Tomography and generative training with quantum Boltzmann machines
Open accessAbstract
The promise of quantum neural nets, which utilize quantum effects to model complex data sets, has made their development an aspirational goal for quantum machine learning and quantum computing in general. Here we provide methods of training quantum Boltzmann machines. Our work generalizes existing methods and provides additional approaches for training quantum neural networks that compare favorably to existing methods. We further demonstrate that quantum Boltzmann machines enable a form of partial quantum state tomography that further provides a generative model for the input quantum state. Classical Boltzmann machines are incapable of this. This verifies the long-conjectured connection between tomography and quantum machine learning. Finally, we prove that classical computers cannot simulate our training process in general unless $\text{BQP}=\text{BPP}$, provide lower bounds on the complexity of the training procedures and numerically investigate training for small nonstoquastic Hamiltonians.
The claims and statistics of this paper haven’t been checked yet.
Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.
Integrity signals
Registry
Cites
Statistics
Claims the paper makes
The claims and statistics of this paper haven’t been checked yet.
Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.
Claims other papers make about this one
Each is checked against this paper’s own text: does it say what the citing paper says it does?
A QBM outperformed classical Boltzmann machines in generative training to reproduce small Haar-random states.
“(Kieferová and Wiebe, 2017) found that a QBMoutperformed classical Boltzmann machines in generative training to reproduce small Haar-random states.”
From Noisy intermediate-scale quantum algorithms (2022) · cites this paper as Kieferová and Wiebe, 2017
The Numerical Results (p. 4) report that the quantum model fit the generative-training data better than the classical model, but that comparison used noisy step-function data. The tomography results (p. 5) show that the QBM learned two-qubit Haar-random states; they do not compare its performance on those states with classical Boltzmann machines. The claim combines findings from two different tests. (LLM, conf=0.90)
Experimental — accuracy not yet measuredLearning probabilistic generative models is in many cases an intractable task.
“For example, learning probabilistic generative models is in many cases an intractable task, and quantum-assisted algorithms have been proposed for both gate-based 17,18 and quantum annealing devices.[19][20][21][22][23]”
From A generative modeling approach for benchmarking and training shallow quantum circuits (2019) · cites this paper as [18]
Self-check
Percentages, sums and reported differences re-computed from the paper’s own numbers.
The self-check needs the paper’s full text.
Cites
Training Products of Experts by Minimizing Contrastive Divergence
Hinton, G. E. (2002) · Neural Computation · doi:10.1162/089976602760128018 · Cited by 4,879 · cited 2× · see citing works
“Classical algorithms circumvent this problem by approximating it using contrastive divergence [1].”
BackgroundQuantum Support Vector Machine for Big Data Classification
Rebentrost, P., Mohseni, M., & Lloyd, S. (2014) · Physical Review Letters · doi:10.1103/physrevlett.113.130503 · Cited by 1,951 · see citing works
Simulation of electronic structure Hamiltonians using quantum computers
Whitfield, J. D., Biamonte, J., & Aspuru-Guzik, A. (2011) · Molecular Physics · doi:10.1080/00268976.2011.552441 · Cited by 472 · see citing works
Benedetti, M., Realpe-Gómez, J., Biswas, R., & Perdomo-Ortiz, A. (2016) · Physical Review A · doi:10.1103/physreva.94.022308 · Cited by 215 · see citing works
Poulin, D., & Wocjan, P. (2009) · Physical Review Letters · doi:10.1103/physrevlett.103.220502 · Cited by 201 · see citing works
Sample-optimal tomography of quantum states
Haah, J., Harrow, A. W., Ji, Z., Wu, X., & Yu, N. (2017) · IEEE Transactions on Information Theory · doi:10.1109/tit.2017.2719044 · Cited by 194 · see citing works
Application of Quantum Annealing to Training of Deep Neural Networks
Adachi, S. H., & Henderson, M. P. (2015) · arXiv · doi:10.48550/arxiv.1510.06356 · Cited by 176 · see citing works
A quantum–quantum Metropolis algorithm
Yung, M., & Aspuru‐Guzik, A. (2012) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.1111758109 · Cited by 147 · see citing works
Quantum adiabatic machine learning
Pudenz, K. L., & Lidar, D. A. (2012) · Quantum Information Processing · doi:10.1007/s11128-012-0506-4 · Cited by 139 · see citing works
Sample-optimal tomography of quantum states
Haah, J., Harrow, A. W., Ji, Z., Wu, X., & Yu, N. (2016) · ACM Symposium on Theory of Computing (STOC) · doi:10.1145/2897518.2897585 · Cited by 89 · see citing works
Quantum algorithms for Gibbs sampling and hitting-time estimation
Chowdhury, A. C. N., & Somma, R. D. (2017) · Quantum Information and Computation · doi:10.26421/qic17.1-2-3 · Cited by 76 · see citing works
Advances in Artificial Intelligence
Kégl, B., & Lapalme, G. (2005) · Lecture notes in computer science · doi:10.1007/b136277 · Cited by 41 · see citing works
Inductive Supervised Quantum Learning
Monràs, A., Sentís, G., & Wittek, P. (2017) · Physical Review Letters · doi:10.1103/physrevlett.118.190503 · Cited by 39 · see citing works
Toward the Implementation of a Quantum RBM
de Freitas, N. (2011) · Cited by 34 · see citing works
Quantum Inspired Training for Boltzmann MachinesPreprint
Wiebe, N., Kapoor, A., Granade, C., & Svore, K. M. (2015) · arXiv (Cornell University) · doi:10.48550/arxiv.1507.02642 · Cited by 21 · see citing works
Self-citationKulchytskyy, B., Andriyash, E., Amin, M., & Melko, R. G. (2016) · Bulletin of the American Physical Society · Cited by 21 · see citing works
Flagged first, then by citations
Research works printed but not linked: 2
- [4]Marcello Benedetti, John Realpe-G´ omez, Rupak Biswas, and Alejandro Perdomo-Ortiz. Estimation of effective temperatures in a quantum annealer and its impact in sampling applications: A case study towards deep learning applications. arXiv preprint arXiv:1510.07611 , 2015. · arXiv
- [9]Man-Hong Yung and Al´ an Aspuru-Guzik. A quantumquantum metropolis algorithm. Proceedings of the National Academy of Sciences , 109(3):754-759, 2012.
Cited by
Sorted by citations
Biamonte, J., Wittek, P., Pancotti, N., Rebentrost, P., Wiebe, N., & Lloyd, S. (2017) · Nature · doi:10.1038/nature23474 · Cited by 4,779 · see citing works
Self-citationNoisy intermediate-scale quantum algorithms
Bharti, K., Cervera-Lierta, A., Kyaw, T. H., Haug, T., Alperin-Lea, S., Anand, A., Degroote, M., Heimonen, H., Kottmann, J. S., Menke, T., et al. (2022) · Reviews of Modern Physics · doi:10.1103/revmodphys.94.015004 · Cited by 1,726 · see citing works
A Survey on quantum computing technology
Gyongyosi, L., & Imre, S. (2019) · Computer Science Review · doi:10.1016/j.cosrev.2018.11.002 · Cited by 646 · see citing works
Continuous-variable quantum neural networks
Killoran, N., Bromley, T. R., Arrazola, J. M., Schuld, M., Quesada, N., & Lloyd, S. (2019) · Physical Review Research · doi:10.1103/physrevresearch.1.033063 · Cited by 418 · see citing works
A generative modeling approach for benchmarking and training shallow quantum circuits
Benedetti, M., Garcia-Pintos, D., Perdomo, O., Leyton-Ortega, V., Nam, Y., & Perdomo-Ortiz, A. (2019) · npj Quantum Information · doi:10.1038/s41534-019-0157-8 · Cited by 379 · see citing works
Potential of quantum computing for drug discovery
Cao, Y., Romero, J., & Aspuru-Guzik, A. (2018) · IBM Journal of Research and Development · doi:10.1147/jrd.2018.2888987 · Cited by 357 · see citing works
Differentiable learning of quantum circuit Born machines
Liu, J.-G., & Wang, L. (2018) · Physical Review A · doi:10.1103/physreva.98.062324 · Cited by 279 · see citing works
Opportunities and challenges for quantum-assisted machine learning in near-term quantum computers
Perdomo-Ortiz, A., Benedetti, M., Realpe-Gómez, J., & Biswas, R. (2018) · Quantum Science and Technology · doi:10.1088/2058-9565/aab859 · Cited by 274 · see citing works
The prospects of quantum computing in computational molecular biology
Outeiral, C., Strahm, M., Shi, J., Morris, G. M., Benjamin, S. C., & Deane, C. M. (2021) · WIREs Computational Molecular Science · doi:10.1002/wcms.1481 · Cited by 232 · see citing works
Information Perspective to Probabilistic Modeling: Boltzmann Machines versus Born Machines
Cheng, S., Chen, J., & Wang, L. (2018) · Entropy · doi:10.3390/e20080583 · Cited by 104 · see citing works
Entanglement-Induced Barren Plateaus
Ortiz Marrero, C., Kieferová, M., & Wiebe, N. (2021) · PRX Quantum · doi:10.1103/prxquantum.2.040316 · Cited by 39 · see citing works
Self-citationBarthe, A., Grossi, M., Vallecorsa, S., Tura, J., & Dunjko, V. (2025) · npj Quantum Information · doi:10.1038/s41534-025-01064-3 · Cited by 11 · see citing works
- Authors
- Kieferová, M., & Wiebe, N.
- Affiliations
- University of Waterloo (CA)
- Microsoft (United States) (US)
- Venue
- Physical Review A
- Publisher
- American Physical Society (APS)
- Published
- 2017-12-22
- Volume
- 96
- Issue
- 6
- ISSN
- 0556-2791 (ISSN-L)
- 2469-9926
- 2469-9934
- OpenAlex
- W2579928628
- Cited by
- 170
- References
- 16
- Citations by year
- 2026: 10 · 2025: 17 · 2024: 21 · 2023: 25 · 2022: 12 · 2021: 31 · 2020: 19 · 2019: 18 · 2018: 16 · 2017: 1
- FWCI
- 8.69
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Physical Review A (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- arXiv (Cornell University)
- Topics
- Physical Sciences › Computer Science › Artificial Intelligence › Quantum Computing Algorithms and Architecture
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum many-body systems
- Physical Sciences › Computer Science › Artificial Intelligence › Quantum Information and Cryptography
- Keywords
- quantum Boltzmann machines, quantum neural networks, quantum state tomography, quantum machine learning, training complexity
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
Are you an author, or do you see a problem with this page? Request a review
