 11, 959-975 (2017).
  * (21) P. Jaccard, The distribution of the flora in the Alpine zone. New Phythologist, 11, 37-50 (1912).
  * (22) V. D. Blondel, J.-L. Guillaume, R. Lambiotte, E. Lefebvre, Fast unfolding of communities in large networks. Journal of Statistical Mechanics, 2008, P10008 (2008).
  * (23) K. Fujii, Quantum computation with topological codes: From qubit to topological fault-tolerance, 138 pages. Springer, Singapore (2015).
  * (24) R. P. Feynman, Simulating physics with computers. International Journal of Theoretical Physics, 21, 467-488 (1982).
  * (25) D. Deutsch, Quantum theory, the Church-Turing principle and the universal quantum computer. Proceedings of the Royal Society London A, 400, 97-117 (1985).
  * (26) A. Peres, Reversible logic and quantum computers. Physical Review A, 32, 3266-3276 (1985).
  * (27) J. Maddox, Towards the quantum computer? Nature, 316, 573 (1985).
  * (28) A.Yu. Kitaev, Fault-tolerant quantum computation by anyons. Annals of Physics, 303, 2-30 (2003).
  * (29) R. Barends, J. Kelly, A. Megrant, et al., Superconducting quantum circuits at the surface code threshold for fault tolerance. Nature, 508, 500-503 (2014).
  * (30) P. W. Shor, Scheme for reducing decoherence in quantum computer memory. Physical Review A, 52, R2493-R2496 (1995).
  * (31) A. Steane, Multiple-particle interference and quantum error correction. Proceedings of the Royal Society London A, 452, 2551-2577 (1996).
  * (32) L. K. Grover, Quantum computers can search rapidly by using almost any transformation. Physical Review Letters, 80, 4329-4332 (1998).
  * (33) P. Wittek, Quantum machine learning, 178 pages. Elsevier, Amsterdam-Boston (2014).
  * (34) M. Schuld, and F. Petruccione, Supervised learning with quantum computers, 306 pages. Springer, Switzerland (2018).
  * (35) J. Preskill, Quantum computing in the NISQ era and beyond. Quantum, 2, 79 (2018).
  * (36) G. Melancon, Just how dense are dense graphs in the real world? A methodological note. Proceedings of the 2006 AVI Workshop on Beyond Time and Errors: Novel Evaluation Methods for Information Visualization, 1-7 (2006).
  * (37) J. Gehrke, P. Ginsparg, and J. M. Kleinberg, Overview of the 2003 KDD Cup. SIGKDD Explorations 5(2), 149-151 (2003).
  * (38) J. Leskovec, J. Kleinberg, and C. Faloutsos, Graphs over Time: Densification Laws, Shrinking Diameters and Possible Explanations. ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (KDD), 177-187 (2005).
  * (39) Y. Kikuo, N. Ono, and Y. Fujiwara, A Gap in the Community-Size Distribution of a Large-Scale Social Networking Site. arXiv:physics/0701168.
  * (40) D. P. DiVincenzo, The Physical Implementation of Quantum Computation. Fortschritte der Phisyk, 48, 771-783 (2000).


# Multi-GPU-Enabled Hybrid Quantum-Classical Workflow in Quantum-HPC
Middleware:
Applications in Quantum Simulations ††thanks: The first two authors
contributed equally to this work.

Kuan-Cheng Chen126, Xiaoren Li3, Xiaotian Xu12, Yun-Yuan Wang4 Chen-Yu Liu5
1Centre for Quantum Engineering, Science and Technology (QuEST), Imperial
College London, SW7 2BX London, UK
2Department of Materials, Imperial College London, SW7 2BX, London, UK
3Department of Physics and Astronomy, University of Waterloo, ON N2L 3G1
Waterloo, Canada  4NVidia AI Technology Center (NVAITC), NVIDIA Corp., 114
Taipei, Taiwan  5Graduate Institute of Applied Physics, National Taiwan
University, 10663 Taipei, Taiwan  6Email<EMAIL_ADDRESS>
###### Abstract

Achieving high-performance computation on quantum systems presents a
formidable challenge that necessitates bridging the capabilities between
quantum hardware and classical computing resources. This study introduces an
innovative distribution-aware Quantum-Classical-Quantum (QCQ) architecture,
which integrates cutting-edge quantum software frameworks with high-
performance classical computing resources to address challenges in quantum
simulation for materials and condensed matter physics, including the
prediction of quantum phase transitions. At the heart of this architecture is
the seamless integration of Variational Quantum Eigensolver (VQE) algorithms
running on Quantum Processing Units (QPUs) for efficient quantum state
preparation, Tensor Network states, and Quantum Convolutional Neural Networks
(QCNNs) for classifying quantum states on classical hardware.

For benchmarking quantum simulators, the QCQ architecture utilizes the
cuQuantum SDK to leverage multi-GPU acceleration, integrated with PennyLane’s
Lightning plugin, demonstrating up to tenfold increases in computational speed
for complex phase transition classification tasks compared to traditional CPU-
based methods. This significant acceleration enables models such as the
transverse field Ising and XXZ systems to accurately predict phase transitions
with a 99.5% accuracy. The architecture’s ability to distribute computation
between QPUs and classical resources addresses critical bottlenecks in quantum
High-Performance Computing (HPC), paving the way for scalable quantum
simulation.

The QCQ framework embodies a synergistic combination of quantum algorithms,
machine learning, and Quantum-HPC capabilities, enhancing its potential to
provide transformative insights into the behavior of quantum systems across
different scales. As quantum hardware continues to improve, this hybrid
distribution-aware framework will play a crucial role in realizing the full
potential of quantum computing by seamlessly integrating distributed quantum
resources with the state-of-the-art classical computing infrastructure.

###### Index Terms:

Quantum Machine Learning, Distributed Quantum Computing, Quantum Software,
Quantum Simulation, cuQuantum SDK, Tensor Network

##  I Introduction

Quantum Machine Learning (QML) integrates the disciplines of machine learning
and quantum computing[1], employing parameterized quantum circuits as
statistical models[2]. This technology has seen an increasing array of
applications in the natural sciences[3], generative modelling[4], and
classification problems[5, 6]. Due to its high expressivity[7], QML
demonstrates superior performance over conventional models across numerous
domains. This includes stellar classification within large datasets[8],
leveraging an architecture that effectively bridges classical data with
quantum algorithms. However, current applications are constrained by the
limited number of Quantum Processing Unit (QPU) qubits and the fidelity of
qubits in the Noisy Intermediate-Scale Quantum (NISQ) era[9]. Despite these
challenges, recent research conducted by Q-CTRL and IBM Quantum has made
progress in enhancing the success probability of algorithms through an
automated deterministic error-suppression workflow and quantum error
mitigation technique[10, 11]. Nonetheless, the availability of qubits remains
a significant issue in the design of quantum machine learning algorithms[9].

On the other hand, the work of Huang et al. has demonstrated quantum advantage
through the utilization of quantum data with a quantum computer[12]. For
instance, Monaco et al.’s application of a Variational Quantum Eigensolver
(VQE)-based quantum neural network model for quantum phase detection in the
axial next-nearest-neighbour Ising (ANNNI) model illustrates its
superiority[13]. However, such architectures necessitate an increased number
of qubits for learning more complex models[14]. The demand for a large number
of qubits can be addressed by integrating architectures through distributed
quantum computing[15], also referred to as quantum-centric supercomputing by
IBM Quantum[16]. In the realm of compact quantum devices, achieving high-
fidelity qubit operations and their subsequent verification is relatively
manageable. Nonetheless, the shift towards modular approaches mandates the
transfer of quantum information between QPUs, thereby fostering effective
qubit interactions across various modules. While this approach offers certain
benefits, the ensuing inter-module communication is often characterized by
slower speeds and reduced reliability, leading to the emergence of a challenge
known as the quantum interconnect bottleneck (QIB)[17]. Furthermore, the
adoption of distributed quantum algorithms introduces vulnerabilities that
necessitate the formulation of robust quantum protocols to ensure system
integrity and security[18].

In this study, we endeavor to develop a scheme that is conducive to high-
performance computing (HPC) within the quantum domain [19], specifically
designed to enable distributed quantum computing through a hybrid quantum-
classical workflow. This framework incorporates multi-GPU acceleration (via
the cuQuantum SDK[20]) to support quantum simulators (which serve as
counterparts to noiseless QPUs, if QPUs are not available). Additionally, we
integrate the latest Pennylane Lightning plugins[21]. This component leverages
CUDA-aware MPI (Message Passing Interface) boosted by NVLink’s 600 GB/s
bidirectional bandwidth for optimized GPU-to-GPU communication, enabling rapid
distributed simulation tasks[22]. This middleware scheme offers a synergy
between classical GPU acceleration and quantum computing acceleration[23, 19].
The relevant conceptual architectures for Quantum-HPC middleware are discussed
by Saurabh et. al[19].

Echoing the architectural paradigm introduced by Monaco et al.[13], our
methodology adopts a VQE-based ansatz to explore quantum phase transitions in
both the transverse field Ising and the XXZ models[24]. Different from the
quantum-to-quantum or the quantum-to-classical architecture[13, 24, 25, 26],
we propose an quantum-classical-quantum sandwich (QSandwich) architecture.
This architecture capitalizes on the capabilities of classical convolutional
neural networks (CNNs) to significantly reduce the qubit requirements for the
quantum classifier.

For the implementation of the VQE-based ansatz, we also explore the potential
of distributed-VQE approaches[27, 28, 25] or the employment of multi-GPU
configurations[29] to meet the demands of large qubit requirements in complex
scenarios. The QSandwich framework is designed to ensure that each quantum
layer is effectively managed and utilized within a distributed quantum
computing environment, leveraging small, yet high-fidelity QPU resources. This
approach underscores our commitment to advancing the frontier of quantum
computing by optimizing computational resources and fidelity in the execution
of quantum simulations.

##  II Applications and Algorithms

###  II-A QSandwitch: Quantum-Classical-Quantum framework for Quantum-HPC
Processing

In the domain of Quantum-HPC[19], we proposed a novel distribution-aware
hybrid quantum-classical-quantum (QCQ) processing framework shown in Fig. 1,
representing an optimized approach to the computational challenges of
distributed quantum computing for classifying phase transition. Initiated by
the VQE algorithm, this framework begins its operation within the quantum
realm, where a specific quantum state is prepared to encode potential
solutions. This quantum state is then converted into classical information via
a process termed state feature selection, during which essential
characteristics of the quantum state are identified and extracted for
subsequent processing in classical CNN layers. This methodology introduces a
structured integration of quantum and classical computing techniques, aiming
to optimize the solution-finding process through the strategic manipulation
and analysis of quantum data.

The classical segment of the QCQ framework harnesses the capabilities of CNNs
to scrutinize the quantum state prepared by the VQE algorithm. This involves
the application of convolutional layers to refine the data, augmented by
pooling layers that distil the essence of the information. The refined data is
then adeptly re-encoded into quantum form, traversing another quantum layer
tasked with extracting a scalar output to classify the phase transition
problem.

The scalar output generated by the process is not the end point but rather an
intermediate stage requiring additional refinement via a sigmoid layer to
improve the accuracy of the final result. The following section will provide a
detailed explanation of this hybrid QCQ architecture, focusing on the
interaction between its quantum and classical components. The integration of
these computational realms aims to enhance the capabilities of the near-term
Quantum-HPC ecosystem, addressing challenges with efficiency beyond what
quantum or classical computing can achieve individually.

Figure 1: Pipeline of the Hybrid QSandwich Architecture.

####  II-A1 VQE for State Preparation

VQE is a common approach in ground state preparation in quantum computing[30].
In our study, we focus on preparing the ground state for the phase-transition
problem with VQE and tensor network ansatz[24]. These methods enable efficient
and accurate approximation of quantum states. Here are the key points of our
approach:

  * •

Ground State Preparation: We employ the VQE to approximate the ground states
of given quantum Hamiltonians. The advantage of using VQE for state
preparation, instead of loading state information from the Pennylane
dataset[31], is that, by knowing the circuit structure, we can reproduce the
ground state completely. On the other hand, the classical-shadow information
from the Pennylane dataset will be helpful to reproduce the state more
accurately[32], however, it could not be the same state. Knowing the circuit
structure also reduces the size of datasets which speeds up the data reading
process in quantum machine learning later on. A n-qubits states information
from pennylane datasets contains $2^{n}$ floats. On the other hand, our VQE
states preparation datasets only records the parameters for the ansatz
circuits. The number of parameters varies depending on the structure and depth
of the ansatz. For example, we used 100 float parameters for a 10-qubit
circuit in state preparation which reduced the size of datasets (otherwise
1024 floats needed) and performance well in classification.

Additionally, building the state preparation ourselves allows us to prepare
more ground states (100 states from the Pennylane dataset[31] and 1000 states
from our preparation) and makes it possible to apply data augmentation.

  * •

Tensor Network Ansatz: Our approach replaces the standard unitary coupled
cluster ansatz with tensor network ansatz states introduced by Uvarov et.
al[24], which was inspired by tensor networks, to prepare the ground states
efficiently. There are several families of variational ansatz states,
including rank-one circuits, tree tensor network circuits, and checkerboard-
shaped circuits with varying depths. The structure of the checkerboard-shaped
circuit is shown in Fig. 2 (b). These ansatz circuits are designed to capture
different amounts of entanglement and are critical for accurately
approximating the ground states of various Hamiltonians. We choose the
checkerboard-shaped circuits in our VQE due to the best performance in
reducing the error of ground state energy[24]. For each entangled blocks in a
checkerboard-shaped circuit, we used Ising entangle gates shown in Fig. 2 (c).

  * •

Hamiltonian Decomposition: We represent the Hamiltonian as a sum of tensor
products of Pauli operators. This decomposition allows us to estimate
individual terms of the Hamiltonian expectation value variational and minimize
the energy using a hybrid process. More math details are shown in II-B

  * •

Optimization: The parameters of the ansatz states are optimized using
classical algorithms to minimize the expectation value of the Hamiltonian.
This process iterates until the ground state is approximated within the
desired accuracy.

Figure 2: (a) Quantum circuit depicted as a tensor network with bonds of
dimension 2. (b) Checkerboard tensor network circuit. [20] Each green block
refers to a two-qubit entangler circuit. (c) The entangled block in the
checkerboard circuit

####  II-A2 Data Augmentation

To enhance the robustness of our model and improve its accuracy, we apply data
augmentation techniques to the quantum states data. This involves applying
rotations and spin flips to the VQE-prepared states, creating additional valid
data points for training the classifier. This technique exploits the
symmetries of the Hamiltonians to generate a more diverse training dataset.

Our methodology represents an approach to predicting phase transitions in
quantum systems by merging the capabilities of quantum simulation and quantum
machine learning. Through the strategic use of tensor network ansatz states, a
quantum classifier, and the computational power of multi-GPU frameworks
facilitated by PennyLane, we achieve significant improvements in speed and
scalability, paving the way for advanced studies in quantum phase transitions.

####  II-A3 Quantum Convolutional Neural Network Classifier (Felix)

Quantum Convolutional Neural Network (QCNN) is utilized to predict phase
transitions, harnessing the combined strengths of convolutional layers and
quantum computing layers [33]. This approach allows for efficient feature
extraction from extensive databases and the natural simulation of quantum
data. The QCNN algorithm, as utilized in the QCQ architecture, incorporates a
single convolutional layer connected with a pooling layer, multiple fully
connected layers, and a quantum circuit layer. This architecture is
illustrated in Fig. 1 and is explained below:

  * •

Convolutional Layer: The model initiates its computational flow with a single
convolutional layer. It utilizes a one-dimensional convolution, generating 30
output channels, with a kernel size identical to that of the feature. This
design is specifically tailored for the initial extraction of features
directly from the input data.

  * •

Max Pooling Layer: One max pooling layer, configured with a kernel size of 1,
is placed strategically to reduce spatial dimensions and hence, the complexity
of the data passing through the network.

  * •

Fully Connected Layers: After flattening the feature map of the max pooling
layer, the model incorporates two fully connected layers that serve to
interpret the features extracted by the convolutional and pooling layers,
culminating in the model’s ability to make predictions. Notably, the second
fully connected layer is specifically designed to interface with the quantum
circuit layer, mapping the classical data into a quantum-compatible format.

  * •

Quantum Circuit Layer: At the heart of QCNN approach is the quantum circuit
layer, invoked through a predefined Pennylane circuit. The circuit is
illustrated in Fig. 3. This layer signifies the model’s capacity to perform
quantum computations, potentially exploiting quantum parallelism and
entanglement to enhance the model’s predictive capabilities. This quantum
layer is followed by another fully connected layer.

  * •

Dropout: A dropout layer with a probability (p) of 0.5 is integrated to
prevent overfitting by randomly omitting a subset of features during the
training phase.

  * •

Prediction: The sigmoid activation function is employed to transform the
feature map of the final fully connected layer into a probability value
ranging between 0 and 1. This mapping facilitates a clear decision-making
process for binary classification tasks. The computation of the loss involves
measuring the discrepancy between the sigmoid function’s output and the
designated target values. Upon completion of the training process, the model
makes predictions based on a threshold criterion: outputs exceeding 0.5 are
classified as 1, indicating the presence of phase transition, while those
below 0.5 are classified as 0.

Figure 3: The quantum circuit used in QCNN. In this circuit, a Hadamard gate
is first applied to each qubit, followed by a rotational-Y gate. The angles of
the rotational gates are the trainable parameters (Theta1, Theta2 and Theta3).
At the end of this circuit, measurement is executed and the possibilities of
finding 7 out of 8 quantum states (except “111”) are the output of this
circuit. Thus, when regarded as a layer in the QCNN, this circuit has 3 input
channels and 7 output channels.

###  II-B Hamiltonians for Solving Phase Transition

In our study, we introduce a novel approach that combines quantum simulation
with QML to classify phases of matter, addressing the computational challenges
faced by classical simulation methods. By employing a variational quantum
algorithm, we leverage the capabilities of quantum computers to prepare and
classify labelled states derived from the VQE algorithm. This method
effectively bypasses the data reading slowdown typically encountered in
quantum-enhanced machine learning applications, presenting a significant
advancement in the field.

Our work utilizes families of variational ansatz circuits inspired by tensor
networks, enabling us to exploit tensor network theory to elucidate properties
of phase diagrams. This approach is instrumental in our development of a
quantum neural network. These results underscore the potential of integrating
quantum simulation with QML to provide deep computational insights into
quantum systems.

We represent the following Hamiltonian as a sum of tensor products of Pauli
operators:

$H=\sum_{\alpha_{1}\alpha_{2}...\alpha_{n}}J_{\alpha_{1}\alpha_{2}...\alpha_{n}}\sigma_{\alpha_{1}}\otimes\sigma_{\alpha_{2}}\otimes\cdots\otimes\sigma_{\alpha_{n}},$
(1)

where $\alpha_{i}\in\\{0,1,2,3\\}$ enumerate the Pauli matrices
$\\{1,X,Y,Z\\}$. With the decomposition (1), individual terms of
$\langle\psi(\theta)|H|\psi(\theta)\rangle$ can be estimated and variationally
minimized elementwise using a classical-to-quantum process. In each iteration,
one prepares the state $|\psi(\theta)\rangle$ and measures each qubit in the
local $X$, $Y$, or $Z$ basis, estimates the energy and updates $\theta$. This
method can become scalable only if the number of terms in the Hamiltonian is
polynomially bounded in the number of spins and the coefficients
$J_{\alpha_{1}\alpha_{2}...\alpha_{n}}$ are defined up to poly(n) digits.

In particular, we use the transverse field Ising model (TFIM):

$H_{TFIM}=J\sum_{i}\sigma_{i}^{z}\sigma_{i+1}^{z}+h\sum_{i}\sigma_{i}^{x},(J>0,h>0)$
(2)

where $\sigma_{i}^{z}$ are Pauli Z matrix acting on the $i$th spin. Constant
$J$ is known as the energy scale, we set $J=1$ in our study which means no
loss of generality. Dimensionless constant $h$ corresponds to the transverse
magnetic field.

When $J>h$, the system has two ground states, with opposite signs of
magnetization.When $J<h$ the system has one ground state with zero
magnetization. The changing of ground states refers to the phase changing of
the system, which happens at $J=h$.

The other model we used is antiferromagnetic XXZ spin chain model:

$H_{xxz}=-\sum_{i}J_{\perp}(\sigma_{i}^{x}\sigma_{i+1}^{x}+\sigma_{i}^{y}\sigma_{i+1}^{y})+J_{z}\sigma_{i}^{z}\sigma_{i+1}^{z},(J_{z}<0)$
(3)

where the sign of $J_{z}$ determines if the system is ferromagnetic
($J_{z}>0$) or antiferromagnetic ($J_{z}<0$). The sign of $J_{\perp}$ is not
important in bipartite lattice because the states can be redefined by
$\sigma^{x}\rightarrow-\sigma^{x},\sigma^{y}\rightarrow-\sigma^{y}$. We set
$J_{\perp}=1$ in our study.

For antiferromagnetic XXZ model, when $|J_{z}|>|J_{\perp}|$, the ground states
spins each takes their opposite values to their neighbours, and the system is
in antiferromagnetic Ising state; When $|J_{z}|<|J_{\perp}|$, the system is in
the planar phase. A Berezinsky–Kosterlitz–Thouless type phase transition
happens at $|J_{z}|=|J_{\perp}|$ [34].

Conclusively, our research demonstrates the feasibility and effectiveness of
using quantum simulation and QML to classify phases of matter. By preparing
approximate ground states variationally and employing them as inputs to a
quantum classifier, we avoid the limitations of traditional Monte Carlo
sampling methods. The success of our nearest-neighbour quantum neural network
in accurately identifying phases of matter highlights the promising future of
this interdisciplinary approach. This synergy between quantum computing and
machine learning opens new avenues for exploring quantum systems,
significantly impacting the fields of condensed matter physics and quantum
computing.

###  II-C Towrad Distributed Quantum Computing: a Conceptual Architecture for
a Distributed-Quantum-HPC Middleware

To develop a conceptual architecture for distributed quantum computing with a
QCQ workflow, we expand upon the concept based on the work of Saurabh et
al[19]. This approach enables a deeper integration of quantum computing
processes within distributed computing frameworks. Based on Fig. 4 delineates
an integrated architecture designed to leverage distributed quantum resources
for advanced computational tasks. The schematic begins with a classical
scheduling layer, where a CPU disaggregates a complex problem into sub-
problems. These are then converted into quantum states through GPUs in an
encoding layer, facilitating the classical-to-quantum transition. The middle
of the system can lie in the distributed-VQE layer[35, 25], where QPUs (linked
by quantum internet and augmented by quantum repeaters) execute quantum
algorithms. Finally, a hybrid quantum-classical neural network analyzes the
quantum states, yielding a scalar output via a quantum classifier. This
streamlined, multi-layered framework illustrates a scalable approach for
executing computationally intensive tasks across distributed quantum systems,
aiming to unlock new potentials in near-term quantum information processing.

Figure 4: The schematic represents a distributed quantum computing
architecture, illustrating the hybrid QCQ framework for classifying phase
transitions. This architecture combines CPU-based scheduling with quantum
processing across multiple layers and employs GPUs for state encoding. The
dashed lines represent omitted $n$ blocks in parallel due to size constraints
of the image

##  III Result

Figure 5: (a)Predicted phases as a function of h for the TFIM model. (b) The
predicted probability of phases as a function of Jz for the XXZ model.
Positive prediction of label II represents phase II, which is above the dashed
lines. The theoretically phase II (disorder phase) is the areas on the right-
hand side of the red lines (shown in red color).

In Fig. 5, we have successfully implemented a quantum machine learning
classifier for phase transition using the PennyLane package, enhanced by the
computational acceleration of the cuQuantum SDK. In our classification task,
we observed an increase in accuracy near the phase transition point when
utilizing checkerboard states with increasing depth in our VQE algorithm.
Notably, even lower-depth ansatz states such as rank-one, tree tensor network,
and single-layer checkerboard states delivered comparable results due to the
relatively simple nature of the Hamiltonian involved.

Figure 6: The evolution of loss (left) and accuracy (right) in the QCNN model.
As the number of epochs increases, the model’s loss decreases exponentially.
The model’s accuracy improves significantly in the initial epochs and then
stabilizes at around 98%. And test accuracy reaches 99.5%.

We prepared a dataset of 100 data points employing a VQE with a four-layered
checkerboard ansatz state. After shuffling, this dataset was divided into an
80% training set and a 20% test set. Impressively, the classifier achieved a
98.0% prediction accuracy rate. For the antiferromagnetic XXZ spin chain
model, we generated 4000 data points to train the classifier, achieving a
94.6% accuracy rate on the test data after enhancing the classifier circuit
with two additional layers.

The QCNN yields results that are both robust and satisfactory (Fig. 6). In the
initial phases of the training epochs, the accuracy of the model escalates
significantly, demonstrating a rapid learning curve which subsequently reaches
a plateau at approximately 98.0%. Further evaluation on a test set reveals
that the model attains a remarkable accuracy of 99.5%.

The dataset employed for the QCNN model was generated utilizing the VQE on XXZ
model. The dataset comprised a total of 1000 data points. In adherence to
standard machine learning practices, the dataset was partitioned in an 80-20
split, with 80% utilized for the training phase and the remaining 20% reserved
for testing.

Our approach demonstrates the potential of quantum-enhanced machine learning
to classify phases of matter with high accuracy, outperforming classical
Monte-Carlo sampling-based methods. This technique is versatile and can be
applied to any model expressible as a spin model, with the capability to be
extended to multi-class classification problems. Our results underscore the
advantages of integrating quantum simulations with advanced machine learning
techniques, paving the way for new insights into the study of quantum phase
transitions.

###  III-A GPU Acceleration with cuQuantum for VQE and QCNN

In this research work, our main task is to develop and optimize a Hybrid QML
model with a distributed-quantum-computing framework for classifying phase
transitions. In our demonstration, we use the PennyLane package, with a focus
on computational efficiency through GPU acceleration with cuQuantum SDK. The
inherent challenge in QML is the intensive computational requirement, often
necessitating QPUs with high-fidelity qubits. To address this, we employ
classical hardware for initial data processing and feature extraction—a step
integral to our QML model’s training process. Our strategy utilizes the
cuQuantum SDK, which offers a GPU-optimized collection of low-level primitives
supporting Pennylane lightning backend. This approach accelerates quantum
machine learning model execution on NVIDIA GPUs, substantially shortens
training durations, and reduces computational costs. By running our
applications on Denvr Dataworks’s CUDA-compatible platform, we benefit from
the enhanced processing capabilities of NVIDIA GPUs, which are designed for
parallel computing and can therefore handle large datasets and complex
operations with greater speed than conventional CPUs. The expected outcome of
our work is a threefold acceleration in the training of our quantum machine
learning models compared to CPU-only execution. This improvement is
attributable to the GPUs’ parallel processing power, which allows for quicker
data processing and computation. The integration of cuQuantum with CUDA not
only augments computational speed but also offers cost-effective solutions by
eliminating the need for expensive quantum hardware and extensive
infrastructure. Our objective is to make QML models more scalable and
accessible, leveraging these technologies to advance research in quantum phase
transitions and establish new performance benchmarks in the field of quantum
computing.

Figure 7: Benchmarking of computation times for VQE, Hybrid QCNN, and Total
processes, comparing performance across Intel CPU (i7-13700KF), Apple M1 pro
(10-core CPU), and 1, 2, and 4 NVIDIA A100 GPUs as quantum simulators.

In line with our objectives, the benchmarking results showcase a comparative
analysis of computational performance across different hardware configurations
for these quantum simulations. The bar plot distinctly illustrates the time
taken to execute specific tasks like the VQE (in this case with 16 qubits),
Hybrid QCNN (in this case with 3 qubits), and the aggregate computational
process across Intel CPUs, Apple M1 chips, and NVIDIA A100 GPUs. The
acceleration impact is significant when utilizing the A100 GPUs, with the most
notable performance improvement observed with a configuration of four GPUs,
suggesting a near-linear scaling in this particular benchmarking scenario.
This benchmarking also agrees with the result shown in Vallero et. al’s
work[36].

Figure 8: The line graph conducts a linearity performance check of multi-GPU
settings for quantum machine learning training.

The accompanying line plot further elucidates the speedup achieved with the
use of 1, 2, and 4 A100 GPUs compared to CPU benchmarks. It becomes evident
that the total speedup gained from GPU acceleration does not fully align with
an ideal linear progression, which is visually represented by a grey dashed
line. This deviation implies that while GPU acceleration substantially
benefits the computational speed, the returns diminish with the addition of
more GPUs, possibly due to overhead associated with parallelization and inter-
GPU communication. The less efficient may be because in our case the qubit
number is not very large. In Vallero et. al’s benchmarking[36], with more
qubits we run, the more acceleration we will observe. These findings
underscore the A100 GPUs’ profound capabilities in enhancing performance for
quantum simulation tasks, thereby underscoring their potential to facilitate
and expedite complex quantum computations in research settings.

##  IV Conclusion

In this study, we propose an innovative QCQ architecture aimed at enhancing
applications in distributed quantum computing and Quantum-HPC systems. This
architecture integrates VQE algorithms and QCNNs, enabling rapid and accurate
quantum simulations, exemplified by phase transition classification within
this research. The hybrid QCQ algorithm demonstrated an exceptional 99.5% test
accuracy in predicting phase transitions for models such as the transverse
field Ising and XXZ. The architecture efficiently utilizes QPUs with a limited
number of high-fidelity qubits, such as superconducting circuits[37], ion
traps[38], neutral atoms[39], and NV center diamonds[40], to scale up the
qubit count and achieve distributed quantum computing.

At the heart of the QCQ framework is the seamless integration of variational
quantum eigensolver algorithms, tensor network states, and convolutional
neural networks. This synergistic amalgamation allows for the effective
preparation of quantum states through quantum simulations, followed by their
classification using advanced machine learning techniques. The architecture
finds applications in quantum reinforcement learning[41] and can optimize the
hybrid neural network structure using an $polylog(M)$ approach[42] or the fast
weights approach[43].

The success of this QCQ framework highlights the immense potential of merging
quantum computing with machine learning. By surmounting the limitations
inherent in classical algorithms, this method paves the way for deeper
insights into the behavior of quantum systems across various sizes and
configurations. It finds applications in fields such as materials science,
condensed matter physics, and sustainability research.

As research in quantum hardware and algorithms progresses, architectures like
QCQ will play a critical role in unlocking the transformative capabilities of
quantum-enhanced computing. While challenges in further enhancing accuracy and
extending applicability remain, the scalability and efficiency exhibited by
this framework open up exciting avenues for future exploration. Integrating
the strengths of both quantum and classical computing, the QCQ architecture
represents a significant advancement towards fully leveraging the potential of
quantum technologies in diverse scientific and technological domains.

##  V Future work

Our upcoming research aims to enhance the QCQ architecture by harnessing the
CUDA Quantum platform[44], delivering substantial speedup and flexibility for
quantum-classical computing workflows through distributed quantum simulation
and seamless integration with machine learning frameworks.

## Acknowledgment

The authors extend their gratitude to Andreas Hehn, Tobi Odufeso, and Tom
Lubowe for invaluable assistance and insights, which have been pivotal to the
success of this research work. This research was supported by Xanadu, NVIDIA,
Amazon, and Denvr Dataworks under QHack 2024. K.C. is grateful for the
financial support from both the Turing Scheme for the Imperial Global Fellows
Fund and the Imperial QuEST Seed Fund. Special acknowledgement goes to the
Quantum Open Source Foundation (QOSF) and QuantumPedia AI for their
contributions and their quantum computing mentorship program.

## Code Avalibility

The source code for generating the dataset and figures presented in this
research work is openly accessible at
https://github.com/Louisanity/cuPhastLearn.

## References

  * [1] J. Biamonte, P. Wittek, N. Pancotti, P. Rebentrost, N. Wiebe and S. Lloyd, Quantum machine learning, Nature 549 (2017), pp. 195–202.
  * [2] M. Benedetti, E. Lloyd, S. Sack and M. Fiorentini, Parameterized quantum circuits as machine learning models, Quantum Science and Technology 4 (2019), p. 043001.
  * [3] V. S. Ngairangbam, M. Spannowsky and M. Takeuchi, Anomaly detection in high-energy physics using a quantum autoencoder, Physical Review D 105 (2022), p. 095004.
  * [4] C. Zoufal, A. Lucchi and S. Woerner, Quantum generative adversarial networks for learning and loading random distributions, npj Quantum Information 5 (2019), p. 103.
  * [5] A. Mari, T. R. Bromley, J. Izaac, M. Schuld and N. Killoran, Transfer learning in hybrid classical-quantum neural networks, Quantum 4 (2020), p. 340.
  * [6] S. Y.-C. Chen, T.-C. Wei, C. Zhang, H. Yu and S. Yoo, Quantum convolutional neural networks for high energy physics data analysis, Physical Review Research 4 (2022), p. 013231.
  * [7] A. Abbas, D. Sutter, C. Zoufal, A. Lucchi, A. Figalli and S. Woerner, The power of quantum neural networks, Nature Computational Science 1 (2021), pp. 403–409.
  * [8] 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).
  * [9] S. C. Marshall, C. Gyurik and V. Dunjko, High dimensional quantum machine learning with small quantum computers, Quantum 7 (2023), p. 1078.
  * [10] P. S. Mundada et al., Experimental benchmarking of an automated deterministic error-suppression workflow for quantum algorithms, Physical Review Applied 20 (2023), p. 024034.
  * [11] 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.
  * [12] H.-Y. Huang et al., Quantum advantage in learning from experiments, Science 376 (2022), pp. 1182–1186.
  * [13] S. Monaco, O. Kiss, A. Mandarino, S. Vallecorsa and M. Grossi, Quantum phase detection generalization from marginal quantum neural network models, Physical Review B 107 (2023), p. L081105.
  * [14] N. Grzesiak et al., Efficient arbitrary simultaneously entangling gates on a trapped-ion quantum computer, Nature communications 11 (2020), p. 2963.
  * [15] D. Cuomo, M. Caleffi and A. S. Cacciapuoti, Towards a distributed quantum computing ecosystem, IET Quantum Communication 1 (2020), pp. 3–8.
  * [16] Y. Alexeev et al., Quantum-centric supercomputing for materials science: A perspective on challenges and future directions, arXiv preprint arXiv:2312.09733 (2023).
  * [17] D. Awschalom et al., Development of quantum interconnects (quics) for next-generation information technologies, PRX Quantum 2 (2021), p. 017002.
  * [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] N. Saurabh, S. Jha and A. Luckow, A conceptual architecture for a quantum-hpc middleware, in 2023 IEEE International Conference on Quantum Software (QSW), IEEE, 2023, pp. 116–127.
  * [20] NVIDIA, cuQuantum SDK: Simulating quantum circuits on GPUs.  https://developer.nvidia.com/cuquantum-sdk, 2021.  Accessed: February 23, 2023.
  * [21] V. Bergholm et al., Pennylane: Automatic differentiation of hybrid quantum-classical computations, arXiv preprint arXiv:1811.04968 (2018).
  * [22] H. Gao and N. Sakharnykh, Scaling joins to a thousand gpus., in ADMS@ VLDB, 2021, pp. 55–64.
  * [23] K. Wintersperger, H. Safi and W. Mauerer, Qpu-system co-design for quantum hpc accelerators, in International Conference on Architecture of Computing Systems, Springer, 2022, pp. 100–114.
  * [24] A. Uvarov, A. Kardashin and J. D. Biamonte, Machine learning phase transitions with a quantum processor, Physical Review A 102 (2020), p. 012415.
  * [25] I. Khait, E. Tham, D. Segal and A. Brodutch, Variational quantum eigensolvers in the era of distributed quantum computers, arXiv preprint arXiv:2302.14067 (2023).
  * [26] C.-Y. Liu, C.-H. A. Lin and K.-C. Chen, Learning quantum phase estimation by variational quantum circuits, arXiv preprint arXiv:2311.04690 (2023).
  * [27] S. DiAdamo, M. Ghibaudi and J. Cruise, Distributed quantum computing and network control for accelerated vqe, IEEE Transactions on Quantum Engineering 2 (2021), pp. 1–21.
  * [28] Y. Du, Y. Qian, X. Wu and D. Tao, A distributed learning scheme for variational quantum algorithms, IEEE Transactions on Quantum Engineering 3 (2022), pp. 1–16.
  * [29] M. Modani, A. Banerjee and A. Das, Multi-gpu-enabled quantum circuit simulations on hpc-ai system, in International Conference on Data Management, Analytics & Innovation, Springer, 2023, pp. 845–854.
  * [30] J.-G. Liu, Y.-H. Zhang, Y. Wan and L. Wang, Variational quantum eigensolver with fewer qubits, Physical Review Research 1 (2019), p. 023025.
  * [31] D. G. Utkarsh Azad, Pennylane spin datasets.  https://pennylane.ai/datasets/qspin/ising-model, 2023.
  * [32] H.-Y. Huang, R. Kueng and J. Preskill, Predicting many properties of a quantum system from very few measurements, Nature Physics 16 (2020), pp. 1050–1057.
  * [33] I. Cong, S. Choi and M. D. Lukin, Quantum convolutional neural networks, Nature Physics 15 (2019), pp. 1273–1278.
  * [34] F. Franchini, An Introduction to Integrable Techniques for One-Dimensional Quantum Systems, Springer International Publishing, 2017.  url: http://dx.doi.org/10.1007/978-3-319-48487-7, doi:10.1007/978-3-319-48487-7.
  * [35] R. Parekh, A. Ricciardi, A. Darwish and S. DiAdamo, Quantum algorithms and simulation for parallel and distributed quantum computing, in 2021 IEEE/ACM Second International Workshop on Quantum Computing Software (QCS), IEEE, 2021, pp. 9–19.
  * [36] M. Vallero, F. Vella and P. Rech, State of practice: evaluating gpu performance of state vector and tensor network methods, arXiv preprint arXiv:2401.06188 (2024).
  * [37] J. Clarke and F. K. Wilhelm, Superconducting quantum bits, Nature 453 (2008), pp. 1031–1042.
  * [38] D. Kielpinski, C. Monroe and D. J. Wineland, Architecture for a large-scale ion-trap quantum computer, Nature 417 (2002), pp. 709–711.
  * [39] D. S. Weiss and M. Saffman, Quantum computing with neutral atoms, Physics Today 70 (2017), pp. 44–50.
  * [40] L. Childress and R. Hanson, Diamond nv centers for quantum computing and quantum networks, MRS bulletin 38 (2013), pp. 134–138.
  * [41] S. Y.-C. Chen, C.-H. H. Yang, J. Qi, P.-Y. Chen, X. Ma and H.-S. Goan, Variational quantum circuits for deep reinforcement learning, IEEE Access 8 (2020), pp. 141007–141024.
  * [42] C.-Y. Liu et al., Training classical neural networks by quantum machine learning, arXiv preprint arXiv:2402.16465 (2024).
  * [43] S. Yen-Chi Chen, Learning to program variational quantum circuits with fast weights, arXiv e-prints (2024), pp. arXiv–2402.
  * [44] 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.