Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

  • Perspective
  • Published:

A survey on the complexity of learning quantum states

Nature Reviews Physics volume 6, pages 59–69 (2024)Cite this article

Subjects

Abstract

Quantum learning theory is a new and very active area of research at the intersection of quantum computing and machine learning. Important breakthroughs in the past two years have rapidly solidified its foundations and led to a need for an encompassing survey that can be read by seasoned and early-career researchers in quantum computing. In this Perspective, we survey various results that rigorously study the complexity of learning quantum states. These include progress on quantum tomography, learning physical quantum states, alternative learning models to tomography, and learning classical functions encoded as quantum states. We highlight how these results are leading towards a successful theory with a range of exciting open questions, some of which we list throughout the text.

This is a preview of subscription content, access via your institution

Access options

Buy this article

  • Purchase on Springer Link
  • Instant access to full article PDF
Buy now

Prices may be subject to local taxes which are calculated during checkout

Fig. 1: Gradients of strongly convex functions.
Fig. 2: Marginal of Gibbs states.
Fig. 3: Quantum kernel estimation.

Similar content being viewed by others

References

  1. Jumper, J. et al. Highly accurate protein structure prediction with AlphaFold. Nature 596, 583–589 (2021).

    ADS  Google Scholar 

  2. Vaswani, A. et al. Attention is all you need. Adv. Neural Inf. Process. Syst. 30, 5998–6008 (2017).

    Google Scholar 

  3. Silver, D. et al. Mastering the game of Go without human knowledge. Nature 550, 354–359 (2017).

    ADS  Google Scholar 

  4. Valiant, L. G. A theory of the learnable. Commun. ACM 27, 1134–1142 (1984).

    Google Scholar 

  5. Eisert, J., Cramer, M. & Plenio, M. B. Colloquium: Area laws for the entanglement entropy. Rev. Mod. Phys. 82, 277–306 (2010).

    ADS  MathSciNet  Google Scholar 

  6. Aaronson, S. The complexity of quantum states and transformations: from quantum money to black holes. Preprint at https://arxiv.org/abs/1607.05256 (2016).

  7. Kokail, C., van Bijnen, R., Elben, A., Vermersch, B. & Zoller, P. Entanglement Hamiltonian tomography in quantum simulation. Nat. Phys. 17, 936–942 (2021).

    Google Scholar 

  8. Cramer, M. et al. Efficient quantum state tomography. Nat. Commun. 1, 149 (2010).

    ADS  Google Scholar 

  9. Lowe, A. & Nayak, A. Lower bounds for learning quantum states with single-copy measurements. Preprint at https://arxiv.org/abs/2207.14438 (2022).

  10. Flammia, S. T., Gross, D., Liu, Y.-K. & Eisert, J. Quantum tomography via compressed sensing: error bounds, sample complexity and efficient estimators. New J. Phys. 14, 095022 (2012).

    ADS  Google Scholar 

  11. Kueng, R., Rauhut, H. & Terstiege, U. Low rank matrix recovery from rank one measurements. Appl. Comput. Harmon. Anal. 42, 88–116 (2017).

    MathSciNet  Google Scholar 

  12. Haah, J., Harrow, A. W., Ji, Z., Wu, X. & Yu, N. Sample-optimal tomography of quantum states. IEEE Trans. Inf. Theory 63, 5628–5641 (2017).

    MathSciNet  Google Scholar 

  13. O’Donnell, R. & Wright, J. Efficient quantum tomography II. In Proc. 49th Annual ACM SIGACT Symposium on Theory of Computing 962–974 (ACM, 2017).

  14. Wright, J. How to Learn a Quantum State. PhD thesis (Carnegie Mellon Univ., 2016).

  15. O’Donnell, R. & Wright, J. A primer on the statistics of longest increasing subsequences and quantum states. SIGACT News 48, 37–59 (2017).

    Google Scholar 

  16. Yuen, H. An improved sample complexity lower bound for quantum state tomography. Preprint at https://arxiv.org/abs/2206.11185 (2022).

  17. Chen, S., Huang, B., Li, J., Liu, A. & Sellke, M. Tight bounds for state tomography with incoherent measurements. Preprint at https://arxiv.org/abs/2206.05265 (2022).

  18. Gottesman, D. Theory of fault-tolerant quantum computation. Phys. Rev. A 57, 127–137 (1998).

    ADS  Google Scholar 

  19. Aaronson, S. & Gottesman, D. Improved simulation of stabilizer circuits. Phys. Rev. A 70, 052328 (2004).

    ADS  Google Scholar 

  20. Aaronson, S. & Gottesman, D. Identifying Stabilizer States (PIRSA, 2008); https://pirsa.org/08080052.

  21. Montanaro, A. Learning stabilizer states by Bell sampling. Preprint at https://arxiv.org/abs/1707.04012 (2017).

  22. Low, R. A. Learning and testing algorithms for the Clifford group. Phys. Rev. A 80, 052314 (2009).

    ADS  MathSciNet  Google Scholar 

  23. Bravyi, S. B. & Kitaev, A. Yu. Quantum codes on a lattice with boundary. Preprint at https://arxiv.org/abs/quant-ph/9811052 (1998).

  24. Fowler, A. G., Mariantoni, M., Martinis, J. M. & Cleland, A. N. Surface codes: towards practical large-scale quantum computation. Phys. Rev. A 86, 032324 (2012).

    ADS  Google Scholar 

  25. Kliuchnikov, V., Maslov, D. & Mosca, M. Fast and efficient exact synthesis of single qubit unitaries generated by Clifford and t gates. Quantum Inf. Comput. 13, 607–630 (2013).

    MathSciNet  Google Scholar 

  26. Selinger, P. Efficient Clifford + T approximation of single-qubit operators. Quantum Inf. Comput. 15, 159–180 (2015).

    MathSciNet  Google Scholar 

  27. Ross, N. J. & Selinger, P. Optimal ancilla-free Clifford + T approximation of z-rotations. Quantum Inf. Comput. 16, 901–953 (2016).

    MathSciNet  Google Scholar 

  28. Bravyi, S., Smith, G. & Smolin, J. A. Trading classical and quantum computational resources. Phys. Rev. X 6, 021043 (2016).

    Google Scholar 

  29. Bravyi, S. & Gosset, D. Improved classical simulation of quantum circuits dominated by Clifford gates. Phys. Rev. Lett. 116, 250501 (2016).

    ADS  Google Scholar 

  30. Bravyi, S. et al. Simulation of quantum circuits by low-rank stabilizer decompositions. Quantum 3, 181 (2019).

    Google Scholar 

  31. Lai, C.-Y. & Cheng, H.-C. Learning quantum circuits of some T gates. IEEE Trans. Inf. Theory 68, 3951–3964 (2022).

    MathSciNet  Google Scholar 

  32. Hinsche, M. et al. A single T-gate makes distribution learning hard. Preprint at https://arxiv.org/abs/2207.03140 (2022).

  33. Grewal, S., Iyer, V., Kretschmer, W. & Liang, D. Low-stabilizer-complexity quantum states are not pseudorandom. In 4th Innovations in Theoretical Computer Science Conference (ITCS 2023), LIPIcs Vol. 251 of (ed. Kalai, Y. T.) 64:1–64:20 (Schloss Dagstuhl — Leibniz-Zentrum für Informatik, 2023).

  34. Grewal, S., Iyer, V., Kretschmer, W. & Liang, D. Improved stabilizer estimation via Bell difference sampling. Preprint at https://arxiv.org/abs/2304.13915 (2023).

  35. Grewal, S., Iyer, V., Kretschmer, W. & Liang, D. Efficient learning of quantum states prepared with few non-Clifford gates. Preprint at https://arxiv.org/abs/2305.13409 (2023).

  36. Leone, L., Oliviero, S. F. & Hamma, A. Learning t-doped stabilizer states. Preprint at https://arxiv.org/abs/2305.15398 (2023).

  37. Ji, Z., Liu, Y.-K. & Song, F. Pseudorandom quantum states. In Annual International Cryptology Conference, 126–152 (Springer, 2018).

  38. Brakerski, Z. & Shmueli, O. (Pseudo) random quantum states with binary phase. In Theory of Cryptography Conference, 229–250 (Springer, 2019).

  39. Irani, S., Natarajan, A., Nirkhe, C., Rao, S. & Yuen, H. Quantum search-to-decision reductions and the state synthesis problem. In 37th Computational Complexity Conference (CCC), LIPIcs Vol. 234, 5:1–5:19 (Schloss Dagstuhl — Leibniz-Zentrum für Informatik, 2022).

  40. Ananth, P., Qian, L. & Yuen, H. Cryptography from pseudorandom quantum states. Preprint at https://arxiv.org/abs/2112.10020 (2021).

  41. Rossi, M., Huber, M., Bruß, D. & Macchiavello, C. Quantum hypergraph states. New J. Phys. 15, 113022 (2013).

    ADS  MathSciNet  Google Scholar 

  42. Takeuchi, Y., Morimae, T. & Hayashi, M. Quantum computational universality of hypergraph states with Pauli-X and Z basis measurements. Sci. Rep. 9, 1–14 (2019).

    Google Scholar 

  43. Arunachalam, S., Bravyi, S., Dutt, A. & Yoder, T. J. Optimal algorithms for learning quantum phase states. Preprint at https://arxiv.org/abs/2208.07851v1 (2022).

  44. Gottesman, D. & Chuang, I. L. Demonstrating the viability of universal quantum computation using teleportation and single-qubit operations. Nature 402, 390–393 (1999).

    ADS  Google Scholar 

  45. Cui, S. X., Gottesman, D. & Krishna, A. Diagonal gates in the Clifford hierarchy. Phys. Rev. A 95, 012329 (2017).

    ADS  MathSciNet  Google Scholar 

  46. Bernstein, E. & Vazirani, U. Quantum complexity theory. SIAM J. Comput. 26, 1411–1473 (1997).

    MathSciNet  Google Scholar 

  47. Rötteler, M. Quantum algorithms to solve the hidden shift problem for quadratics and for functions of large Gowers norm. In International Symposium on Mathematical Foundations of Computer Science, 663–674 (Springer, 2009).

  48. Montanaro, A. Quantum circuits and low-degree polynomials over \({{\mathbb{F}}}_{2}\). J. Phys. A 50, 084002 (2017).

  49. Bremner, M. J., Jozsa, R. & Shepherd, D. J. Classical simulation of commuting quantum computations implies collapse of the polynomial hierarchy. Proc. R. Soc. A 467, 459–472 (2011).

    ADS  MathSciNet  Google Scholar 

  50. Liang, D. Clifford circuits can be properly PAC learned if and only if RP = NP. Preprint at https://arxiv.org/abs/2204.06638 (2022).

  51. Chow, C. & Liu, C. Approximating discrete probability distributions with dependence trees. IEEE Trans. Inf. Theory 14, 462–467 (1968).

    Google Scholar 

  52. Hinton, G. E. & Sejnowski, T. J. in Parallel Distributed Processing: Explorations in the Microstructure of Cognition Vol. 1, 282–317 (MIT Press, 1986).

  53. Tanaka, T. Mean-field theory of Boltzmann machine learning. Phys. Rev. E 58, 2302–2310 (1998).

    ADS  Google Scholar 

  54. Albert, J. & Swendsen, R. H. The inverse Ising problem. Phys. Procedia 57, 99–103 (2014.

    ADS  Google Scholar 

  55. Bresler, G. Efficiently learning Ising models on arbitrary graphs. In STOC’15 — Proc. 2015 ACM Symposium on Theory of Computing, 771–782 (ACM, 2015).

  56. Vuffray, M., Misra, S., Lokhov, A. & Chertkov, M. Interaction screening: efficient and sample-optimal learning of Ising models. Adv. Neural Inf. Process. Syst. 29, 2595–2603 (2016).

  57. Klivans, A. R. & Meka, R. Learning graphical models using multiplicative weights. In 58th IEEE Annual Symposium on Foundations of Computer Science (FOCS), (ed. Umans, C.) 343–354 (IEEE Computer Society, 2017).

  58. Jiang, Y.-X. et al. Unconventional chiral charge order in kagome superconductor KV3Sb5. Nat. Mater. 20, 1353–1357 (2021).

    ADS  Google Scholar 

  59. Schlomer, H. et al. Quantifying hole-motion-induced frustration in doped antiferromagnets by Hamiltonian reconstruction. Preprint at https://arxiv.org/abs/2210.02440 (2022).

  60. Anshu, A., Arunachalam, S., Kuwahara, T. & Soleimanifar, M. Sample-efficient learning of interacting quantum systems. Nat. Phys. 17, 931–935 (2021).

    Google Scholar 

  61. Anshu, A., Arunachalam, S., Kuwahara, T. & Soleimanifar, M. Efficient learning of commuting Hamiltonians on lattices. http://anuraganshu.seas.harvard.edu/links (2021).

  62. Kuwahara, T., Kato, K. & Brandão, F. G. S. L. Clustering of conditional mutual information for quantum Gibbs states above a threshold temperature. Phys. Rev. Lett. 124, 220601 (2020).

    ADS  MathSciNet  Google Scholar 

  63. Haah, J., Kothari, R. & Tang, E. Optimal learning of quantum Hamiltonians from high-temperature Gibbs states. In 63rd IEEE Annual Symposium on Foundations of Computer Science (FOCS 2022), 135–146 (IEEE, 2022).

  64. Kuwahara, T. & Saito, K. Gaussian concentration bound and ensemble equivalence in generic quantum many-body systems including long-range interactions. Ann. Phys. 421, 168278 (2020).

    MathSciNet  Google Scholar 

  65. Wild, D. S. & Alhambra, A. M. Classical simulation of short-time quantum dynamics. PRX Quantum 4, 020340 (2023).

  66. Bairey, E., Arad, I. & Lindner, N. H. Learning a local Hamiltonian from local measurements. Phys. Rev. Lett. 122, 020504 (2019).

    ADS  Google Scholar 

  67. Qi, X.-L. & Ranard, D. Determining a local Hamiltonian from a single eigenstate. Quantum 3, 159 (2019).

    Google Scholar 

  68. Kato, K. & Brandão, F. G. S. L. Quantum approximate Markov chains are thermal. Commun. Math. Phys. 370, 117–149 (2019).

    ADS  MathSciNet  Google Scholar 

  69. Aaronson, S. The learnability of quantum states. Proc. R. Soc. A 463, 3089–3114 (2007).

    ADS  MathSciNet  Google Scholar 

  70. Cheng, H., Hsieh, M. & Yeh, P. The learnability of unknown quantum measurements. Quantum Inf. Comput. 16, 615–656 (2016).

    MathSciNet  Google Scholar 

  71. Arunachalam, S. & de Wolf, R. Guest column: a survey of quantum learning theory. SIGACT News 48, 41–67 (2017).

    MathSciNet  Google Scholar 

  72. Rocchetto, A. Stabiliser states are efficiently PAC-learnable. Preprint at https://arxiv.org/abs/1705.00345 (2017).

  73. Gollakota, A. & Liang, D. On the hardness of PAC-learning stabilizer states with noise. Quantum 6, 640 (2022).

    Google Scholar 

  74. Blum, A., Kalai, A. & Wasserman, H. Noise-tolerant learning, the parity problem, and the statistical query model. J. ACM 50, 506–519 (2003).

    MathSciNet  Google Scholar 

  75. Aaronson, S., Chen, X., Hazan, E., Kale, S. & Nayak, A. Online learning of quantum states. Adv. Neural Inf. Process. Syst. 31, 8962–8972 (2018).

    Google Scholar 

  76. Chen, X. et al. Adaptive online learning of quantum states. Preprint at https://arxiv.org/abs/2206.00220 (2022).

  77. Rakhlin, A., Sridharan, K. & Tewari, A. Online learning via sequential complexities. J. Mach. Learn. Res. 16, 155–186 (2015).

    MathSciNet  Google Scholar 

  78. Aaronson, S. & Rothblum, G. N. Gentle measurement of quantum states and differential privacy. In Proc. 51st Annual ACM SIGACT Symposium on Theory of Computing, 322–333 (ACM, 2019).

  79. Dwork, C. Differential privacy. In Proc. Automata, Languages and Programming: 33rd International Colloquium (ICALP 2006), Part II 33, 1–12 (Springer, 2006).

  80. Dwork, C. & Roth, A. et al. The algorithmic foundations of differential privacy. Found. Trends Theor. Comput. Sci. 9, 211–407 (2014).

    MathSciNet  Google Scholar 

  81. Kasiviswanathan, S. P., Lee, H. K., Nissim, K., Raskhodnikova, S. & Smith, A. D. What can we learn privately? SIAM J. Comput. 40, 793–826 (2011).

    MathSciNet  Google Scholar 

  82. Bun, M., Livni, R. & Moran, S. An equivalence between private classification and online prediction. In 61st IEEE Annual Symposium on Foundations of Computer Science (FOCS), 389–402 (IEEE, 2020).

  83. Arunachalam, S., Quek, Y. & Smolin, J. A. Private learning implies quantum stability. Adv. Neural Inf. Process. Syst. 34, 20503–20515 (2021).

    Google Scholar 

  84. Aaronson, S. Shadow tomography of quantum states. In Proc. 50th Annual ACM SIGACT Symposium on Theory of Computing (STOC), 325–338 (ACM, 2018).

  85. Huang, H.-Y., Kueng, R. & Preskil, J. Predicting many properties of a quantum system from very few measurements. Nat. Phys. 16, 1050–1057 (2020).

    Google Scholar 

  86. Badescu, C. & O’Donnell, R. Improved quantum data analysis. In STOC ’21: 53rd Annual ACM SIGACT Symposium on Theory of Computing, 1398–1411 (ACM, 2021).

  87. Chung, K. & Lin, H. Sample efficient algorithms for learning quantum channels in PAC model and the approximate state discrimination problem. In 16th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2021), LIPIcs Vol. 197 (ed. Hsieh, M.) 3:1–3:22 (Schloss Dagstuhl — Leibniz-Zentrum für Informatik, 2021).

  88. Fanizza, M., Quek, Y. & Rosati, M. Learning quantum processes without input control. Preprint at https://arxiv.org/abs/2211.05005 (2022).

  89. Linial, N., Mansour, Y. & Nisan, N. Constant depth circuits, Fourier transform, and learnability. J. ACM 40, 607–620 (1993).

    MathSciNet  Google Scholar 

  90. Gopalan, P., Kalai, A. T. & Klivans, A. R. Agnostically learning decision trees. In Proc. 40th Annual ACM Symposium on Theory of Computing, 527–536 (ACM, 2008).

  91. Chen, S., Cotler, J., Huang, H. & Li, J. Exponential separations between learning with and without quantum memory. In 62nd IEEE Annual Symposium on Foundations of Computer Science (FOCS), 574–585 (IEEE, 2021).

  92. Gong, W. & Aaronson, S. Learning distributions over quantum measurement outcomes. Preprint at https://arxiv.org/abs/2209.03007 (2022).

  93. Bshouty, N. H. & Jackson, J. C. Learning DNF over the uniform distribution using a quantum example oracle. In Proc. 8th Annual Conference on Computational Learning Theory, COLT (ed. Maass, W.) 118–127 (ACM, 1995).

  94. Atıcı, A. & Servedio, R. A. Quantum algorithms for learning and testing juntas. Quantum Inf. Process. 6, 323–348 (2007).

    MathSciNet  Google Scholar 

  95. Arunachalam, S., Chakraborty, S., Lee, T., Paraashar, M. & De Wolf, R. Two new results about quantum exact learning. Quantum 5, 587 (2021).

    Google Scholar 

  96. Arunachalam, S. & de Wolf, R. Optimal quantum sample complexity of learning algorithms. J. Mach. Learn. Res. 19, 1–36 (2018).

    MathSciNet  Google Scholar 

  97. Kanade, V., Rocchetto, A. & Severini, S. Learning DNFs under product distributions via μ-biased quantum Fourier sampling. Quantum Inf. Comput. 19, 1261–1278 (2018).

    MathSciNet  Google Scholar 

  98. Arunachalam, S., Grilo, A. B., Gur, T., Oliveira, I. C. & Sundaram, A. Quantum learning algorithms imply circuit lower bounds. In 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS), 562–573 (IEEE, 2022).

  99. Grilo, A. B., Kerenidis, I. & Zijlstra, T. Learning-with-errors problem is easy with quantum samples. Phys. Rev. A 99, 032314 (2019).

    ADS  Google Scholar 

  100. Arunachalam, S., Grilo, A. B. & Yuen, H. Quantum statistical query learning. Preprint at https://arxiv.org/abs/2002.08240 (2020).

  101. Kearns, M. J. Efficient noise-tolerant learning from statistical queries. J. ACM 45, 983–1006 (1998).

    MathSciNet  Google Scholar 

  102. Reyzin, L. Statistical queries and statistical algorithms: foundations and applications. Preprint at https://arxiv.org/abs/2004.00557 (2020).

  103. Feldman, V., Grigorescu, E., Reyzin, L., Vempala, S. S. & Xiao, Y. Statistical algorithms and a lower bound for detecting planted cliques. J. ACM 64, 1–37 (2017).

    MathSciNet  Google Scholar 

  104. Hinsche, M. et al. Learnability of the output distributions of local quantum circuits. Preprint at https://arxiv.org/abs/2110.05517 (2021).

  105. Du, Y., Hsieh, M.-H., Liu, T., You, S. & Tao, D. Learnability of quantum neural networks. PRX Quantum 2, 040337 (2021).

    ADS  Google Scholar 

  106. Harrow, A. W., Hassidim, A. & Lloyd, S. Quantum algorithm for linear systems of equations. Phys. Rev. Lett. 103, 150502 (2009).

    ADS  MathSciNet  Google Scholar 

  107. Wiebe, N., Braun, D. & Lloyd, S. Quantum algorithm for data fitting. Phys. Rev. Lett. 109, 050505 (2012).

    ADS  Google Scholar 

  108. Lloyd, S., Mohseni, M. & Rebentrost, P. Quantum algorithms for supervised and unsupervised machine learning. Preprint at https://arxiv.org/abs/1307.0411 (2013).

  109. Lloyd, S., Mohseni, M. & Rebentrost, P. Quantum principal component analysis. Nat. Phys. 10, 631–633 (2014).

    Google Scholar 

  110. Rebentrost, P., Mohseni, M. & Lloyd, S. Quantum support vector machine for big data classification. Phys. Rev. Lett. 113, 130503 (2014).

    ADS  Google Scholar 

  111. Lloyd, S., Garnerone, S. & Zanardi, P. Quantum algorithms for topological and geometric analysis of data. Nat. Commun. 7, 1–7 (2016).

    Google Scholar 

  112. Cong, I. & Duan, L. Quantum discriminant analysis for dimensionality reduction and classification. New J. Phys. 18, 073011 (2016).

    ADS  Google Scholar 

  113. Kerenidis, I. & Prakash, A. Quantum recommendation systems. In 8th Innovations in Theoretical Computer Science Conference (ITCS), LIPIcs Vol. 67, 49:1–49:21 (Schloss Dagstuhl — Leibniz-Zentrum für Informatik, 2017).

  114. Brandão, F. G. S. L. et al. Quantum SDP solvers: large speed-ups, optimality, and applications to quantum learning. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019), LIPIcs Vol. 132, 27:1–27:14 (2019).

  115. Rebentrost, P., Steffens, A., Marvian, I. & Lloyd, S. Quantum singular-value decomposition of nonsparse low-rank matrices. Phys. Rev. A 97, 012327 (2018).

    ADS  MathSciNet  Google Scholar 

  116. Zhao, Z., Fitzsimons, J. K. & Fitzsimons, J. F. Quantum-assisted Gaussian process regression. Phys. Rev. A 99, 052331 (2019).

    ADS  Google Scholar 

  117. Aaronson, S. Read the fine print. Nat. Phys. 11, 291–293 (2015).

    Google Scholar 

  118. Tang, E. A quantum-inspired classical algorithm for recommendation systems. In Proc. 51st Annual ACM SIGACT Symposium on Theory of Computing (STOC), 217–228 (2019).

  119. Tang, E. Quantum principal component analysis only achieves an exponential speedup because of its state preparation assumptions. Phys. Rev. Lett. 127, 060503 (2021).

    ADS  MathSciNet  Google Scholar 

  120. Gilyén, A., Lloyd, S. & Tang, E. Quantum-inspired low-rank stochastic regression with logarithmic dependence on the dimension. Preprint at https://arxiv.org/abs/1811.04909 (2018).

  121. Chia, N.-H., Lin, H.-H. & Wang, C. Quantum-inspired sublinear classical algorithms for solving low-rank linear systems. Preprint at https://arxiv.org/abs/1811.04852 (2018).

  122. Ding, C., Bao, T.-Y. & Huang, H.-L. Quantum-inspired support vector machine. Preprint at https://arxiv.org/abs/1906.08902 (2019).

  123. Chia, N.-H. et al. Sampling-based sublinear low-rank matrix arithmetic framework for dequantizing quantum machine learning. In Proc. 52nd Annual ACM SIGACT Symposium on Theory of Computing (STOC) 387–400 (ACM, 2020).

  124. Havlíček, V. et al. Supervised learning with quantum-enhanced feature spaces. Nature 567, 209–212 (2019).

    ADS  Google Scholar 

  125. Schuld, M. & Killoran, N. Quantum machine learning in feature Hilbert spaces. Phys. Rev. Lett. 122, 040504 (2019).

    ADS  Google Scholar 

  126. Vapnik, V. The Nature of Statistical Learning Theory (Springer, 2013).

  127. Glick, J. R. et al. Covariant quantum kernels for data with group structure. Preprint at https://arxiv.org/abs/2105.03406 (2021).

  128. Liu, Y., Arunachalam, S. & Temme, K. A rigorous and robust quantum speed-up in supervised machine learning. Nat. Phys. 17, 1013–1017 (2021).

    Google Scholar 

  129. Huang, H.-Y., Chen, S. & Preskill, J. Learning to predict arbitrary quantum processes. Preprint at https://arxiv.org/abs/2210.14894 (2022).

  130. Caro, M. C. Learning quantum processes and Hamiltonians via the Pauli transfer matrix. Preprint at https://arxiv.org/abs/2212.04471 (2022).

  131. Caro, M. C. Binary classification with classical instances and quantum labels. Quantum Mach. Intell. 3, 18 (2021).

    Google Scholar 

  132. Caro, M. C. & Datta, I. Pseudo-dimension of quantum circuits. Quantum Mach. Intell. 2, 14 (2020).

    Google Scholar 

  133. Haah, J., Kothari, R., O’Donnell, R. & Tang, E. Query-optimal estimation of unitary channels in diamond distance. Preprint at https://arxiv.org/abs/2302.14066 (2023).

  134. Lewis, L. et al. Improved machine learning algorithm for predicting ground state properties. Preprint at https://arxiv.org/abs/2301.13169 (2023).

  135. Onorati, E., Rouzé, C., França, D. S. & Watson, J. D. Efficient learning of ground & thermal states within phases of matter. Preprint at https://arxiv.org/abs/2301.12946 (2023).

  136. Huang, H.-Y., Kueng, R., Torlai, G., Albert, V. V. & Preskill, J. Provably efficient machine learning for quantum many-body problems. Science 377, eabk3333 (2022).

    MathSciNet  Google Scholar 

  137. Caro, M. C., Gil-Fuster, E., Meyer, J. J., Eisert, J. & Sweke, R. Encoding-dependent generalization bounds for parametrized quantum circuits. Quantum 5, 582 (2021).

    Google Scholar 

  138. Caro, M. C. et al. Generalization in quantum machine learning from few training data. Nat. Commun. 13, 4919 (2022).

    ADS  Google Scholar 

  139. Gibbs, J. et al. Dynamical simulation via quantum machine learning with provable generalization. Preprint at https://arxiv.org/abs/2204.10269 (2022).

  140. Caro, M. C. et al. Out-of-distribution generalization for learning quantum dynamics. Preprint at https://arxiv.org/abs/2204.10268 (2022).

  141. Becker, S., Datta, N., Lami, L. & Rouzé, C. Classical shadow tomography for continuous variables quantum systems. Preprint at https://arxiv.org/abs/2211.07578 (2022).

  142. Gebhart, V. et al. Learning quantum systems. Nat. Rev. Phys. 5, 141–156 (2023).

    Google Scholar 

  143. Khavari, B. & Rabusseau, G. Lower and upper bounds on the pseudo-dimension of tensor network models. Adv. Neural Inf. Process. 34, 10931–10943 (2021).

    Google Scholar 

  144. Lanyon, B. et al. Efficient tomography of a quantum many-body system. Nat. Phys. 13, 1158–1162 (2017).

    Google Scholar 

  145. Aaronson, S. & Grewal, S. Efficient tomography of non-interacting fermion states. Preprint at https://arxiv.org/abs/2102.10458 (2023).

  146. O’Gorman, B. Fermionic tomography and learning. Preprint at https://arxiv.org/abs/2207.14787 (2022).

  147. Coyle, B., Mills, D., Danos, V. & Kashefi, E. The Born supremacy: quantum advantage and training of an Ising born machine. NPJ Quantum Inf. 6, 60 (2020).

    ADS  Google Scholar 

  148. Zhong, W., Gao, X., Yelin, S. F. & Najafi, K. Many-body localized hidden Born machine. Preprint at https://arxiv.org/abs/2207.02346 (2022).

  149. Gomez, A. M., Yelin, S. F. & Najafi, K. Reconstructing quantum states using basis-enhanced Born machines. Preprint at https://arxiv.org/abs/2206.01273 (2022).

  150. Wiebe, N., Granade, C., Ferrie, C. & Cory, D. G. Hamiltonian learning and certification using quantum resources. Phys. Rev. Lett. 112, 190501 (2014).

    ADS  Google Scholar 

  151. Verdon, G., Marks, J., Nanda, S., Leichenauer, S. & Hidary, J. Quantum Hamiltonian-based models and the variational quantum thermalizer algorithm. Preprint at https://arxiv.org/abs/1910.02071 (2019).

  152. Abbas, A. et al. The power of quantum neural networks. Nat. Comput. Sci. 1, 403–409 (2021).

    Google Scholar 

  153. Beer, K. et al. Training deep quantum neural networks. Nat. Commun. 11, 808 (2020).

    ADS  Google Scholar 

  154. Huang, H.-Y., Tong, Y., Fang, D. & Su, Y. Learning many-body Hamiltonians with Heisenberg-limited scaling. Phys. Rev. Lett. 130, 200403 (2023).

    ADS  MathSciNet  Google Scholar 

  155. Dutt, A. et al. Active learning of quantum system Hamiltonians yields query advantage. Preprint at https://arxiv.org/abs/2112.14553 (2021).

  156. Franca, D. S., Markovich, L. A., Dobrovitski, V., Werner, A. H. & Borregaard, J. Efficient and robust estimation of many-qubit Hamiltonians. Preprint at https://arxiv.org/abs/2205.09567 (2022).

  157. Gu, A., Cincio, L. & Coles, P. J. Practical black box Hamiltonian learning. Preprint at https://arxiv.org/abs/2206.15464 (2022).

  158. Anshu, A., Landau, Z. & Liu, Y. Distributed quantum inner product estimation. In Proc. 54th Annual ACM SIGACT Symposium on Theory of Computing (STOC 2022) 44–51 (ACM, 2022).

  159. Bacon, D., Childs, A. M. & van Dam, W. Optimal measurements for the dihedral hidden subgroup problem. Chic. J. Theor. Comput. Sci. https://doi.org/10.4086/cjtcs.2006.002 (2006).

  160. Hallgren, S., Russell, A. & Ta-Shma, A. Normal subgroup reconstruction and quantum computation using group representations. In Proc. 32nd Annual ACM Symposium on Theory of Computing, 627–635 (ACM, 2000).

  161. Kitaev, A. Y. Quantum measurements and the Abelian stabilizer problem. Electron. Colloquium Comput. Complex. https://api.semanticscholar.org/CorpusID:17023060 (1996).

  162. Roetteler, M. & Beth, T. Polynomial-time solution to the hidden subgroup problem for a class of non-Abelian groups. Preprint at https://arxiv.org/abs/quant-ph/9812070 (1998).

  163. Friedl, K., Ivanyos, G., Magniez, F., Santha, M. & Sen, P. Hidden translation and translating coset in quantum computing. SIAM J. Comput. 43, 1–24 (2014).

    MathSciNet  Google Scholar 

  164. Kimmel, S., Lin, C. Y.-Y., Low, G. H., Ozols, M. & Yoder, T. J. Hamiltonian simulation with optimal sample complexity. NPJ Quantum Inf. 3, 13 (2017).

    ADS  Google Scholar 

Download references

Acknowledgements

We thank A. Deshpande for comments. A.A. acknowledges support through the National Science Foundation (NSF) Career Award no. 2238836 and NSF Award QCIS-FF (Quantum Computing & Information Science Faculty Fellow) at Harvard University (NSF 2013303).

Author information

Authors and Affiliations

  1. School of Engineering and Applied Sciences, Harvard University, Allston, MA, USA

    Anurag Anshu

  2. IBM Quantum, Almaden Research Center, San Jose, CA, USA

    Srinivasan Arunachalam

Authors
  1. Anurag Anshu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Srinivasan Arunachalam
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

The authors contributed equally to all aspects of the article.

Corresponding authors

Correspondence to Anurag Anshu or Srinivasan Arunachalam.

Ethics declarations

Competing interests

The authors declare no competing interests.

Peer review

Peer review information

Nature Reviews Physics thanks Matthias Caro, Marco Cerezo and Ewin Tang for their contribution to the peer review of this work.

Additional information

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Anshu, A., Arunachalam, S. A survey on the complexity of learning quantum states. Nat Rev Phys 6, 59–69 (2024). https://doi.org/10.1038/s42254-023-00662-4

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/s42254-023-00662-4

This article is cited by

Search

Advanced search

Quick links

Nature Briefing AI and Robotics

Sign up for the Nature Briefing: AI and Robotics newsletter — what matters in AI and robotics research, free to your inbox weekly.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing: AI and Robotics