The Annals of Statistics

Optimal large-scale quantum state tomography with Pauli measurements

Tony Cai, Donggyu Kim, Yazhen Wang, Ming Yuan, and Harrison H. Zhou

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Quantum state tomography aims to determine the state of a quantum system as represented by a density matrix. It is a fundamental task in modern scientific studies involving quantum systems. In this paper, we study estimation of high-dimensional density matrices based on Pauli measurements. In particular, under appropriate notion of sparsity, we establish the minimax optimal rates of convergence for estimation of the density matrix under both the spectral and Frobenius norm losses; and show how these rates can be achieved by a common thresholding approach. Numerical performance of the proposed estimator is also investigated.

Article information

Source
Ann. Statist., Volume 44, Number 2 (2016), 682-712.

Dates
Received: January 2015
Revised: August 2015
First available in Project Euclid: 17 March 2016

Permanent link to this document
https://projecteuclid.org/euclid.aos/1458245732

Digital Object Identifier
doi:10.1214/15-AOS1382

Mathematical Reviews number (MathSciNet)
MR3476614

Zentralblatt MATH identifier
1341.62116

Subjects
Primary: 62H12: Estimation 81P50: Quantum state estimation, approximate cloning
Secondary: 62C20: Minimax procedures 62P35: Applications to physics 81P45: Quantum information, communication, networks [See also 94A15, 94A17] 81P68: Quantum computation [See also 68Q05, 68Q12]

Keywords
Compressed sensing density matrix Pauli matrices quantum measurement quantum probability quantum statistics sparse representation spectral norm minimax estimation

Citation

Cai, Tony; Kim, Donggyu; Wang, Yazhen; Yuan, Ming; Zhou, Harrison H. Optimal large-scale quantum state tomography with Pauli measurements. Ann. Statist. 44 (2016), no. 2, 682--712. doi:10.1214/15-AOS1382. https://projecteuclid.org/euclid.aos/1458245732


Export citation

References

  • Alquier, P., Butucea, C., Hebiri, M. and Meziani, K. (2013). Rank penalized estimation of a quantum system. Phys. Rev. A 88 032133.
  • Artiles, L. M., Gill, R. D. and Guţă, M. I. (2005). An invitation to quantum tomography. J. R. Stat. Soc. Ser. B. Stat. Methodol. 67 109–134.
  • Aspuru-Guzik, A., Dutoi, A. D., Love, P. J. and Head-Gordon, M. (2005). Simulated quantum computation of molecular energies. Science 309 1704–1707.
  • Aubry, J.-M., Butucea, C. and Meziani, K. (2009). State estimation in quantum homodyne tomography with noisy data. Inverse Probl. 25 015003, 22.
  • Benenti, G., Casati, G. and Strini, G. (2004). Principles of Quantum Computation and Information. Vol. I: Basic Concepts. World Scientific, River Edge, NJ.
  • Benenti, G., Casati, G. and Strini, G. (2007). Principles of Quantum Computation and Information. Vol. II: Basic Tools and Special Topics. World Scientific, Hackensack, NJ.
  • Berry, D. W., Childs, A. M., Cleve, R., Kothari, R. and Somma, R. D. (2014). Exponential improvement in precision for simulating sparse Hamiltonians. In STOC’14—Proceedings of the 2014 ACM Symposium on Theory of Computing 283–292. ACM, New York.
  • Boixo, S., Rnnow, T. F., Isakov, S. V., Wang, Z., Wecker, D., Lidar, D. A., Martinis, J. M. and Troyer, M. (2014). Evidence for quantum annealing with more than one hundred qubits. Nature Physics 10 218–224.
  • Britton, J. W., Sawyer, B. C., Keith, A., Wang, C.-C. J., Freericks, J. K., Uys, H., Biercuk, M. J. and Bollinger, J. J. (2012). Engineered 2D Ising interactions on a trapped-ion quantum simulator with hundreds of spins. Nature 484 489–492.
  • Brumfiel, G. (2012). Simulation: Quantum leaps. Nature 491 322–324.
  • Bunea, F., She, Y. and Wegkamp, M. H. (2011). Optimal selection of reduced rank estimators of high-dimensional matrices. Ann. Statist. 39 1282–1309.
  • Bunea, F., She, Y. and Wegkamp, M. H. (2012). Joint variable and rank selection for parsimonious estimation of high-dimensional matrices. Ann. Statist. 40 2359–2388.
  • Butucea, C., Guţă, M. and Artiles, L. (2007). Minimax and adaptive estimation of the Wigner function in quantum homodyne tomography with noisy data. Ann. Statist. 35 465–494.
  • Cai, T. T. and Zhang, A. (2015). ROP: Matrix recovery via rank-one projections. Ann. Statist. 43 102–138.
  • Cai, T. T. and Zhou, H. H. (2012). Optimal rates of convergence for sparse covariance matrix estimation. Ann. Statist. 40 2389–2420.
  • Candès, E. J. and Plan, Y. (2009). Matrix completion with noise. Proceedings of the IEEE 98 925–936.
  • Candès, E. J. and Plan, Y. (2011). Tight oracle inequalities for low-rank matrix recovery from a minimal number of noisy random measurements. IEEE Trans. Inform. Theory 57 2342–2359.
  • Candès, E. J. and Recht, B. (2009). Exact matrix completion via convex optimization. Found. Comput. Math. 9 717–772.
  • Candès, E. J. and Tao, T. (2010). The power of convex relaxation: Near-optimal matrix completion. IEEE Trans. Inform. Theory 56 2053–2080.
  • Donoho, D. L. (2006). Compressed sensing. IEEE Trans. Inform. Theory 52 1289–1306.
  • Donoho, D. L. and Johnstone, I. M. (1994). Minimax risk over $l_{p}$-balls for $l_{q}$-error. Probab. Theory Related Fields 99 277–303.
  • Flammia, S. T., Gross, D., Liu, Y. K. and Eisert, J. (2012). Quantum tomography via compressed sensing: Error bounds, sample complexity and efficient estimators. New J. Phys. 14 095022.
  • Gross, D. (2011). Recovering low-rank matrices from few coefficients in any basis. IEEE Trans. Inform. Theory 57 1548–1566.
  • Gross, D., Liu, Y.-K., Flammia, S. T., Becker, S. and Eisert, J. (2010). Quantum state tomography via compressed sensing. Phys. Rev. Lett. 105 150401.
  • Guţă, M. and Artiles, L. (2007). Minimax estimation of the Wigner function in quantum homodyne tomography with ideal detectors. Math. Methods Statist. 16 1–15.
  • Häffner, H., Hänsel, W., Roos, C. F., Benhelm, J., Chek-al-Kar, D., Chwalla, M., Körber, T., Rapol, U. D., Riebe, M., Schmidt, P. O., Becher, C., Gühne, O., Dür, W. and Blatt, R. (2005). Scalable multiparticle entanglement of trapped ions. Nature 438 643–646.
  • Holevo, A. S. (1982). Probabilistic and Statistical Aspects of Quantum Theory. North-Holland Series in Statistics and Probability 1. North-Holland, Amsterdam.
  • Johnson, M. W., Amin, M. H. S., Gildert, S., Lanting, T., Hamze, F., Dickson, N., Harris, R., Berkley, A. J., Johansson, J., Bunyk, P., Chapple, E. M., Enderud, C., Hilton, J. P., Karimi, K., Ladizinsky, E., Ladizinsky, N., Oh, T., Perminov, I., Rich1, C., Thom, M. C., Tolkacheva, E., Truncik, C. J. S., Uchaikin, S., Wang, J., Wilson, B. and Rose, G. (2011). Quantum annealing with manufactured spins. Nature 473 194–198.
  • Jones, N. (2013). Computing: The quantum company. Nature 498 286–288.
  • Keshavan, R. H., Montanari, A. and Oh, S. (2010). Matrix completion from noisy entries. J. Mach. Learn. Res. 11 2057–2078.
  • Klopp, O. (2011). Rank penalized estimators for high-dimensional matrices. Electron. J. Stat. 5 1161–1183.
  • Klopp, O. (2012). Noisy low-rank matrix completion with general sampling distribution. Manuscript.
  • Koltchinskii, V. (2011). Von Neumann entropy penalization and low-rank matrix estimation. Ann. Statist. 39 2936–2973.
  • Koltchinskii, V., Lounici, K. and Tsybakov, A. B. (2011). Nuclear-norm penalization and optimal rates for noisy low-rank matrix completion. Ann. Statist. 39 2302–2329.
  • Lanyon, B. P., Whitfield, J. D., Gillett, G. G., Goggin, M. E., Almeida, M. P., Kassal, I., Biamonte, J. D., Mohseni, M., Powell, B. J., Barbieri, M., Aspuru-Guzik, A. and White, A. G. (2010). Towards quantum chemistry on a quantum computer. Nat. Chem. 2 106–111.
  • Le Cam, L. (1973). Convergence of estimates under dimensionality restrictions. Ann. Statist. 1 38–53.
  • Liu, Y. K. (2011). Universal low-rank matrix recovery from Pauli measurements. Unpublished manuscript.
  • Negahban, S. and Wainwright, M. J. (2011). Estimation of (near) low-rank matrices with noise and high-dimensional scaling. Ann. Statist. 39 1069–1097.
  • Nielsen, M. A. and Chuang, I. L. (2000). Quantum Computation and Quantum Information. Cambridge Univ. Press, Cambridge.
  • Recht, B. (2011). A simpler approach to matrix completion. J. Mach. Learn. Res. 12 3413–3430.
  • Recht, B., Fazel, M. and Parrilo, P. A. (2010). Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization. SIAM Rev. 52 471–501.
  • Rohde, A. and Tsybakov, A. B. (2011). Estimation of high-dimensional low-rank matrices. Ann. Statist. 39 887–930.
  • Sakurai, J. J. and Napolitano, J. (2010). Modern Quantum Mechanics, 2nd ed. Addison-Wesley, Reading, MA.
  • Senko, C., Smith, J., Richerme, P., Lee, A., Campbell, W. C. and Monroe, C. (2014). Coherent imaging spectroscopy of a quantum many-body spin system. Science 345 430–433.
  • Shankar, R. (1994). Principles of Quantum Mechanics, 2nd ed. Plenum Press, New York.
  • Tao, M., Wang, Y. and Zhou, H. H. (2013). Optimal sparse volatility matrix estimation for high-dimensional Itô processes with measurement errors. Ann. Statist. 41 1816–1864.
  • Wang, Y. (2011). Quantum Monte Carlo simulation. Ann. Appl. Stat. 5 669–683.
  • Wang, Y. (2012). Quantum computation and quantum information. Statist. Sci. 27 373–394.
  • Wang, Y. (2013). Asymptotic equivalence of quantum state tomography and noisy matrix completion. Ann. Statist. 41 2462–2504.
  • Wang, Y. and Xu, C. (2015). Density matrix estimation in quantum homodyne tomography. Statist. Sinica 25 953–973.
  • Yu, B. (1997). Assouad, Fano, and Le Cam. In Festschrift for Lucien Le Cam 423–435. Springer, New York.
  • Zhang, C. (2012). Minimax $\ell_{q}$ risk in $\ell_{p}$ balls. Inst. Math. Stat. Collect. 8 78–89.