Efficient qubit phase estimation using adaptive measurements

Marco A. Rodríguez-García1, Isaac Pérez Castillo2, and P. Barberis-Blostein1

1Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, Universidad Nacional Autónoma de México, Ciudad Universitaria, Ciudad de México 04510, Mexico
2Departamento de Física, Universidad Autónoma Metropolitana-Iztapalapa, San Rafael Atlixco 186, Ciudad de México 09340, Mexico

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Abstract

Estimating correctly the quantum phase of a physical system is a central problem in quantum parameter estimation theory due to its wide range of applications from quantum metrology to cryptography. Ideally, the optimal quantum estimator is given by the so-called quantum Cramér-Rao bound, so any measurement strategy aims to obtain estimations as close as possible to it. However, more often than not, the current state-of-the-art methods to estimate quantum phases fail to reach this bound as they rely on maximum likelihood estimators of non-identifiable likelihood functions. In this work we thoroughly review various schemes for estimating the phase of a qubit, identifying the underlying problem which prohibits these methods to reach the quantum Cramér-Rao bound, and propose a new adaptive scheme based on covariant measurements to circumvent this problem. Our findings are carefully checked by Monte Carlo simulations, showing that the method we propose is both mathematically and experimentally more realistic and more efficient than the methods currently available.

Many applications in the second quantum revolution we are presently living in, from quantum computing, quantum metrology to quantum cryptography can be recast as that of estimating the quantum phase of qubits. Unfortunately, using state-of-the-art methods, it is generally not possible to produce efficient estimators of this phase due to a non identifiability problem. In our work, we propose a novel adaptive approach, using covariant estimators and confidence intervals, which is able to achieve the theoretical minimal error, the so-called Cramér-Rao bound, thus offering an alternative and efficient method to the currently available techniques.

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[1] O. E. Barndorff-Nielsen and R. D. Gill. Fisher information in quantum statistics. J. Phys. A, 33 (24): 4481–4490, 2000. 10.1088/​0305-4470/​33/​24/​306.
https:/​/​doi.org/​10.1088/​0305-4470/​33/​24/​306

[2] D. W. Berry and H. M. Wiseman. Adaptive measurements and optimal states for quantum interferometry. Technical Digest - Summaries of Papers Presented at the Quantum Electronics and Laser Science Conference, (5): 60–61, 2001. 10.1109/​QELS.2001.961853.
https:/​/​doi.org/​10.1109/​QELS.2001.961853

[3] D. W. Berry, B. L. Higgins, S. D. Bartlett, M. W. Mitchell, G. J. Pryde, and H. M. Wiseman. How to perform the most accurate possible phase measurements. Phys. Rev. A, 80 (5): 1–22, 2009. 10.1103/​PhysRevA.80.052114.
https:/​/​doi.org/​10.1103/​PhysRevA.80.052114

[4] S. Boixo and R. D. Somma. Parameter estimation with mixed-state quantum computation. Phys. Rev. A, 77: 052320, 2008. 10.1103/​PhysRevA.77.052320.
https:/​/​doi.org/​10.1103/​PhysRevA.77.052320

[5] S. L. Braunstein and C. M. Caves. Statistical distance and the geometry of quantum states. Phys. Rev. Lett., 72: 3439–3443, 1994. 10.1103/​PhysRevLett.72.3439.
https:/​/​doi.org/​10.1103/​PhysRevLett.72.3439

[6] G. Casella and R.L. Berger. Statistical Inference. Duxbury advanced series in statistics and decision sciences. Thomson Learning, 2002. ISBN 9780534243128.

[7] F. Chapeau-Blondeau. Optimizing qubit phase estimation. Phys. Rev. A, 94: 022334, 2016. 10.1103/​PhysRevA.94.022334.
https:/​/​doi.org/​10.1103/​PhysRevA.94.022334

[8] D. Dee and A. Da Silva. Maximum-likelihood estimation of forecast and observation error covariance parameters. part I: Methodology. Mon. Weather Rev., 127, 09 1998. 10.1175/​1520-0493(1999)127<1822:MLEOFA>2.0.CO;2.
https:/​/​doi.org/​10.1175/​1520-0493(1999)127<1822:MLEOFA>2.0.CO;2

[9] M. H. DeGroot and M.J. Schervish. Probability and Statistics. Addison-Wesley, 2012. ISBN 9780321500465. 10.1080/​09332480.2013.845457.
https:/​/​doi.org/​10.1080/​09332480.2013.845457

[10] J. P. Dowling. Quantum optical metrology – the lowdown on high-N00N states. Contemporary Physics, 49 (2): 125–143, 2008. 10.1080/​00107510802091298.
https:/​/​doi.org/​10.1080/​00107510802091298

[11] R. Engle and D. McFadden. Handbook of Econometrics, volume 4. North Holland, 1994. ISBN 0444887660,9780444887665.

[12] F. Fröwis, M. Skotiniotis, B. Kraus, and W. Dür. Optimal quantum states for frequency estimation. New Journal of Physics, 16 (8): 083010, 2014. 10.1088/​1367-2630/​16/​8/​083010.
https:/​/​doi.org/​10.1088/​1367-2630/​16/​8/​083010

[13] A. Fujiwara. Strong consistency and asymptotic efficiency for adaptive quantum estimation problems. J. Phys. A, 39 (40): 12489–12504, sep 2006. 10.1088/​0305-4470/​39/​40/​014.
https:/​/​doi.org/​10.1088/​0305-4470/​39/​40/​014

[14] V. Giovannetti, S. Lloyd, and L. Maccone. Quantum-enhanced measurements: Beating the standard quantum limit. Science, 306 (5700): 1330–1336, 2004. 10.1126/​science.1104149.
https:/​/​doi.org/​10.1126/​science.1104149

[15] C. W. Helstrom. Quantum detection and estimation theory. J. Stat. Phys., 1 (2): 231–252, 1969. 10.1007/​BF01007479.
https:/​/​doi.org/​10.1007/​BF01007479

[16] A. S. Holevo. Probabilistic and Statistical Aspects of Quantum Theory. Springer Science & Business Media, 1982. 10.1007/​978-88-7642-378-9.
https:/​/​doi.org/​10.1007/​978-88-7642-378-9

[17] A. S. Holevo. Asymptotic estimation of a shift parameter of a quantum state. Theory of Probability and its Applications, 49 (2): 207–220, 2005. 10.1137/​S0040585X97981044.
https:/​/​doi.org/​10.1137/​S0040585X97981044

[18] A.S Holevo. Statistical decision theory for quantum systems. Journal of Multivariate Analysis, 3 (4): 337 – 394, 1973. 10.1016/​0047-259X(73)90028-6.
https:/​/​doi.org/​10.1016/​0047-259X(73)90028-6

[19] Z. Huang, K. R. Motes, P. M. Anisimov, J. P. Dowling, and D. W. Berry. Adaptive phase estimation with two-mode squeezed vacuum and parity measurement. Phys. Rev. A, 95: 053837, 2017. 10.1103/​PhysRevA.95.053837.
https:/​/​doi.org/​10.1103/​PhysRevA.95.053837

[20] E. L. Lehmann and G. Casella. Theory of Point Estimation. Springer-Verlag, New York, NY, USA, second edition, 1998.

[21] M. Manisera and P. Zuccolotto. Identifiability of a model for discrete frequency distributions with a multidimensional parameter space. Journal of Multivariate Analysis, 140: 302–316, 2015. 10.1016/​j.jmva.2015.05.011.
https:/​/​doi.org/​10.1016/​j.jmva.2015.05.011

[22] L. S. Martin, W. P. Livingston, S. Hacohen-Gourgy, and I. Wiseman, H. M.and Siddiqi. Implementation of a canonical phase measurement with quantum feedback. Nature Physics, 2020. 10.1038/​s41567-020-0939-0.
https:/​/​doi.org/​10.1038/​s41567-020-0939-0

[23] E. Merzbacher. Quantum Mechanics. North Holland, 3 edition, 1998. ISBN 0471887021, 9780471887027.

[24] A. Monras. Optimal phase measurements with pure Gaussian states. Phys. Rev. A, 73: 033821, 2006. 10.1103/​PhysRevA.73.033821.
https:/​/​doi.org/​10.1103/​PhysRevA.73.033821

[25] H. Nagaoka. On the Parameter Estimation Problem for Quantum Statistical Models, pages 125–132. 2005. 10.1142/​9789812563071_0011.
https:/​/​doi.org/​10.1142/​9789812563071_0011

[26] M. A. Nielsen and I. L. Chuang. Quantum Computation and Quantum Information: 10th Anniversary Edition. Cambridge University Press, 2010. 10.1017/​CBO9780511976667.
https:/​/​doi.org/​10.1017/​CBO9780511976667

[27] C. Oh, C. Lee, C. Rockstuhl, H. Jeong, J. Kim, H. Nha, and S. Lee. Optimal gaussian measurements for phase estimation in single-mode gaussian metrology. npj Quantum Information, 5 (1): 1–9, 2019. 10.1038/​s41534-019-0124-4.
https:/​/​doi.org/​10.1038/​s41534-019-0124-4

[28] R. Okamoto, M. Iefuji, S. Oyama, K. Yamagata, H. Imai, A. Fujiwara, and S. Takeuchi. Experimental demonstration of adaptive quantum state estimation. Phys. Rev. Lett., 109: 130404, 2012. 10.1103/​PhysRevLett.109.130404.
https:/​/​doi.org/​10.1103/​PhysRevLett.109.130404

[29] M. G. A. Paris. Quantum estimation for quantum technology. International Journal of Quantum Information, 7: 125–137, 2009. 10.1142/​S0219749909004839.
https:/​/​doi.org/​10.1142/​S0219749909004839

[30] Y. Peng and H. Fan. Feedback ansatz for adaptive-feedback quantum metrology training with machine learning. Phys. Rev. A, 101: 022107, 2020. 10.1103/​PhysRevA.101.022107.
https:/​/​doi.org/​10.1103/​PhysRevA.101.022107

[31] L. Pezzè and A. Smerzi. Quantum theory of phase estimation, pages 691–741. 2014. 10.3254/​978-1-61499-448-0-691.
https:/​/​doi.org/​10.3254/​978-1-61499-448-0-691

[32] M. A. Rodríguez-García. Qubit-phase-estimation. https:/​/​github.com/​Gateishion/​Quantum-Phase-Estimation.git, 2020.
https:/​/​github.com/​Gateishion/​Quantum-Phase-Estimation.git

[33] F. Toscano, W. P. Bastos, and R. L. de Matos Filho. Attainability of the quantum information bound in pure-state models. Phys. Rev. A, 95: 042125, 2017. 10.1103/​PhysRevA.95.042125.
https:/​/​doi.org/​10.1103/​PhysRevA.95.042125

[34] E. P. Wigner. Symmetries and reflections. Ox Bow Press., reprint edition edition, 1979.

[35] K. Yamagata, A. Fujiwara, and R. D. Gill. Quantum local asymptotic normality based on a new quantum likelihood ratio. Ann. Statist., (4): 2197–2217, 08 . 10.1214/​13-AOS1147.
https:/​/​doi.org/​10.1214/​13-AOS1147

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