Abstract
Fuelled by increasing computer power and algorithmic advances, machine learning techniques have become powerful tools for finding patterns in data. Quantum systems produce atypical patterns that classical systems are thought not to produce efficiently, so it is reasonable to postulate that quantum computers may outperform classical computers on machine learning tasks. The field of quantum machine learning explores how to devise and implement quantum software that could enable machine learning that is faster than that of classical computers. Recent work has produced quantum algorithms that could act as the building blocks of machine learning programs, but the hardware and software challenges are still considerable.
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.References
Rosenblatt, F. The perceptron: a probabilistic model for information storage and organization in the brain. Psychol. Rev. 65, 386 (1958)
LeCun, Y., Bengio, Y. & Hinton, G. Deep learning. Nature 521, 436–444 (2015)
Le, Q. V. Building high-level features using large scale unsupervised learning. In IEEE Int. Conf. on Acoustics, Speech and Signal Processing (ICASSP) 8595–8598 (IEEE, 2013)
Schuld, M., Sinayskiy, I. & Petruccione, F. An introduction to quantum machine learning. Contemp. Phys. 56, 172–185 (2015)
Wittek, P. Quantum Machine Learning: What Quantum Computing Means to Data Mining (Academic Press, New York, NY, USA, 2014)
Adcock, J. et al. Advances in quantum machine learning. Preprint at https://arxiv.org/abs/1512.02900 (2015)
Arunachalam, S. & de Wolf, R. A survey of quantum learning theory. Preprint at https://arxiv.org/abs/1701.06806 (2017)
Harrow, A. W., Hassidim, A. & Lloyd, S. Quantum algorithm for linear systems of equations. Phys. Rev. Lett. 103, 150502 (2009)
Wiebe, N., Braun, D. & Lloyd, S. Quantum algorithm for data fitting. Phys. Rev. Lett. 109, 050505 (2012)
Childs, A. M., Kothari, R. & Somma, R. D. Quantum linear systems algorithm with exponentially improved dependence on precision. Preprint at https://arxiv.org/abs/1511.02306 (2015)
Lloyd, S., Mohseni, M. & Rebentrost, P. Quantum principal component analysis. Nat. Phys. 10, 631–633 (2014)
Kimmel, S., Lin, C. Y.-Y., Low, G. H., Ozols, M. & Yoder, T. J. Hamiltonian simulation with optimal sample complexity. Preprint at https://arxiv.org/abs/1608.00281 (2016)
Rebentrost, P., Mohseni, M. & Lloyd, S. Quantum support vector machine for big data classification. Phys. Rev. Lett. 113, 130503 (2014). This study applies quantum matrix inversion in a supervised discriminative learning algorithm.
Lloyd, S., Garnerone, S. & Zanardi, P. Quantum algorithms for topological and geometric analysis of data. Nat. Commun. 7, 10138 (2016)
Dridi, R. & Alghassi, H. Homology computation of large point clouds using quantum annealing. Preprint at https://arxiv.org/abs/1512.09328 (2015)
Rebentrost, P., Steffens, A. & Lloyd, S. Quantum singular value decomposition of non-sparse low-rank matrices. Preprint at https://arxiv.org/abs/1607.05404 (2016)
Schuld, M., Sinayskiy, I. & Petruccione, F. Prediction by linear regression on a quantum computer. Phys. Rev. A 94, 022342 (2016)
Brandao, F. G. & Svore, K. Quantum speed-ups for semidefinite programming. Preprint at https://arxiv.org/abs/1609.05537 (2016)
Rebentrost, P., Schuld, M., Petruccione, F. & Lloyd, S. Quantum gradient descent and Newton’s method for constrained polynomial optimization. Preprint at https://arxiv.org/abs/1612.01789 (2016)
Wiebe, N., Kapoor, A. & Svore, K. M. Quantum deep learning. Preprint at https://arxiv.org/abs/1412.3489 (2014)
Adachi, S. H. & Henderson, M. P. Application of quantum annealing to training of deep neural networks. Preprint at https://arxiv.org/abs/arXiv:1510.06356 (2015)
Amin, M. H., Andriyash, E., Rolfe, J., Kulchytskyy, B. & Melko, R. Quantum Boltzmann machine. Preprint at https://arxiv.org/abs/arXiv:1601.02036 (2016)
Sasaki, M., Carlini, A. & Jozsa, R. Quantum template matching. Phys. Rev. A 64, 022317 (2001)
Bisio, A., Chiribella, G., D’Ariano, G. M., Facchini, S. & Perinotti, P. Optimal quantum learning of a unitary transformation. Phys. Rev. A 81, 032324 (2010)
Bisio, A., D’Ariano, G. M., Perinotti, P. & Sedlák, M. Quantum learning algorithms for quantum measurements. Phys. Lett. A 375, 3425–3434 (2011)
Sentís, G., Calsamiglia, J., Muñoz-Tapia, R. & Bagan, E. Quantum learning without quantum memory. Sci. Rep. 2, 708 (2012)
Sentís, G., Gut¸a˘, M. & Adesso, G. Quantum learning of coherent states. EPJ Quant. Technol. 2, 17 (2015)
Paparo, G. D., Dunjko, V., Makmal, A., Martin-Delgado, M. A. & Briegel, H. J. Quantum speedup for active learning agents. Phys. Rev. X 4, 031002 (2014)
Dunjko, V., Friis, N. & Briegel, H. J. Quantum-enhanced deliberation of learning agents using trapped ions. New J. Phys. 17, 023006 (2015)
Dunjko, V., Taylor, J. M. & Briegel, H. J. Quantum-enhanced machine learning. Phys. Rev. Lett. 117, 130501 (2016). This paper investigates the theoretical maximum speedup achievable in reinforcement learning in a closed quantum system, which proves to be Grover-like if we wish to obtain classical verification of the learning process.
Sentís, G., Bagan, E., Calsamiglia, J., Chiribella, G. & Muñoz Tapia, R. Quantum change point. Phys. Rev. Lett. 117, 150502 (2016)
Faccin, M., Migdał, P., Johnson, T. H., Bergholm, V. & Biamonte, J. D. Community detection in quantum complex networks. Phys. Rev. X 4, 041012 (2014). This paper defines closeness measures and then maximizes modularity with hierarchical clustering to partition quantum data.
Clader, B. D., Jacobs, B. C. & Sprouse, C. R. Preconditioned quantum linear system algorithm. Phys. Rev. Lett. 110, 250504 (2013)
Lloyd, S., Mohseni, M. & Rebentrost, P. Quantum algorithms for supervised and unsupervised machine learning. Preprint at https://arxiv.org/abs/1307.0411 (2013)
Wiebe, N., Kapoor, A. & Svore, K. M. Quantum algorithms for nearest-neighbor methods for supervised and unsupervised learning. Quantum Inf. Comput. 15, 316–356 (2015)
Lau, H.-K., Pooser, R., Siopsis, G. & Weedbrook, C. Quantum machine learning over infinite dimensions. Phys. Rev. Lett. 118, 080501 (2017)
Aïmeur, E ., Brassard, G . & Gambs, S. in Machine Learning in a Quantum World 431–442 (Springer, 2006)
Shor, P. W. Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM J. Comput. 26, 1484–1509 (1997)
Nielsen, M. A . & Chuang, I. L. Quantum Computation and Quantum Information (Cambridge Univ. Press, 2000)
Wossnig, L., Zhao, Z. & Prakash, A. A quantum linear system algorithm for dense matrices. Preprint at https://arxiv.org/abs/1704.06174 (2017)
Giovannetti, V., Lloyd, S. & Maccone, L. Quantum random access memory. Phys. Rev. Lett. 100, 160501 (2008)
Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996)
Vapnik, V. The Nature of Statistical Learning Theory (Springer, 1995)
Anguita, D., Ridella, S., Rivieccio, F. & Zunino, R. Quantum optimization for training support vector machines. Neural Netw. 16, 763–770 (2003)
Dürr, C. & Høyer, P. A quantum algorithm for finding the minimum. Preprint at https://arxiv.org/abs/quant-ph/9607014 (1996)
Chatterjee, R. & Yu, T. Generalized coherent states, reproducing kernels, and quantum support vector machines. Preprint at https://arxiv.org/abs/1612.03713 (2016)
Zhao, Z., Fitzsimons, J. K. & Fitzsimons, J. F. Quantum assisted Gaussian process regression. Preprint at https://arxiv.org/abs/1512.03929 (2015)
Li, Z., Liu, X., Xu, N. & Du, J. Experimental realization of a quantum support vector machine. Phys. Rev. Lett. 114, 140504 (2015)
Whitfield, J. D., Faccin, M. & Biamonte, J. D. Ground-state spin logic. Europhys. Lett. 99, 57004 (2012)
Farhi, E., Goldstone, J. & Gutmann, S. A quantum approximate optimization algorithm. Preprint at https://arxiv.org/abs/1411.4028 (2014)
Aaronson, S. Read the fine print. Nat. Phys. 11, 291–293 (2015)
Arunachalam, S., Gheorghiu, V., Jochym-O’Connor, T., Mosca, M. & Srinivasan, P. V. On the robustness of bucket brigade quantum RAM. New J. Phys. 17, 123010 (2015)
Scherer, A. et al. Concrete resource analysis of the quantum linear system algorithm used to compute the electromagnetic scattering cross section of a 2D target. Preprint at https://arxiv.org/abs/1505.06552 (2015)
Denil, M . & De Freitas, N. Toward the implementation of a quantum RBM. In Neural Information Processing Systems (NIPS) Conf. on Deep Learning and Unsupervised Feature Learning Workshop Vol. 5 (2011)
Dumoulin, V., Goodfellow, I. J., Courville, A. & Bengio, Y. On the challenges of physical implementations of RBMs. Preprint at https://arxiv.org/abs/1312.5258 (2013)
Benedetti, M., Realpe-Gómez, J., Biswas, R. & Perdomo-Ortiz, A. Estimation of effective temperatures in quantum annealers for sampling applications: a case study with possible applications in deep learning. Phys. Rev. A 94, 022308 (2016)
Biamonte, J. D. & Love, P. J. Realizable Hamiltonians for universal adiabatic quantum computers. Phys. Rev. A 78, 012352 (2008). This study established the contemporary experimental target for non-stoquastic (that is, non-quantum stochastic) D-Wave quantum annealing hardware able to realize universal quantum Boltzmann machines.
Temme, K., Osborne, T. J., Vollbrecht, K. G., Poulin, D. & Verstraete, F. Quantum metropolis sampling. Nature 471, 87–90 (2011)
Yung, M.-H. & Aspuru-Guzik, A. A quantum–quantum metropolis algorithm. Proc. Natl Acad. Sci. USA 109, 754–759 (2012)
Chowdhury, A. N. & Somma, R. D. Quantum algorithms for Gibbs sampling and hitting-time estimation. Quant. Inf. Comput. 17, 41–64 (2017)
Kieferova, M. & Wiebe, N. Tomography and generative data modeling via quantum Boltzmann training. Preprint at https://arxiv.org/abs/1612.05204 (2016)
Lloyd, S. & Terhal, B. Adiabatic and Hamiltonian computing on a 2D lattice with simple 2-qubit interactions. New J. Phys. 18, 023042 (2016)
Ventura, D. & Martinez, T. Quantum associative memory. Inf. Sci. 124, 273–296 (2000)
Granade, C. E., Ferrie, C., Wiebe, N. & Cory, D. G. Robust online Hamiltonian learning. New J. Phys. 14, 103013 (2012)
Wiebe, N., Granade, C., Ferrie, C. & Cory, D. G. Hamiltonian learning and certification using quantum resources. Phys. Rev. Lett. 112, 190501 (2014)
Wiebe, N., Granade, C. & Cory, D. G. Quantum bootstrapping via compressed quantum Hamiltonian learning. New J. Phys. 17, 022005 (2015)
Marvian, I. & Lloyd, S. Universal quantum emulator. Preprint at https://arxiv.org/abs/1606.02734 (2016)
Dolde, F. et al. High-fidelity spin entanglement using optimal control. Nat. Commun. 5, 3371 (2014)
Zahedinejad, E., Ghosh, J. & Sanders, B. C. Designing high-fidelity single-shot three-qubit gates: a machine-learning approach. Phys. Rev. Appl. 6, 054005 (2016)
Zahedinejad, E., Ghosh, J. & Sanders, B. C. High-fidelity single-shot Toffoli gate via quantum control. Phys. Rev. Lett. 114, 200502 (2015)
Zeidler, D., Frey, S., Kompa, K.-L. & Motzkus, M. Evolutionary algorithms and their application to optimal control studies. Phys. Rev. A 64, 023420 (2001)
Las Heras, U., Alvarez-Rodriguez, U., Solano, E. & Sanz, M. Genetic algorithms for digital quantum simulations. Phys. Rev. Lett. 116, 230504 (2016)
Banchi, L., Pancotti, N. & Bose, S. Quantum gate learning in qubit networks: Toffoli gate without time-dependent control. npj Quant. Inf. 2, 16019 (2016)
August, M. & Ni, X. Using recurrent neural networks to optimize dynamical decoupling for quantum memory. Preprint at https://arxiv.org/abs/1604.00279 (2016)
Amstrup, B., Toth, G. J., Szabo, G., Rabitz, H. & Loerincz, A. Genetic algorithm with migration on topology conserving maps for optimal control of quantum systems. J. Phys. Chem. 99, 5206–5213 (1995)
Hentschel, A. & Sanders, B. C. Machine learning for precise quantum measurement. Phys. Rev. Lett. 104, 063603 (2010)
Lovett, N. B., Crosnier, C., Perarnau-Llobet, M. & Sanders, B. C. Differential evolution for many-particle adaptive quantum metrology. Phys. Rev. Lett. 110, 220501 (2013)
Palittapongarnpim, P., Wittek, P., Zahedinejad, E., Vedaie, S. & Sanders, B. C. Learning in quantum control: high-dimensional global optimization for noisy quantum dynamics. Neurocomputing https://doi.org/10.1016/j.neucom.2016.12.087 (in the press)
Carrasquilla, J. & Melko, R. G. Machine learning phases of matter. Nat. Phys. 13, 431–434 (2017)
Broecker, P., Carrasquilla, J., Melko, R. G. & Trebst, S. Machine learning quantum phases of matter beyond the fermion sign problem. Preprint at https://arxiv.org/abs/1608.07848 (2016)
Carleo, G. & Troyer, M. Solving the quantum many-body problem with artificial neural networks. Science 355, 602–606 (2017)
Brunner, D., Soriano, M. C., Mirasso, C. R. & Fischer, I. Parallel photonic information processing at gigabyte per second data rates using transient states. Nat. Commun. 4, 1364 (2013)
Cai, X.-D. et al. Entanglement-based machine learning on a quantum computer. Phys. Rev. Lett. 114, 110504 (2015)
Hermans, M., Soriano, M. C., Dambre, J., Bienstman, P. & Fischer, I. Photonic delay systems as machine learning implementations. J. Mach. Learn. Res. 16, 2081–2097 (2015)
Tezak, N. & Mabuchi, H. A coherent perceptron for all-optical learning. EPJ Quant. Technol. 2, 10 (2015)
Neigovzen, R., Neves, J. L., Sollacher, R. & Glaser, S. J. Quantum pattern recognition with liquid-state nuclear magnetic resonance. Phys. Rev. A 79, 042321 (2009)
Pons, M. et al. Trapped ion chain as a neural network: error resistant quantum computation. Phys. Rev. Lett. 98, 023003 (2007)
Neven, H . et al. Binary classification using hardware implementation of quantum annealing. In 24th Ann. Conf. on Neural Information Processing Systems (NIPS-09) 1–17 (2009). This paper was among the first experimental demonstrations of machine learning using quantum annealing.
Denchev, V. S ., Ding, N ., Vishwanathan, S . & Neven, H. Robust classification with adiabatic quantum optimization. In Proc. 29th Int. Conf. on Machine Learning (ICML-2012) (2012)
Karimi, K. et al. Investigating the performance of an adiabatic quantum optimization processor. Quantum Inform. Process. 11, 77–88 (2012)
O’Gorman, B. A. et al. Bayesian network structure learning using quantum annealing. EPJ Spec. Top. 224, 163–188 (2015)
Denchev, V. S., Ding, N., Matsushima, S., Vishwanathan, S. V. N. & Neven, H. Totally corrective boosting with cardinality penalization. Preprint at https://arxiv.org/abs/1504.01446 (2015)
Kerenidis, I. & Prakash, A. Quantum recommendation systems. Preprint at https://arxiv.org/abs/1603.08675 (2016)
Alvarez-Rodriguez, U., Lamata, L., Escandell-Montero, P., Martín-Guerrero, J. D. & Solano, E. Quantum machine learning without measurements. Preprint at https://arxiv.org/abs/1612.05535 (2016)
Wittek, P. & Gogolin, C. Quantum enhanced inference in Markov logic networks. Sci. Rep. 7, 45672 (2017)
Lamata, L. Basic protocols in quantum reinforcement learning with superconducting circuits. Sci. Rep. 7, 1609 (2017)
Schuld, M., Fingerhuth, M. & Petruccione, F. Quantum machine learning with small-scale devices: implementing a distance-based classifier with a quantum interference circuit. Preprint at https://arxiv.org/abs/1703.10793 (2017)
Monràs, A., Sentís, G. & Wittek, P. Inductive supervised quantum learning. Phys. Rev. Lett. 118, 190503 (2017). This paper proves that supervised learning protocols split into a training and application phase in both the classical and the quantum cases.
Tiersch, M., Ganahl, E. J. & Briegel, H. J. Adaptive quantum computation in changing environments using projective simulation. Sci. Rep. 5, 12874 (2015)
Zahedinejad, E., Ghosh, J. & Sanders, B. C. Designing high-fidelity single-shot three-qubit gates: a machine learning approach. Preprint at https://arxiv.org/abs/1511.08862 (2015)
Palittapongarnpim, P ., Wittek, P . & Sanders, B. C. Controlling adaptive quantum phase estimation with scalable reinforcement learning. In Proc. 24th Eur. Symp. Artificial Neural Networks (ESANN-16) on Computational Intelligence and Machine Learning 327–332 (2016)
Wan, K. H., Dahlsten, O., Kristjánsson, H., Gardner, R. & Kim, M. S. Quantum generalisation of feedforward neural networks. Preprint at https://arxiv.org/abs/1612.01045 (2016)
Lu, D. et al. Towards quantum supremacy: enhancing quantum control by bootstrapping a quantum processor. Preprint at https://arxiv.org/abs/1701.01198 (2017)
Mavadia, S., Frey, V., Sastrawan, J., Dona, S. & Biercuk, M. J. Prediction and real-time compensation of qubit decoherence via machine learning. Nat. Commun. 8, 14106 (2017)
Rønnow, T. F. et al. Defining and detecting quantum speedup. Science 345, 420–424 (2014)
Low, G. H., Yoder, T. J. & Chuang, I. L. Quantum inference on Bayesian networks. Phys. Rev. A 89, 062315 (2014)
Wiebe, N. & Granade, C. Can small quantum systems learn? Preprint at https://arxiv.org/abs/1512.03145 (2015)
Wiebe, N., Kapoor, A. & Svore, K. M. Quantum perceptron models. Adv. Neural Inform. Process. Syst. 29, 3999–4007 (2016)
Scherer, A. et al. Concrete resource analysis of the quantum linear-system algorithm used to compute the electromagnetic scattering cross section of a 2D target. Quantum Inform. Process. 16, 60 (2017)
Acknowledgements
J.B. acknowledges financial support from AFOSR grant FA9550-16-1-0300, Models and Protocols for Quantum Distributed Computation. P.W. acknowledges financial support from the ERC (Consolidator Grant QITBOX), Spanish Ministry of Economy and Competitiveness (Severo Ochoa Programme for Centres of Excellence in R&D SEV-2015-0522 and QIBEQI FIS2016-80773-P), Generalitat de Catalunya (CERCA Programme and SGR 875), and Fundacio Privada Cellex. P.R. and S.L. acknowledge funding from ARO and AFOSR under MURI programmes. We thank L. Zheglova for producing Fig. 1.
Author information
Authors and Affiliations
Contributions
All authors designed the study, analysed data, interpreted data, produced Box 3 Figure and wrote the article.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing financial interests.
Additional information
Reviewer Information Nature thanks L. Lamata and the other anonymous reviewer(s) for their contribution to the peer review of this work.
Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
PowerPoint slides
Rights and permissions
About this article
Cite this article
Biamonte, J., Wittek, P., Pancotti, N. et al. Quantum machine learning. Nature 549, 195–202 (2017). https://doi.org/10.1038/nature23474
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1038/nature23474
- Springer Nature Limited
This article is cited by
-
Quantum computing for nonlinear differential equations and turbulence
Nature Reviews Physics (2025)
-
Experimental benchmarking of quantum state overlap estimation strategies with photonic systems
Light: Science & Applications (2025)
-
Generalizability of machine learning models for diabetes detection a study with nordic islet transplant and PIMA datasets
Scientific Reports (2025)
-
Error mitigation in brainbox quantum autoencoders
Scientific Reports (2025)
-
Hybrid quantum neural networks show strongly reduced need for free parameters in entity matching
Scientific Reports (2025)