1
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Sanavio C, Tignone E, Ercolessi E. Hybrid Classical-Quantum Branch-and-Bound Algorithm for Solving Integer Linear Problems. Entropy (Basel) 2024; 26:345. [PMID: 38667899 PMCID: PMC11048856 DOI: 10.3390/e26040345] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/10/2024] [Revised: 04/10/2024] [Accepted: 04/17/2024] [Indexed: 04/28/2024]
Abstract
Quantum annealers are suited to solve several logistic optimization problems expressed in the QUBO formulation. However, the solutions proposed by the quantum annealers are generally not optimal, as thermal noise and other disturbing effects arise when the number of qubits involved in the calculation is too large. In order to deal with this issue, we propose the use of the classical branch-and-bound algorithm, that divides the problem into sub-problems which are described by a lower number of qubits. We analyze the performance of this method on two problems, the knapsack problem and the traveling salesman problem. Our results show the advantages of this method, that balances the number of steps that the algorithm has to make with the amount of error in the solution found by the quantum hardware that the user is willing to risk. The results are obtained using the commercially available quantum hardware D-Wave Advantage, and they outline the strategy for a practical application of the quantum annealers.
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Affiliation(s)
- Claudio Sanavio
- Center for Life Nano-Neuroscience at la Sapienza, Fondazione Istituto Italiano di Tecnologia, Viale Regina Elena 291, I-00161 Rome, Italy;
| | - Edoardo Tignone
- Leithà S.r.l. | Unipol Group, Via Stalingrado 37, I-40128 Bologna, Italy;
| | - Elisa Ercolessi
- Dipartimento di Fisica e Astronomia “Augusto Righi”, Alma Mater Studiorum Università di Bologna, Via Irnerio 46, I-40127 Bologna, Italy
- Istituto Nazionale di Fisica Nucleare, Sezione di Bologna, Viale Berti-Pichat 6/2, I-40127 Bologna, Italy
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2
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Chen Y, Huang JH, Sun Y, Zhang Y, Li Y, Xu X. Haplotype-resolved assembly of diploid and polyploid genomes using quantum computing. Cell Rep Methods 2024:100754. [PMID: 38614089 DOI: 10.1016/j.crmeth.2024.100754] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/08/2023] [Revised: 01/03/2024] [Accepted: 03/20/2024] [Indexed: 04/15/2024]
Abstract
Precision medicine's emphasis on individual genetic variants highlights the importance of haplotype-resolved assembly, a computational challenge in bioinformatics given its combinatorial nature. While classical algorithms have made strides in addressing this issue, the potential of quantum computing remains largely untapped. Here, we present the vehicle routing problem (VRP) assembler: an approach that transforms this task into a vehicle routing problem, an optimization formulation solvable on a quantum computer. We demonstrate its potential and feasibility through a proof of concept on short synthetic diploid and triploid genomes using a D-Wave quantum annealer. To tackle larger-scale assembly problems, we integrate the VRP assembler with Google's OR-Tools, achieving a haplotype-resolved local assembly across the human major histocompatibility complex (MHC) region. Our results show encouraging performance compared to Hifiasm with phasing accuracy approaching the theoretical limit, underscoring the promising future of quantum computing in bioinformatics.
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Affiliation(s)
- Yibo Chen
- BGI Research, Shenzhen 518083, China
| | | | - Yuhui Sun
- BGI Research, Shenzhen 518083, China
| | - Yong Zhang
- BGI Research, Wuhan 430047, China; Guangdong Bigdata Engineering Technology Research Center for Life Sciences, BGI Research, Shenzhen 518083, China.
| | - Yuxiang Li
- BGI Research, Wuhan 430047, China; Guangdong Bigdata Engineering Technology Research Center for Life Sciences, BGI Research, Shenzhen 518083, China.
| | - Xun Xu
- BGI Research, Shenzhen 518083, China; BGI Research, Wuhan 430047, China.
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3
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Haga A. Quantum annealing-based computed tomography using variational approach for a real-number image reconstruction. Phys Med Biol 2024; 69:04NT02. [PMID: 38252994 DOI: 10.1088/1361-6560/ad2155] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/08/2023] [Accepted: 01/22/2024] [Indexed: 01/24/2024]
Abstract
Objective. Despite recent advancements in quantum computing, the limited number of available qubits has hindered progress in CT reconstruction. This study investigates the feasibility of utilizing quantum annealing-based computed tomography (QACT) with current quantum bit levels.Approach. The QACT algorithm aims to precisely solve quadratic unconstrained binary optimization problems. Furthermore, a novel approach is proposed to reconstruct images by approximating real numbers using the variational method. This approach allows for accurate CT image reconstruction using a small number of qubits. The study examines the impact of projection data quantity and noise on various image sizes ranging from 4 × 4 to 24 × 24 pixels. The reconstructed results are compared against conventional reconstruction algorithms, namely maximum likelihood expectation maximization (MLEM) and filtered back projection (FBP).Main result. By employing the variational approach and utilizing two qubits for each pixel of the image, accurate reconstruction was achieved with an adequate number of projections. Under conditions of abundant projections and lower noise levels, the image quality in QACT algorithm outperformed that of MLEM and FBP algorithms. However, in situations with limited projection data and in the presence of noise, the image quality in QACT was inferior to that in MLEM.Significance. This study developed the QACT reconstruction algorithm using the variational approach for real-number reconstruction. Remarkably, only 2 qubits were required for each pixel representation, demonstrating their sufficiency for accurate reconstruction.
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Affiliation(s)
- Akihiro Haga
- Graduate School of Biomedical Sciences, Tokushima University, Tokushima 770-8503, Japan
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4
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Nau MA, Vija AH, Gohn W, Reymann MP, Maier AK. Exploring the Limitations of Hybrid Adiabatic Quantum Computing for Emission Tomography Reconstruction. J Imaging 2023; 9:221. [PMID: 37888328 PMCID: PMC10607451 DOI: 10.3390/jimaging9100221] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/29/2023] [Revised: 10/07/2023] [Accepted: 10/08/2023] [Indexed: 10/28/2023] Open
Abstract
Our study explores the feasibility of quantum computing in emission tomography reconstruction, addressing a noisy ill-conditioned inverse problem. In current clinical practice, this is typically solved by iterative methods minimizing a L2 norm. After reviewing quantum computing principles, we propose the use of a commercially available quantum annealer and employ corresponding hybrid solvers, which combine quantum and classical computing to handle more significant problems. We demonstrate how to frame image reconstruction as a combinatorial optimization problem suited for these quantum annealers and hybrid systems. Using a toy problem, we analyze reconstructions of binary and integer-valued images with respect to their image size and compare them to conventional methods. Additionally, we test our method's performance under noise and data underdetermination. In summary, our method demonstrates competitive performance with traditional algorithms for binary images up to an image size of 32×32 on the toy problem, even under noisy and underdetermined conditions. However, scalability challenges emerge as image size and pixel bit range increase, restricting hybrid quantum computing as a practical tool for emission tomography reconstruction until significant advancements are made to address this issue.
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Affiliation(s)
- Merlin A. Nau
- Pattern Recognition Lab, Department of Computer Science, Friedrich-Alexander Universität Erlangen-Nürnberg (FAU), Martensstrasse 3, 91058 Erlangen, Germany
- Siemens Healthineers GmbH, Siemensstrasse 1, 91301 Forchheim, Germany
| | - A. Hans Vija
- Siemens Medical Solutions USA, Inc., 2501 Barrington Rd, Hoffman Estates, IL 60192, USA
| | - Wesley Gohn
- Siemens Medical Solutions USA, Inc., 2501 Barrington Rd, Hoffman Estates, IL 60192, USA
| | - Maximilian P. Reymann
- Pattern Recognition Lab, Department of Computer Science, Friedrich-Alexander Universität Erlangen-Nürnberg (FAU), Martensstrasse 3, 91058 Erlangen, Germany
- Siemens Healthineers GmbH, Siemensstrasse 1, 91301 Forchheim, Germany
| | - Andreas K. Maier
- Pattern Recognition Lab, Department of Computer Science, Friedrich-Alexander Universität Erlangen-Nürnberg (FAU), Martensstrasse 3, 91058 Erlangen, Germany
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5
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Kubacki M, Niranjan M. Quantum annealing-based clustering of single cell RNA-seq data. Brief Bioinform 2023; 24:bbad377. [PMID: 37874950 PMCID: PMC10597635 DOI: 10.1093/bib/bbad377] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/05/2022] [Revised: 07/18/2023] [Accepted: 09/14/2023] [Indexed: 10/26/2023] Open
Abstract
Cluster analysis is a crucial stage in the analysis and interpretation of single-cell gene expression (scRNA-seq) data. It is an inherently ill-posed problem whose solutions depend heavily on hyper-parameter and algorithmic choice. The popular approach of K-means clustering, for example, depends heavily on the choice of K and the convergence of the expectation-maximization algorithm to local minima of the objective. Exhaustive search of the space for multiple good quality solutions is known to be a complex problem. Here, we show that quantum computing offers a solution to exploring the cost function of clustering by quantum annealing, implemented on a quantum computing facility offered by D-Wave [1]. Out formulation extracts minimum vertex cover of an affinity graph to sub-sample the cell population and quantum annealing to optimise the cost function. A distribution of low-energy solutions can thus be extracted, offering alternate hypotheses about how genes group together in their space of expressions.
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Affiliation(s)
- Michal Kubacki
- Faculty of Engineering and Physical Sciences, University of Southampton
| | - Mahesan Niranjan
- Faculty of Engineering and Physical Sciences, University of Southampton
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6
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Miyahara H, Roychowdhury V. Quantum advantage in variational Bayes inference. Proc Natl Acad Sci U S A 2023; 120:e2212660120. [PMID: 37490536 PMCID: PMC10400996 DOI: 10.1073/pnas.2212660120] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/27/2022] [Accepted: 02/14/2023] [Indexed: 07/27/2023] Open
Abstract
Variational Bayes (VB) inference algorithm is used widely to estimate both the parameters and the unobserved hidden variables in generative statistical models. The algorithm-inspired by variational methods used in computational physics-is iterative and can get easily stuck in local minima, even when classical techniques, such as deterministic annealing (DA), are used. We study a VB inference algorithm based on a nontraditional quantum annealing approach-referred to as quantum annealing variational Bayes (QAVB) inference-and show that there is indeed a quantum advantage to QAVB over its classical counterparts. In particular, we show that such better performance is rooted in key quantum mechanics concepts: i) The ground state of the Hamiltonian of a quantum system-defined from the given data-corresponds to an optimal solution for the minimization problem of the variational free energy at very low temperatures; ii) such a ground state can be achieved by a technique paralleling the quantum annealing process; and iii) starting from this ground state, the optimal solution to the VB problem can be achieved by increasing the heat bath temperature to unity, and thereby avoiding local minima introduced by spontaneous symmetry breaking observed in classical physics based VB algorithms. We also show that the update equations of QAVB can be potentially implemented using ⌈logK⌉ qubits and 𝒪(K) operations per step, where K is the number of values hidden categorical variables can take. Thus, QAVB can match the time complexity of existing VB algorithms, while delivering higher performance.
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Affiliation(s)
- Hideyuki Miyahara
- Department of Electrical and Computer Engineering, Henry Samueli School of Engineering and Applied Science, University of California, Los Angeles, CA 90095
| | - Vwani Roychowdhury
- Department of Electrical and Computer Engineering, Henry Samueli School of Engineering and Applied Science, University of California, Los Angeles, CA 90095
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7
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Tučs A, Berenger F, Yumoto A, Tamura R, Uzawa T, Tsuda K. Quantum Annealing Designs Nonhemolytic Antimicrobial Peptides in a Discrete Latent Space. ACS Med Chem Lett 2023; 14:577-582. [PMID: 37197452 PMCID: PMC10184305 DOI: 10.1021/acsmedchemlett.2c00487] [Citation(s) in RCA: 2] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/17/2022] [Accepted: 04/10/2023] [Indexed: 05/19/2023] Open
Abstract
Increasing the variety of antimicrobial peptides is crucial in meeting the global challenge of multi-drug-resistant bacterial pathogens. While several deep-learning-based peptide design pipelines are reported, they may not be optimal in data efficiency. High efficiency requires a well-compressed latent space, where optimization is likely to fail due to numerous local minima. We present a multi-objective peptide design pipeline based on a discrete latent space and D-Wave quantum annealer with the aim of solving the local minima problem. To achieve multi-objective optimization, multiple peptide properties are encoded into a score using non-dominated sorting. Our pipeline is applied to design therapeutic peptides that are antimicrobial and non-hemolytic at the same time. From 200 000 peptides designed by our pipeline, four peptides proceeded to wet-lab validation. Three of them showed high anti-microbial activity, and two are non-hemolytic. Our results demonstrate how quantum-based optimizers can be taken advantage of in real-world medical studies.
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Affiliation(s)
- Andrejs Tučs
- Graduate
School of Frontier Sciences, The University
of Tokyo, 5-1-5 Kashiwa-no-ha, Kashiwa, Chiba 277-8561, Japan
| | - Francois Berenger
- Graduate
School of Frontier Sciences, The University
of Tokyo, 5-1-5 Kashiwa-no-ha, Kashiwa, Chiba 277-8561, Japan
| | - Akiko Yumoto
- Emergent
Bioengineering Materials Research Team, RIKEN Center for Emergent Matter Science, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan
| | - Ryo Tamura
- Graduate
School of Frontier Sciences, The University
of Tokyo, 5-1-5 Kashiwa-no-ha, Kashiwa, Chiba 277-8561, Japan
- International
Center for Materials Nanoarchitectonics (WPI-MANA), National Institute for Materials Science (NIMS), Tsukuba 305−0044, Japan
- Research
and Services Division of Materials Data and Integrated System, National Institute for Materials Science (NIMS), Tsukuba 305-0044, Japan
- RIKEN
Center for Advanced Intelligence Project, RIKEN, 1-4-1 Nihombashi, Chuo-ku, Tokyo 103-0027, Japan
| | - Takanori Uzawa
- Emergent
Bioengineering Materials Research Team, RIKEN Center for Emergent Matter Science, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan
- Nano Medical
Engineering Laboratory, RIKEN Cluster for
Pioneering Research, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan
- E-mail:
| | - Koji Tsuda
- Graduate
School of Frontier Sciences, The University
of Tokyo, 5-1-5 Kashiwa-no-ha, Kashiwa, Chiba 277-8561, Japan
- Research
and Services Division of Materials Data and Integrated System, National Institute for Materials Science (NIMS), Tsukuba 305-0044, Japan
- RIKEN
Center for Advanced Intelligence Project, RIKEN, 1-4-1 Nihombashi, Chuo-ku, Tokyo 103-0027, Japan
- E-mail:
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8
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Sampei H, Saegusa K, Chishima K, Higo T, Tanaka S, Yayama Y, Nakamura M, Kimura K, Sekine Y. Quantum Annealing Boosts Prediction of Multimolecular Adsorption on Solid Surfaces Avoiding Combinatorial Explosion. JACS Au 2023; 3:991-996. [PMID: 37124301 PMCID: PMC10131206 DOI: 10.1021/jacsau.3c00018] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 01/09/2023] [Revised: 03/11/2023] [Accepted: 03/17/2023] [Indexed: 05/03/2023]
Abstract
Quantum annealing has been used to predict molecular adsorption on solid surfaces. Evaluation of adsorption, which takes place in all solid surface reactions, is a crucially important subject for study in various fields. However, predicting the most stable coordination by theoretical calculations is challenging for multimolecular adsorption because there are numerous candidates. This report presents a novel method for quick adsorption coordination searches using the quantum annealing principle without combinatorial explosion. This method exhibited much faster search and more stable molecular arrangement findings than conventional methods did, particularly in a high coverage region. We were able to complete a configurational prediction of the adsorption of 16 molecules in 2286 s (including 2154 s for preparation, only required once), whereas previously it has taken 38 601 s. This approach accelerates the tuning of adsorption behavior, especially in composite materials and large-scale modeling, which possess more combinations of molecular configurations.
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Affiliation(s)
- Hiroshi Sampei
- Department
of Applied Chemistry, Waseda University, Tokyo 169-8555, Japan
| | - Koki Saegusa
- Department
of Applied Chemistry, Waseda University, Tokyo 169-8555, Japan
| | - Kenshin Chishima
- Department
of Applied Chemistry, Waseda University, Tokyo 169-8555, Japan
| | - Takuma Higo
- Department
of Applied Chemistry, Waseda University, Tokyo 169-8555, Japan
| | - Shu Tanaka
- Department
of Applied Physics and Physico-Informatics, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan
- Green
Computing System Research Organization, Waseda University, Wasedamachi-27,
Shinjuku-ku, Tokyo 162-0042, Japan
| | - Yoshihiro Yayama
- Central
Technical Research Laboratory, ENEOS Corporation, 231-0815, 8 Chidoricho, Naka-ku, Yokohama, Kanagawa 100-8162, Japan
| | - Makoto Nakamura
- Quantum
Research Center, Fujitsu Ltd., 4-1-1 Kamiodanaka, Kawasaki, Kanagawa 211-8588, Japan
| | - Koichi Kimura
- Quantum
Research Center, Fujitsu Ltd., 4-1-1 Kamiodanaka, Kawasaki, Kanagawa 211-8588, Japan
| | - Yasushi Sekine
- Department
of Applied Chemistry, Waseda University, Tokyo 169-8555, Japan
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9
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Xu H, Dasgupta S, Pothen A, Banerjee A. Dynamic Asset Allocation with Expected Shortfall via Quantum Annealing. Entropy (Basel) 2023; 25:541. [PMID: 36981429 PMCID: PMC10047987 DOI: 10.3390/e25030541] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 02/02/2023] [Revised: 03/01/2023] [Accepted: 03/17/2023] [Indexed: 06/18/2023]
Abstract
Recent advances in quantum hardware offer new approaches to solve various optimization problems that can be computationally expensive when classical algorithms are employed. We propose a hybrid quantum-classical algorithm to solve a dynamic asset allocation problem where a target return and a target risk metric (expected shortfall) are specified. We propose an iterative algorithm that treats the target return as a constraint in a Markowitz portfolio optimization model, and dynamically adjusts the target return to satisfy the targeted expected shortfall. The Markowitz optimization is formulated as a Quadratic Unconstrained Binary Optimization (QUBO) problem. The use of the expected shortfall risk metric enables the modeling of extreme market events. We compare the results from D-Wave's 2000Q and Advantage quantum annealers using real-world financial data. Both quantum annealers are able to generate portfolios with more than 80% of the return of the classical optimal solutions, while satisfying the expected shortfall. We observe that experiments on assets with higher correlations tend to perform better, which may help to design practical quantum applications in the near term.
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Affiliation(s)
- Hanjing Xu
- Department of Computer Science, Purdue University, West Lafayette, IN 47906, USA; (H.X.)
| | - Samudra Dasgupta
- Department of Physics, Purdue University, West Lafayette, IN 47906, USA
- Oak Ridge National Laboratory, Quantum Computing Institute, Oak Ridge, TN 37831, USA
- Bredesen Center, University of Tennessee, Knoxville, TN 37996, USA
| | - Alex Pothen
- Department of Computer Science, Purdue University, West Lafayette, IN 47906, USA; (H.X.)
| | - Arnab Banerjee
- Department of Physics, Purdue University, West Lafayette, IN 47906, USA
- Oak Ridge National Laboratory, Quantum Computing Institute, Oak Ridge, TN 37831, USA
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10
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Sinha A. Development of research network on Quantum Annealing Computation and Information using Google Scholar data. Philos Trans A Math Phys Eng Sci 2023; 381:20210413. [PMID: 36463919 DOI: 10.1098/rsta.2021.0413] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/05/2022] [Accepted: 09/08/2022] [Indexed: 06/17/2023]
Abstract
We build and analyse the network of 100 top-cited nodes (research papers and books from Google Scholar; the strength or citation of the nodes range from about 44 000 up to 100) starting in early 1980 until last year. These searched publications (papers and books) are based on Quantum Annealing Computation and Information categorized into four different sets: (A) Quantum/Transverse Field Spin Glass Model, (B) Quantum Annealing, (C) Quantum Adiabatic Computation and (D) Quantum Computation Information in the title or abstract of the searched publications. We fitted the growth in the annual number of publication ([Formula: see text]) in each of these four categories, A-D, to the form [Formula: see text] where [Formula: see text] denotes the time in years. We found the scaling time [Formula: see text] to be of the order of about 10 years for categories A and C, whereas [Formula: see text] is of the order of about 5 years for categories B and D. This article is part of the theme issue 'Quantum annealing and computation: challenges and perspectives'.
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Affiliation(s)
- Antika Sinha
- Department of Computer Science, Asutosh College, Kolkata, West Bengal 700026, India
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11
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Mukherjee S. Role of quantum fluctuation in inducing ergodicity in the spin glass phase and its effect in quantum annealing. Philos Trans A Math Phys Eng Sci 2023; 381:20210406. [PMID: 36463924 DOI: 10.1098/rsta.2021.0406] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/16/2022] [Accepted: 09/30/2022] [Indexed: 06/17/2023]
Abstract
We review the numerical studies on the critical behaviour of the quantum Sherrington-Kirkpatrick (SK) spin glass model, which indicate that a quantum critical behaviour is observed up to a low but non-zero value of temperature. We revisit the numerical investigations on the spin glass order parameter distributions, which identify a low temperature along with high transverse field spin glass phase where the order parameter distribution becomes a delta function in the thermodynamic limit indicating the restoration of replica symmetry and ergodic nature of the system. In the remaining spin glass phase associated with high temperature and low transverse field, the observed distribution is broad akin to the Parisi order parameter distribution. This essentially indicates the non-ergodic behaviour of the system. We further discuss the annealing dynamics studies on the quantum SK model. Such investigations reveal the system size independence of annealing time when the annealing paths go through the ergodic spin glass region. Interestingly, when such dynamics are performed in the non-ergodic spin glass phase the annealing time becomes an increasing function of the system size. Spin autocorrelation shows faster relaxation in the ergodic spin glass region compared with that found in the non-ergodic spin glass region. This article is part of the theme issue 'Quantum annealing and computation: challenges and perspectives'.
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Affiliation(s)
- Sudip Mukherjee
- Department of Physics, Barasat Government College, Kolkata 700124, India
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12
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Suzuki S, Oshiyama H, Shibata N. Statistics of the number of defects after quantum annealing in a thermal environment. Philos Trans A Math Phys Eng Sci 2023; 381:20210411. [PMID: 36463929 DOI: 10.1098/rsta.2021.0411] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/25/2022] [Accepted: 09/30/2022] [Indexed: 06/17/2023]
Abstract
We study the statistics of the kink number generated by quantum annealing in a one-dimensional transverse Ising model coupled to a bosonic thermal bath. Using the freezing ansatz for quantum annealing in the thermal environment, we show the relation between the ratio of the second to the first cumulant of the kink number distribution and the average kink density. The theoretical result is confirmed thoroughly by numerical simulation using the non-Markovian infinite time-evolving block decimation which we proposed recently. The simulation using D-Wave's quantum annealer is also discussed. This article is part of the theme issue 'Quantum annealing and computation: challenges and perspectives'.
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Affiliation(s)
- Sei Suzuki
- Department of Liberal Arts, Saitama Medical University, Moroyama, Saitama, Japan
| | - Hiroki Oshiyama
- Department of Information Sciences, Tohoku University, Sendai, Japan
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13
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Kadowaki T, Nishimori H. Greedy parameter optimization for diabatic quantum annealing. Philos Trans A Math Phys Eng Sci 2023; 381:20210416. [PMID: 36463922 PMCID: PMC9719795 DOI: 10.1098/rsta.2021.0416] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/12/2023]
Abstract
A shorter processing time is desirable for quantum computation to minimize the effects of noise. We propose a simple procedure to variationally determine a set of parameters in the transverse-field Ising model for quantum annealing (QA) appended with a field along the [Formula: see text]-axis. The method consists of greedy optimization of the signs of coefficients of the [Formula: see text]-field term based on the outputs of short annealing processes. We test the idea in the ferromagnetic system with all-to-all couplings and spin-glass problems, and find that the method outperforms the traditional form of QA and simulated annealing in terms of the success probability and the time to solution, in particular, in the case of shorter annealing times, achieving the goal of improved performance while avoiding noise. The non-stoquastic [Formula: see text] term can be eliminated by a rotation in the spin space, resulting in a non-trivial diabatic control of the coefficients in the stoquastic transverse-field Ising model, which may be feasible for experimental realization. This article is part of the theme issue 'Quantum annealing and computation: challenges and perspectives'.
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Affiliation(s)
| | - Hidetoshi Nishimori
- Institute of Innovative Research, Tokyo Institute of Technology, Yokohama, Kanagawa 226-8503, Japan
- Graduate School of Information Sciences, Tohoku University, Sendai, Miyagi 980-8579, Japan
- RIKEN Interdisciplinary Theoretical and Mathematical Sciences Program (iTHEMS), Wako, Saitama 351-0198, Japan
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14
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Okuyama M, Ohki K, Ohzeki M. Threshold theorem in isolated quantum dynamics with stochastic control errors. Philos Trans A Math Phys Eng Sci 2023; 381:20210412. [PMID: 36463918 PMCID: PMC9719793 DOI: 10.1098/rsta.2021.0412] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/17/2022] [Accepted: 10/06/2022] [Indexed: 06/17/2023]
Abstract
We investigate the effect of stochastic control errors in the time-dependent Hamiltonian on isolated quantum dynamics. The control errors are formulated as time-dependent stochastic noise in the Schrödinger equation. For a class of stochastic control errors, we establish a threshold theorem that provides a sufficient condition to obtain the target state, which should be determined in noiseless isolated quantum dynamics, as a relation between the number of measurements and noise strength. The theorem guarantees that if the sum of the noise strengths is less than the inverse of computational time, the target state can be obtained through a constant-order number of measurements. If the opposite is true, the number of measurements to guarantee obtaining the target state increases exponentially with computational time. Our threshold theorem can be applied to any isolated quantum dynamics such as quantum annealing and adiabatic quantum computation. This article is part of the theme issue 'Quantum annealing and computation: challenges and perspectives'.
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Affiliation(s)
- Manaka Okuyama
- Graduate School of Information Sciences, Tohoku University, Sendai 980-8579, Japan
| | - Kentaro Ohki
- Graduate School of Informatics, Kyoto University, Kyoto 606-8501, Japan
| | - Masayuki Ohzeki
- Graduate School of Information Sciences, Tohoku University, Sendai 980-8579, Japan
- Department of Physics, Tokyo Institute of Technology, Oh-okayama, Meguro-ku, Tokyo 152-8551, Japan
- Sigma-i Co., Ltd., Tokyo 108-0075, Japan
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15
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Mozgunov E, Lidar DA. Quantum adiabatic theorem for unbounded Hamiltonians with a cutoff and its application to superconducting circuits. Philos Trans A Math Phys Eng Sci 2023; 381:20210407. [PMID: 36463925 PMCID: PMC9719797 DOI: 10.1098/rsta.2021.0407] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 05/01/2022] [Accepted: 07/18/2022] [Indexed: 06/17/2023]
Abstract
We present a new quantum adiabatic theorem that allows one to rigorously bound the adiabatic timescale for a variety of systems, including those described by originally unbounded Hamiltonians that are made finite-dimensional by a cutoff. Our bound is geared towards the qubit approximation of superconducting circuits and presents a sufficient condition for remaining within the [Formula: see text]-dimensional qubit subspace of a circuit model of [Formula: see text] qubits. The novelty of this adiabatic theorem is that, unlike previous rigorous results, it does not contain [Formula: see text] as a factor in the adiabatic timescale, and it allows one to obtain an expression for the adiabatic timescale independent of the cutoff of the infinite-dimensional Hilbert space of the circuit Hamiltonian. As an application, we present an explicit dependence of this timescale on circuit parameters for a superconducting flux qubit and demonstrate that leakage out of the qubit subspace is inevitable as the tunnelling barrier is raised towards the end of a quantum anneal. We also discuss a method of obtaining a [Formula: see text] effective Hamiltonian that best approximates the true dynamics induced by slowly changing circuit control parameters. This article is part of the theme issue 'Quantum annealing and computation: challenges and perspectives'.
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Affiliation(s)
- Evgeny Mozgunov
- Center for Quantum Information Science and Technology, University of Southern California, Los Angeles, CA 90089, USA
| | - Daniel A. Lidar
- Center for Quantum Information Science and Technology, University of Southern California, Los Angeles, CA 90089, USA
- Department of Electrical and Computer Engineering, University of Southern California, Los Angeles, CA 90089, USA
- Department of Physics and Astronomy, University of Southern California, Los Angeles, CA 90089, USA
- Department of Chemistry, University of Southern California, Los Angeles, CA 90089, USA
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16
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Berwald J, Chancellor N, Dridi R. Understanding domain-wall encoding theoretically and experimentally. Philos Trans A Math Phys Eng Sci 2023; 381:20210410. [PMID: 36463928 DOI: 10.1098/rsta.2021.0410] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/13/2022] [Accepted: 09/14/2022] [Indexed: 06/17/2023]
Abstract
We analyse the method of encoding pairwise interactions of higher-than-binary discrete variables (these models are sometimes referred to as discrete quadratic models) into binary variables based on domain walls on one-dimensional Ising chains. We discuss how this is relevant to quantum annealing, but also many gate model algorithms such as VQE and QAOA. We theoretically show that for problems of practical interest for quantum computing and assuming only quadratic interactions are available between the binary variables, it is not possible to have a more efficient general encoding in terms of number of binary variables per discrete variable. We furthermore use a D-Wave Advantage 1.1 flux qubit quantum annealing computer to show that the dynamics effectively freeze later for a domain-wall encoding compared with a traditional one-hot encoding. This second result could help explain the dramatic performance improvement of domain wall over one-hot, which has been seen in a recent experiment on D-Wave hardware. This is an important result because usually problem encoding and the underlying physics are considered separately, our work suggests that considering them together may be a more useful paradigm. We argue that this experimental result is also likely to carry over to a number of other settings, we discuss how this has implications for gate-model and quantum-inspired algorithms. This article is part of the theme issue 'Quantum annealing and computation: challenges and perspectives'.
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Affiliation(s)
| | - Nicholas Chancellor
- Quantum Computing Inc. Leesburg, VA, USA
- Physics Department, Durham University Physics department Quantum Light and Matter section and Durham-Newcastle Joint Quantum Centre, South Road, Durham, UK
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17
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Domino K, Koniorczyk M, Krawiec K, Jałowiecki K, Deffner S, Gardas B. Quantum Annealing in the NISQ Era: Railway Conflict Management. Entropy (Basel) 2023; 25:191. [PMID: 36832558 PMCID: PMC9955039 DOI: 10.3390/e25020191] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 11/30/2022] [Revised: 01/11/2023] [Accepted: 01/15/2023] [Indexed: 06/18/2023]
Abstract
We are in the noisy intermediate-scale quantum (NISQ) devices' era, in which quantum hardware has become available for application in real-world problems. However, demonstrations of the usefulness of such NISQ devices are still rare. In this work, we consider a practical railway dispatching problem: delay and conflict management on single-track railway lines. We examine the train dispatching consequences of the arrival of an already delayed train to a given network segment. This problem is computationally hard and needs to be solved almost in real time. We introduce a quadratic unconstrained binary optimization (QUBO) model of this problem, which is compatible with the emerging quantum annealing technology. The model's instances can be executed on present-day quantum annealers. As a proof-of-concept, we solve selected real-life problems from the Polish railway network using D-Wave quantum annealers. As a reference, we also provide solutions calculated with classical methods, including the conventional solution of a linear integer version of the model as well as the solution of the QUBO model using a tensor network-based algorithm. Our preliminary results illustrate the degree of difficulty of real-life railway instances for the current quantum annealing technology. Moreover, our analysis shows that the new generation of quantum annealers (the advantage system) does not perform well on those instances, either.
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Affiliation(s)
- Krzysztof Domino
- Institute of Theoretical and Applied Informatics, Polish Academy of Sciences, Bałtycka 5, 44-100 Gliwice, Poland
| | - Mátyás Koniorczyk
- Wigner Research Centre, Konkoly-Thege M. út 29-33, H-1525 Budapest, Hungary
| | - Krzysztof Krawiec
- Faculty of Transport and Aviation Engineering, Silesian University of Technology, 40-019 Katowice, Poland
| | | | - Sebastian Deffner
- Department of Physics, University of Maryland, Baltimore County, Baltimore, MD 21250, USA
- Instituto de Física ‘Gleb Wataghin’, Universidade Estadual de Campinas, Campinas 13083-859, SP, Brazil
| | - Bartłomiej Gardas
- Institute of Theoretical and Applied Informatics, Polish Academy of Sciences, Bałtycka 5, 44-100 Gliwice, Poland
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18
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Yarkoni S, Raponi E, Bäck T, Schmitt S. Quantum annealing for industry applications: introduction and review. Rep Prog Phys 2022; 85:104001. [PMID: 36001953 DOI: 10.1088/1361-6633/ac8c54] [Citation(s) in RCA: 13] [Impact Index Per Article: 6.5] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/20/2021] [Accepted: 08/24/2022] [Indexed: 06/15/2023]
Abstract
Quantum annealing (QA) is a heuristic quantum optimization algorithm that can be used to solve combinatorial optimization problems. In recent years, advances in quantum technologies have enabled the development of small- and intermediate-scale quantum processors that implement the QA algorithm for programmable use. Specifically, QA processors produced by D-Wave systems have been studied and tested extensively in both research and industrial settings across different disciplines. In this paper we provide a literature review of the theoretical motivations for QA as a heuristic quantum optimization algorithm, the software and hardware that is required to use such quantum processors, and the state-of-the-art applications and proofs-of-concepts that have been demonstrated using them. The goal of our review is to provide a centralized and condensed source regarding applications of QA technology. We identify the advantages, limitations, and potential of QA for both researchers and practitioners from various fields.
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Nembrini R, Ferrari Dacrema M, Cremonesi P. Feature Selection for Recommender Systems with Quantum Computing. Entropy (Basel) 2021; 23:970. [PMID: 34441110 DOI: 10.3390/e23080970] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 06/30/2021] [Revised: 07/19/2021] [Accepted: 07/22/2021] [Indexed: 11/16/2022]
Abstract
The promise of quantum computing to open new unexplored possibilities in several scientific fields has been long discussed, but until recently the lack of a functional quantum computer has confined this discussion mostly to theoretical algorithmic papers. It was only in the last few years that small but functional quantum computers have become available to the broader research community. One paradigm in particular, quantum annealing, can be used to sample optimal solutions for a number of NP-hard optimization problems represented with classical operations research tools, providing an easy access to the potential of this emerging technology. One of the tasks that most naturally fits in this mathematical formulation is feature selection. In this paper, we investigate how to design a hybrid feature selection algorithm for recommender systems that leverages the domain knowledge and behavior hidden in the user interactions data. We represent the feature selection as an optimization problem and solve it on a real quantum computer, provided by D-Wave. The results indicate that the proposed approach is effective in selecting a limited set of important features and that quantum computers are becoming powerful enough to enter the wider realm of applied science.
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20
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Liu J, Yao KT, Spedalieri F. Dynamic Topology Reconfiguration of Boltzmann Machines on Quantum Annealers. Entropy (Basel) 2020; 22:E1202. [PMID: 33286970 DOI: 10.3390/e22111202] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [What about the content of this article? (0)] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 09/27/2020] [Revised: 10/16/2020] [Accepted: 10/21/2020] [Indexed: 11/18/2022]
Abstract
Boltzmann machines have useful roles in deep learning applications, such as generative data modeling, initializing weights for other types of networks, or extracting efficient representations from high-dimensional data. Most Boltzmann machines use restricted topologies that exclude looping connectivity, as such connectivity creates complex distributions that are difficult to sample. We have used an open-system quantum annealer to sample from complex distributions and implement Boltzmann machines with looping connectivity. Further, we have created policies mapping Boltzmann machine variables to the quantum bits of an annealer. These policies, based on correlation and entropy metrics, dynamically reconfigure the topology of Boltzmann machines during training and improve performance.
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21
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Cruz-Santos W, Venegas-Andraca SE, Lanzagorta M. A QUBO Formulation of the Stereo Matching Problem for D-Wave Quantum Annealers. Entropy (Basel) 2018; 20:e20100786. [PMID: 33265874 PMCID: PMC7512348 DOI: 10.3390/e20100786] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 08/06/2018] [Revised: 10/08/2018] [Accepted: 10/08/2018] [Indexed: 11/16/2022]
Abstract
In this paper, we propose a methodology to solve the stereo matching problem through quantum annealing optimization. Our proposal takes advantage of the existing Min-Cut/Max-Flow network formulation of computer vision problems. Based on this network formulation, we construct a quadratic pseudo-Boolean function and then optimize it through the use of the D-Wave quantum annealing technology. Experimental validation using two kinds of stereo pair of images, random dot stereograms and gray-scale, shows that our methodology is effective.
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Affiliation(s)
- William Cruz-Santos
- CU-UAEM Valle de Chalco, Hermenegildo Galeana 3, Valle de Chalco 56615, Estado de México, Mexico
| | - Salvador E. Venegas-Andraca
- Tecnologico de Monterrey, Escuela de Ingenieria y Ciencias. Ave., Eugenio Garza Sada 2501, Monterrey 64849, NL, Mexico
- Correspondence: or ; Tel.: +52-55-5864-5555
| | - Marco Lanzagorta
- US Naval Research Laboratory, 4555 Overlook Ave., SW Washington, DC 20375, USA
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22
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Suksmono AB. Finding a Hadamard Matrix by Simulated Quantum Annealing. Entropy (Basel) 2018; 20:e20020141. [PMID: 33265232 PMCID: PMC7512635 DOI: 10.3390/e20020141] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 01/02/2018] [Revised: 02/06/2018] [Accepted: 02/16/2018] [Indexed: 11/16/2022]
Abstract
Hard problems have recently become an important issue in computing. Various methods, including a heuristic approach that is inspired by physical phenomena, are being explored. In this paper, we propose the use of simulated quantum annealing (SQA) to find a Hadamard matrix, which is itself a hard problem. We reformulate the problem as an energy minimization of spin vectors connected by a complete graph. The computation is conducted based on a path-integral Monte-Carlo (PIMC) SQA of the spin vector system, with an applied transverse magnetic field whose strength is decreased over time. In the numerical experiments, the proposed method is employed to find low-order Hadamard matrices, including the ones that cannot be constructed trivially by the Sylvester method. The scaling property of the method and the measurement of residual energy after a sufficiently large number of iterations show that SQA outperforms simulated annealing (SA) in solving this hard problem.
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Affiliation(s)
- Andriyan Bayu Suksmono
- Telecommunication Engineering Scientific and Research Group (TESRG), School of Electrical Engineering and Informatics and The Research Center on Information and Communication Technology (PPTIK-ITB), Institut Teknologi Bandung, Jl. Ganesha No.10, Bandung 40132, Indonesia
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23
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Abstract
Quantum annealers aim at solving nonconvex optimization problems by exploiting cooperative tunneling effects to escape local minima. The underlying idea consists of designing a classical energy function whose ground states are the sought optimal solutions of the original optimization problem and add a controllable quantum transverse field to generate tunneling processes. A key challenge is to identify classes of nonconvex optimization problems for which quantum annealing remains efficient while thermal annealing fails. We show that this happens for a wide class of problems which are central to machine learning. Their energy landscapes are dominated by local minima that cause exponential slowdown of classical thermal annealers while simulated quantum annealing converges efficiently to rare dense regions of optimal solutions.
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24
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Mossi G, Scardicchio A. Ergodic and localized regions in quantum spin glasses on the Bethe lattice. Philos Trans A Math Phys Eng Sci 2017; 375:20160424. [PMID: 29084889 PMCID: PMC5665780 DOI: 10.1098/rsta.2016.0424] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Accepted: 06/19/2017] [Indexed: 06/07/2023]
Abstract
By considering the quantum dynamics of a transverse-field Ising spin glass on the Bethe lattice, we find the existence of a many-body localized (MBL) region at small transverse field and low temperature. The region is located within the thermodynamic spin glass phase. Accordingly, we conjecture that quantum dynamics inside the glassy region is split into a small MBL region and a large delocalized (but not necessarily ergodic) region. This has implications for the analysis of the performance of quantum adiabatic algorithms.This article is part of the themed issue 'Breakdown of ergodicity in quantum systems: from solids to synthetic matter'.
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Affiliation(s)
- G Mossi
- SISSA, Via Bonomea 265, 34136 Trieste, Italy
- INFN, Sezione di Trieste, Via Valerio 2, 34127 Trieste, Italy
| | - A Scardicchio
- INFN, Sezione di Trieste, Via Valerio 2, 34127 Trieste, Italy
- Abdus Salam ICTP Trieste, Strada Costiera 11, 34151 Trieste, Italy
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25
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Liu L, Chen L, Jia H. Social Milieu Oriented Routing: A New Dimension to Enhance Network Security in WSNs. Sensors (Basel) 2016; 16:247. [PMID: 26907277 DOI: 10.3390/s16020247] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [What about the content of this article? (0)] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 01/03/2016] [Revised: 02/14/2016] [Accepted: 02/16/2016] [Indexed: 11/17/2022]
Abstract
In large-scale wireless sensor networks (WSNs), in order to enhance network security, it is crucial for a trustor node to perform social milieu oriented routing to a target a trustee node to carry out trust evaluation. This challenging social milieu oriented routing with more than one end-to-end Quality of Trust (QoT) constraint has proved to be NP-complete. Heuristic algorithms with polynomial and pseudo-polynomial-time complexities are often used to deal with this challenging problem. However, existing solutions cannot guarantee the efficiency of searching; that is, they can hardly avoid obtaining partial optimal solutions during a searching process. Quantum annealing (QA) uses delocalization and tunneling to avoid falling into local minima without sacrificing execution time. This has been proven a promising way to many optimization problems in recently published literatures. In this paper, for the first time, with the help of a novel approach, that is, configuration path-integral Monte Carlo (CPIMC) simulations, a QA-based optimal social trust path (QA_OSTP) selection algorithm is applied to the extraction of the optimal social trust path in large-scale WSNs. Extensive experiments have been conducted, and the experiment results demonstrate that QA_OSTP outperforms its heuristic opponents.
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26
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Schmidt MA, Silevitch DM, Aeppli G, Rosenbaum TF. Using thermal boundary conditions to engineer the quantum state of a bulk magnet. Proc Natl Acad Sci U S A 2014; 111:3689-94. [PMID: 24567389 DOI: 10.1073/pnas.1316070111] [Citation(s) in RCA: 14] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [What about the content of this article? (0)] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022] Open
Abstract
The degree of contact between a system and the external environment can alter dramatically its proclivity to quantum mechanical modes of relaxation. We show that controlling the thermal coupling of cubic-centimeter-sized crystals of the Ising magnet LiHo(x)Y(1-x)F4 to a heat bath can be used to tune the system between a glassy state dominated by thermal excitations over energy barriers and a state with the hallmarks of a quantum spin liquid. Application of a magnetic field transverse to the Ising axis introduces both random magnetic fields and quantum fluctuations, which can retard and speed the annealing process, respectively, thereby providing a mechanism for continuous tuning between the destination states. The nonlinear response of the system explicitly demonstrates quantum interference between internal and external relaxation pathways.
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