Advanced Search
Article Contents
Article Contents

Estimation and uncertainty quantification for the output from quantum simulators

  • * Corresponding author

    * Corresponding author 
Abstract Full Text(HTML) Figure(5) Related Papers Cited by
  • The problem of estimating certain distributions over {0, 1}d is considered here. The distribution represents a quantum system of d qubits, where there are non-trivial dependencies between the qubits. A maximum entropy approach is adopted to reconstruct the distribution from exact moments or observed empirical moments. The Robbins Monro algorithm is used to solve the intractable maximum entropy problem, by constructing an unbiased estimator of the un-normalized target with a sequential Monte Carlo sampler at each iteration. In the case of empirical moments, this coincides with a maximum likelihood estimator. A Bayesian formulation is also considered in order to quantify uncertainty a posteriori. Several approaches are proposed in order to tackle this challenging problem, based on recently developed methodologies. In particular, unbiased estimators of the gradient of the log posterior are constructed and used within a provably convergent Langevin-based Markov chain Monte Carlo method. The methods are illustrated on classically simulated output from quantum simulators.

    Mathematics Subject Classification: Primary: 62C10, 65C05; Secondary: 81P68.


    \begin{equation} \\ \end{equation}
  • 加载中
  • Figure 1.  Truth (left) and reconstruction after $ K = 10^4 $ steps (middle) for known truth. The error as a function of iteration is given in the right plot

    Figure 2.  The reconstruction with $ 1000 $ (left) and $ 50 $ (right) observations are given in the top row, along with the corresponding convergence plots in the second row. The bottom row shows the actual second moments $ \widehat m $ (left), with $ M = 50 $ observations, which are used to train the model, and the moments under the reconstruction (right)

    Figure 3.  Convergence of SGD with the debiased estimator described in Section 6.3 (left), the original unbiased estimator $ \widehat F $ (middle), and the simple consistent but biased estimator (right)

    Figure 4.  Illustration of pairwise marginal UQ for the posterior on the diagonal of $ \Lambda $ with $ M = 1000 $ observations and $ d = 4 $ qubits. The true value of the parameters is indicated in red

    Figure 5.  Illustration of pairwise marginal UQ for the posterior on the diagonal of $ \Lambda $ with $ M = 10^6 $ observations and $ d = 4 $ qubits. The true value of the parameters is indicated in red

  • [1] C. Andrieu and G. O. Roberts, et al., The pseudo-marginal approach for efficient Monte Carlo computations, The Annals of Statistics, 37 (2009), 697-725.  doi: 10.1214/07-AOS574.
    [2] M. A. Beaumont, Estimation of population growth or decline in genetically monitored populations, Genetics, 164 (2003), 1139-1160. 
    [3] A. BeskosD. Crisan and A. Jasra, et al., On the stability of sequential monte carlo methods in high dimensions, The Annals of Applied Probability, 24 (2014), 1396-1445.  doi: 10.1214/13-AAP951.
    [4] J. Bierkens, P. Fearnhead and G. Roberts, The zig-zag process and super-efficient sampling for bayesian analysis of big data, preprint, arXiv: 1607.03188. doi: 10.1214/18-AOS1715.
    [5] R. Blume-Kohout, Optimal, reliable estimation of quantum states, New Journal of Physics, 12 (2010), 043034. doi: 10.1088/1367-2630/12/4/043034.
    [6] A. Bouchard-Côté, S. J. Vollmer and A. Doucet, The bouncy particle sampler: A nonreversible rejection-free Markov chain Monte Carlo method, Journal of the American Statistical Association, 1–13. doi: 10.1080/01621459.2017.1294075.
    [7] S. Boyd and  L. VandenbergheConvex Optimization, Cambridge University Press, 2004.  doi: 10.1017/CBO9780511804441.
    [8] E. J. Candès and B. Recht, Exact matrix completion via convex optimization, Foundations of Computational Mathematics, 9 (2009), 717. doi: 10.1007/s10208-009-9045-5.
    [9] A. M. Childs, R. B. Patterson and D. J. MacKay, Exact sampling from nonattractive distributions using summary states, Physical Review E, 63 (2001), 036113. doi: 10.1103/PhysRevE.63.036113.
    [10] N. Chopin, A sequential particle filter method for static models, Biometrika, 89 (2002), 539-552.  doi: 10.1093/biomet/89.3.539.
    [11] T. M. Cover and J. A. Thomas, Elements of Information Theory, John Wiley & Sons, 2012.
    [12] M. H. Davis, Piecewise-deterministic Markov processes: A general class of non-diffusion stochastic models, Journal of the Royal Statistical Society. Series B (Methodological), 353–388. doi: 10.1111/j.2517-6161.1984.tb01308.x.
    [13] P. Del Moral, Feynman-Kac Formulae, Genealogical and Interacting Particle Systems with Applications, Probability and its Applications (New York), Springer-Verlag, New York, 2004. doi: 10.1007/978-1-4684-9393-1.
    [14] P. Del MoralA. Doucet and A. Jasra, Sequential Monte Carlo samplers, Journal of the Royal Statistical Society: Series B (Statistical Methodology), 68 (2006), 411-436.  doi: 10.1111/j.1467-9868.2006.00553.x.
    [15] P. Del MoralA. Doucet and A. Jasra, Sequential Monte Carlo samplers, J. R. Stat. Soc. Ser. B Stat. Methodol., 68 (2006), 411-436.  doi: 10.1111/j.1467-9868.2006.00553.x.
    [16] J. Franks, A. Jasra, K. Law and M. Vihola, Unbiased inference for discretely observed hidden markov model diffusions, preprint, arXiv: 1807.10259.
    [17] M. Giles, T. Nagapetyan, L. Szpruch, S. Vollmer and K. Zygalakis, Multilevel Monte Carlo for scalable Bayesian computations, preprint, arXiv: 1609.06144. doi: 10.1017/S096249291500001X.
    [18] W. R. Gilks, Markov Chain Monte Carlo, Wiley Online Library, 2005. doi: 10.1002/0470011815.b2a14021.
    [19] M. Girolami and B. Calderhead, Riemann manifold Langevin and Hamiltonian Monte Carlo methods, Journal of the Royal Statistical Society: Series B (Statistical Methodology), 73 (2011), 123-214.  doi: 10.1111/j.1467-9868.2010.00765.x.
    [20] C. Granade, J. Combes and D. Cory, Practical bayesian tomography, New Journal of Physics, 18 (2016), 033024. doi: 10.1088/1367-2630/18/3/033024.
    [21] P. J. GreenK. ŁatuszyńskiM. Pereyra and C. P. Robert, Bayesian computation: a summary of the current state, and samples backwards and forwards, Statistics and Computing, 25 (2015), 835-862.  doi: 10.1007/s11222-015-9574-5.
    [22] D. Gross, Y.-K. Liu, S. T. Flammia, S. Becker and J. Eisert, Quantum state tomography via compressed sensing, Physical Review Letters, 105 (2010), 150401. doi: 10.1103/PhysRevLett.105.150401.
    [23] W. K. Hastings, Monte Carlo sampling methods using Markov chains and their applications, Biometrika, 57 (1970), 97-109.  doi: 10.1093/biomet/57.1.97.
    [24] Z. Hradil, Quantum-state estimation, Physical Review A, 55 (1997), R1561. doi: 10.1103/PhysRevA.55.R1561.
    [25] F. Huszár and N. M. Houlsby, Adaptive bayesian quantum tomography, Physical Review A, 85 (2012), 052120.
    [26] C. Jarzynski, Nonequilibrium equality for free energy differences, Physical Review Letters, 78 (1997), 2690. doi: 10.1103/PhysRevLett.78.2690.
    [27] E. T. Jaynes, Information theory and statistical mechanics, Physical Review, 106 (1957), 620. doi: 10.1103/PhysRev.106.620.
    [28] E. T. Jaynes, Information theory and statistical mechanics. Ⅱ, Physical Review, 108 (1957), 171. doi: 10.1103/PhysRev.108.171.
    [29] K. Jones, Principles of quantum inference, Annals of Physics, 207 (1991), 140-170.  doi: 10.1016/0003-4916(91)90182-8.
    [30] H. Kushner and G. G. Yin, Stochastic Approximation and Recursive Algorithms and Applications, vol. 35, Springer Science & Business Media, 2003.
    [31] A.-M. LyneM. GirolamiY. AtchadéH. Strathmann and D. Simpson, et al., On Russian roulette estimates for bayesian inference with doubly-intractable likelihoods, Statistical Science, 30 (2015), 443-467.  doi: 10.1214/15-STS523.
    [32] D. McLeish, A general method for debiasing a Monte Carlo estimator, Monte Carlo Methods and Applications, 17 (2011), 301–315. doi: 10.1515/mcma.2011.013.
    [33] J. MøllerA. N. PettittR. Reeves and K. K. Berthelsen, An efficient Markov chain Monte Carlo method for distributions with intractable normalising constants, Biometrika, 93 (2006), 451-458.  doi: 10.1093/biomet/93.2.451.
    [34] K. P. Murphy, Machine learning: A probabilistic perspective.
    [35] I. Murray, Z. Ghahramani and D. J. MacKay, Mcmc for doubly-intractable distributions, in Proceedings of the Twenty-Second Conference on Uncertainty in Artificial Intelligence, AUAI Press, 2006, 359–366.
    [36] R. M. Neal, Annealed importance sampling, Statistics and Computing, 11 (2001), 125-139.  doi: 10.1023/A:1008923215028.
    [37] M. Paris and J. Rehacek, Quantum State Estimation, 1st edition, Springer Publishing Company, Incorporated, 2010. doi: 10.1007/b98673.
    [38] S. Patterson and Y. W. Teh, Stochastic gradient riemannian Langevin dynamics on the probability simplex, in Advances in Neural Information Processing Systems, 2013, 3102–3110.
    [39] E. A. Peters, et al., Rejection-free Monte Carlo sampling for general potentials, Physical Review E, 85 (2012), 026703.
    [40] J. Preskill, Quantum computing in the NISQ era and beyond, preprint, arXiv: 1801.00862. doi: 10.1098/rspa.1998.0171.
    [41] J. G. Propp and D. B. Wilson, Exact sampling with coupled markov chains and applications to statistical mechanics, Random Structures & Algorithms, 9 (1996), 223-252.  doi: 10.1002/(SICI)1098-2418(199608/09)9:1/2<223::AID-RSA14>3.0.CO;2-O.
    [42] C.-H. Rhee and P. W. Glynn, A new approach to unbiased estimation for sde's, in Proceedings of the Winter Simulation Conference, Winter Simulation Conference, 2012, 17. doi: 10.1109/WSC.2012.6465150.
    [43] C.-H. Rhee and P. W. Glynn, Unbiased estimation with square root convergence for sde models, Operations Research, 63 (2015), 1026-1043.  doi: 10.1287/opre.2015.1404.
    [44] H. Robbins and S. Monro, A stochastic approximation method, The Annals of Mathematical Statistics, 400–407. doi: 10.1214/aoms/1177729586.
    [45] G. O. Roberts and J. S. Rosenthal, et al., Optimal scaling for various Metropolis-Hastings algorithms, Statistical Science, 16 (2001), 351-367.  doi: 10.1214/ss/1015346320.
    [46] Y. W. TehA. H. Thiery and S. J. Vollmer, Consistency and fluctuations for stochastic gradient Langevin dynamics, The Journal of Machine Learning Research, 17 (2016), 193-225. 
    [47] M. Vihola, Unbiased estimators and multilevel Monte Carlo, Operations Research, 66 (2017), 448-462.  doi: 10.1287/opre.2017.1670.
    [48] S. J. VollmerK. C. Zygalakis and Y. W. Teh, Exploration of the (non-) asymptotic bias and variance of stochastic gradient langevin dynamics, The Journal of Machine Learning Research, 17 (2016), 5504-5548. 
    [49] C. Wei and I. Murray, Markov chain truncation for doubly-intractable inference, in Artificial Intelligence and Statistics, 2017,776–784.
    [50] M. Welling and Y. W. Teh, Bayesian learning via stochastic gradient Langevin dynamics, in Proceedings of the 28th International Conference on Machine Learning (ICML-11), 2011,681–688. doi: 10.4310/CIS.2012.v12.n3.a3.
  • 加载中



Article Metrics

HTML views(2387) PDF downloads(266) Cited by(0)

Access History



    DownLoad:  Full-Size Img  PowerPoint