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Limitations of optimization algorithms on noisy quantum devices

Nature Physics volume 17, pages 1221–1227 (2021)Cite this article

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Abstract

Recent successes in producing intermediate-scale quantum devices have focused interest on establishing whether near-term devices could outperform classical computers for practical applications. A central question is whether noise can be overcome in the absence of quantum error correction or if it fundamentally restricts any potential quantum advantage. We present a transparent way of comparing classical and quantum algorithms running on noisy devices for a large family of tasks that includes optimization and variational eigenstate solving. Our approach is based on entropic inequalities that determine how fast the quantum state converges to the fixed point of the noise model, together with established classical methods of Gibbs state simulation. Our techniques are extremely versatile and so may be applied to a large variety of algorithms, noise models and quantum computing architectures. We use our result to provide estimates for problems within reach of current experiments, such as quantum annealers or variational quantum algorithms. The bounds we obtain indicate that substantial quantum advantages are unlikely for classical optimization unless noise rates are decreased by orders of magnitude or the topology of the problem matches that of the device. This is the case even if the number of available qubits increases substantially.

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Fig. 1: Schematic of the bounds.
Fig. 2: Structure of variational algorithms.
Fig. 3: Example for QAOA.
Fig. 4: Noisy quantum annealers.
Fig. 5: Flowchart and benchmarking of the procedure.

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Data availability

All data that support the plots within this paper and other findings of this study are available from the corresponding authors upon reasonable request.

Code availability

The code for this study can be found at ref. 62.

References

  1. Arute, F. et al. Quantum supremacy using a programmable superconducting processor. Nature 574, 505–510 (2019).

    Article  ADS  Google Scholar 

  2. Preskill, J. Quantum computing in the NISQ era and beyond. Quantum 2, 79 (2018).

    Article  Google Scholar 

  3. Peruzzo, A. et al. A variational eigenvalue solver on a photonic quantum processor. Nat. Commun. 5, 4213 (2014).

    Article  ADS  Google Scholar 

  4. Kokail, C. et al. Self-verifying variational quantum simulation of lattice models. Nature 569, 355–360 (2019).

    Article  ADS  Google Scholar 

  5. Jones, T., Endo, S., McArdle, S., Yuan, X. & Benjamin, S. C. Variational quantum algorithms for discovering Hamiltonian spectra. Phys. Rev. A 99, 062304 (2019).

    Article  ADS  Google Scholar 

  6. LaRose, R., Tikku, A., O’Neel-Judy, É., Cincio, L. & Coles, P. J. Variational quantum state diagonalization. npj Quantum Inf. 5, 57 (2019).

    Article  Google Scholar 

  7. Bravo-Prieto, C. et al. Variational quantum linear solver. Preprint at https://arxiv.org/abs/1909.05820 (2019).

  8. McArdle, S. et al. Variational ansatz-based quantum simulation of imaginary time evolution. npj Quantum Inf. 5, 75 (2019).

    Article  ADS  Google Scholar 

  9. Moll, N. et al. Quantum optimization using variational algorithms on near-term quantum devices. Quantum Sci. Technol. 3, 030503 (2018).

    Article  ADS  Google Scholar 

  10. Kandala, A. et al. Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. Nature 549, 242–246 (2017).

    Article  ADS  Google Scholar 

  11. Wang, D., Higgott, O. & Brierley, S. Accelerated variational quantum eigensolver. Phys. Rev. Lett. 122, 140504 (2019).

    Article  ADS  Google Scholar 

  12. Farhi, E., Goldstone, J. & Gutmann, S. A quantum approximate optimization algorithm. Preprint at https://arxiv.org/abs/1411.4028 (2014).

  13. Farhi, E., Goldstone, J., Gutmann, S. & Sipser, M. Quantum computation by adiabatic evolution. Preprint at https://arxiv.org/abs/quant-ph/0001106 (2000).

  14. Albash, T. & Lidar, D. A. Adiabatic quantum computation. Rev. Mod. Phys. 90, 015002 (2018).

    Article  ADS  MathSciNet  Google Scholar 

  15. Lucas, A. Ising formulations of many NP problems. Front. Phys. https://doi.org/10.3389/fphy.2014.00005 (2014).

  16. Carbone, R. & Martinelli, A. Logarithmic sobolev inequalities in non-commutative algebras. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 18, 1550011 (2015).

    Article  ADS  MathSciNet  Google Scholar 

  17. Kastoryano, M. J. & Brandão, F. G. S. L. Quantum Gibbs samplers: the commuting case. Commun. Math. Phys. 344, 915–957 (2016).

    Article  ADS  MathSciNet  Google Scholar 

  18. Beigi, S., Datta, N. & Rouzé, C. Quantum reverse hypercontractivity: its tensorization and application to strong converses. Commun. Math. Phys. 376, 753–794 (2020).

    Article  ADS  MathSciNet  Google Scholar 

  19. Bardet, I., Capel, Á., Lucia, A., Pérez-García, D. & Rouzé, C. On the modified logarithmic Sobolev inequality for the heat-bath dynamics for 1D systems. J. Math. Phys. 62, 061901 (2021).

    Article  ADS  MathSciNet  Google Scholar 

  20. Capel, Á., Rouzé, C. & França, D. S. The modified logarithmic Sobolev inequality for quantum spin systems: classical and commuting nearest neighbour interactions. Preprint at https://arxiv.org/abs/2009.11817 (2020).

  21. Müller-Hermes, A., França, D. S. & Wolf, M. M. Relative entropy convergence for depolarizing channels. J. Math. Phys. 57, 022202 (2016).

    Article  ADS  MathSciNet  Google Scholar 

  22. Tsuda, K., Rätsch, G. & Warmuth, M. K. Matrix exponentiated gradient updates for on-line learning and Bregman projection. J. Mach. Learn. Res. 6, 995–1018 (2005).

    MathSciNet  MATH  Google Scholar 

  23. Bubeck, S. Convex optimization: algorithms and complexity. Found. Trends Mach. Learn. 8, 231–357 (2015).

    Article  Google Scholar 

  24. Aaronson, S., Chen, X., Hazan, E., Kale, S. & Nayak, A. Online learning of quantum states. J. Stat. Mech. Theory Exp. 2019, 124019 (2019).

    Article  MathSciNet  Google Scholar 

  25. Aaronson, S. Shadow tomography of quantum states. In Proc. 50th Annual ACM SIGACT Symposium on Theory of Computing - STOC 2018 https://doi.org/10.1145%2F3188745.3188802 (ACM Press, 2018).

  26. Brandão, F. G. et al. Quantum SDP solvers: large speed-ups, optimality, and applications to quantum learning. DROPS https://doi.org/10.4230/lipics.icalp.2019.27 (2019).

  27. Brandão, F. G. S. L., Kueng, R. & França, D. S. Faster quantum and classical sdp approximations for quadratic binary optimization. Preprint at https://arxiv.org/abs/1909.04613 (2019).

  28. Youssry, A., Ferrie, C. & Tomamichel, M. Efficient online quantum state estimation using a matrix-exponentiated gradient method. New J. Phys. 21, 033006 (2019).

    Article  ADS  MathSciNet  Google Scholar 

  29. Crosson, E. & Slezak, S. Classical simulation of high temperature quantum Ising models. Preprint at https://arxiv.org/abs/2002.02232 (2020).

  30. Harrow, A., Mehraban, S. & Soleimanifar, M. Classical algorithms, correlation decay, and complex zeros of partition functions of quantum many-body systems. In Proc. 52nd Annual ACM SIGACT Symposium on Theory of Computing (STOC 2020) https://doi.org/10.1145/3357713.3384322 (ACM, 2020).

  31. Kuwahara, T., Kato, K. & Brandão, F. G. Clustering of conditional mutual information for quantum gibbs states above a threshold temperature. Phys. Rev. Lett. https://doi.org/10.1103/PhysRevLett.124.220601 (2020).

  32. Verstraete, F., García-Ripoll, J. J. & Cirac, J. I. Matrix product density operators: Simulation of finite-temperature and dissipative systems. Phys. Rev. Lett. 93, 207204 (2004).

    Article  ADS  Google Scholar 

  33. Foulkes, W. M. C., Mitas, L., Needs, R. J. & Rajagopal, G. Quantum Monte Carlo simulations of solids. Rev. Mod. Phys. 73, 33–83 (2001).

    Article  ADS  Google Scholar 

  34. Rigol, M., Bryant, T. & Singh, R. R. P. Numerical linked-cluster approach to quantum lattice models. Phys. Rev. Lett. 97, 187202 (2006).

    Article  ADS  Google Scholar 

  35. Tang, B., Khatami, E. & Rigol, M. A short introduction to numerical linked-cluster expansions. Comput. Phys. Commun. 184, 557–564 (2013).

    Article  ADS  MathSciNet  Google Scholar 

  36. White, S. R. Minimally entangled typical quantum states at finite temperature. Phys. Rev. Lett. 102, 190601 (2009).

    Article  ADS  MathSciNet  Google Scholar 

  37. Molnar, A., Schuch, N., Verstraete, F. & Cirac, J. I. Approximating Gibbs states of local Hamiltonians efficiently with projected entangled pair states. Phys. Rev. B 91, 045138 (2015).

    Article  ADS  Google Scholar 

  38. Kliesch, M., Gogolin, C., Kastoryano, M., Riera, A. & Eisert, J. Locality of temperature. Phys. Rev. X 4, 031019 (2014).

    Google Scholar 

  39. Harrigan, M. P. et al. Quantum approximate optimization of non-planar graphÿ problems on a planar superconducting processor. Nat. Phys. 17, 332–336 (2021).

    Article  Google Scholar 

  40. Levin, D. A. & Peres, Y. Markov Chains and Mixing Times Vol. 107 (American Mathematical Society, 2017).

  41. Eldan, R., Koehler, F. & Zeitouni, O. A spectral condition for spectral gap: fast mixing in high-temperature Ising models. Preprint at https://arxiv.org/abs/2007.08200 (2020).

  42. Aharonov, D., Ben-Or, M., Impagliazzo, R. & Nisan, N. Limitations of noisy reversible computation. Preprint at http://arxiv.org/abs/quant-ph/9611028 (1996).

  43. Ben-Or, M., Gottesman, D. & Hassidim, A. Quantum refrigerator. Preprint at http://arxiv.org/abs/1301.1995 (2013).

  44. Muller-Hermes, A., Reeb, D. & Wolf, M. M. Quantum subdivision capacities and continuous-time quantum coding. IEEE Trans. Inform. Theory 61, 565–581 (2015).

    Article  MathSciNet  Google Scholar 

  45. Razborov, A. A. An upper bound on the threshold quantum decoherence rate. Preprint at http://arxiv.org/abs/quant-ph/0310136 (2003).

  46. Plenio, M. B. & Virmani, S. Upper bounds on fault tolerance thresholds of noisy Clifford-based quantum computers. New J. Phys. 12, 033012 (2010).

    Article  ADS  MathSciNet  Google Scholar 

  47. Kempe, J., Regev, O., Unger, F. & de Wolf, R. in Automata, Languages and Programming Vol. 5125 (eds Aceto, L. et al.) 845–856 (Springer, 2008).

  48. Buhrman, H. et al. New limits on fault-tolerant quantum computation. In 2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS’06) 411–419 (IEEE, 2006).

  49. Hadlock, F. Finding a maximum cut of a planar graph in polynomial time. SIAM J. Comput. 4, 221–225 (1975).

    Article  MathSciNet  Google Scholar 

  50. Johnson, M. W. et al. Quantum annealing with manufactured spins. Nature 473, 194–198 (2011).

    Article  ADS  Google Scholar 

  51. Technical Description of the D-wave Quantum Processing Unit (D-Wave Systems, 2019).

  52. Ye, Y. Gset Random Graphs (CISE, 2015); https://www.cise.ufl.edu/research/sparse/matrices/Gset/

  53. Yurtsever, A., Tropp, J. A., Fercoq, O., Udell, M. & Cevher, V. Scalable semidefinite programming. J. Math. Data Sci. 3, 171–200 (2021).

    Article  MathSciNet  Google Scholar 

  54. Isakov, S., Zintchenko, I., Rønnow, T. & Troyer, M. Optimised simulated annealing for ising spin glasses. Comput. Phys. Commun. 192, 265–271 (2015).

    Article  ADS  MathSciNet  Google Scholar 

  55. Bardet, I. & Rouzé, C. Hypercontractivity and logarithmic sobolev inequality for non-primitive quantum markov semigroups and estimation of decoherence rates. Preprint at https://arxiv.org/abs/1803.05379 (2018).

  56. Bremner, M. J., Montanaro, A. & Shepherd, D. J. Achieving quantum supremacy with sparse and noisy commuting quantum computations. Quantum 1, 8 (2017).

    Article  Google Scholar 

  57. Job, J. & Lidar, D. Test-driving 1000 qubits. Quantum Sci. Technol. 3, 030501 (2018).

    Article  ADS  Google Scholar 

  58. Kochenberger, G. A., Hao, J.-K., Lü, Z., Wang, H. & Glover, F. Solving large scale Max Cut problems via tabu search. J. Heuristics 19, 565–571 (2013).

    Article  Google Scholar 

  59. Albash, T. & Lidar, D. A. Decoherence in adiabatic quantum computation. Phys. Rev. A 91, 062320 (2015).

    Article  ADS  Google Scholar 

  60. Kastoryano, M. J. & Temme, K. Quantum logarithmic Sobolev inequalities and rapid mixing. J. Math. Phys.54, 052202 (2013).

    Article  ADS  MathSciNet  Google Scholar 

  61. Olkiewicz, R. & Zegarlinski, B. Hypercontractivity in noncommutative Lp spaces. J. Funct. Anal. 161, 246–285 (1999).

    Article  MathSciNet  Google Scholar 

  62. Franca, D. S. Limitations optimization plots. GitHub https://github.com/dsfranca/limitations_optimization_plots (2020).

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Acknowledgements

D.S.F. was supported by VILLUM FONDEN via the QMATH Centre of Excellence under grant no. 10059. R.G.-P. was supported by the Quantum Computing and Simulation Hub, an EPSRC-funded project, part of the UK National Quantum Technologies Programme. We thank H. Guo, J. Brown-Cohen and P.-L. Dallaire-Demers for helpful discussions.

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  1. QMATH, Department of Mathematical Sciences, University of Copenhagen, Copenhagen, Denmark

    Daniel Stilck França

  2. School of Informatics, University of Edinburgh, Edinburgh, UK

    Raul García-Patrón

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  1. Daniel Stilck França
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Both authors discussed the results and contributed to the writing and theory development for the final manuscript.

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Correspondence to Daniel Stilck França or Raul García-Patrón.

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Stilck França, D., García-Patrón, R. Limitations of optimization algorithms on noisy quantum devices. Nat. Phys. 17, 1221–1227 (2021). https://doi.org/10.1038/s41567-021-01356-3

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