Efficient Bayesian phase estimation using mixed priors

Ewout van den Berg

IBM Quantum, IBM T.J. Watson Research Center, Yorktown Heights, NY, USA

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.

Abstract

We describe an efficient implementation of Bayesian quantum phase estimation in the presence of noise and multiple eigenstates. The main contribution of this work is the dynamic switching between different representations of the phase distributions, namely truncated Fourier series and normal distributions. The Fourier-series representation has the advantage of being exact in many cases, but suffers from increasing complexity with each update of the prior. This necessitates truncation of the series, which eventually causes the distribution to become unstable. We derive bounds on the error in representing normal distributions with a truncated Fourier series, and use these to decide when to switch to the normal-distribution representation. This representation is much simpler, and was proposed in conjunction with rejection filtering for approximate Bayesian updates. We show that, in many cases, the update can be done exactly using analytic expressions, thereby greatly reducing the time complexity of the updates. Finally, when dealing with a superposition of several eigenstates, we need to estimate the relative weights. This can be formulated as a convex optimization problem, which we solve using a gradient-projection algorithm. By updating the weights at exponentially scaled iterations we greatly reduce the computational complexity without affecting the overall accuracy.

► BibTeX data

► References

[1] Daniel S. Abrams and Seth Lloyd. Quantum algorithm providing exponential speed increase for finding eigenvalues and eigenvectors. Physical Review Letters, 83 (24): 5162–5165, 1999. 10.1103/​PhysRevLett.83.5162.
https:/​/​doi.org/​10.1103/​PhysRevLett.83.5162

[2] Alán Aspuru-Guzik, Anthony D. Dutoi, Peter J. Love, and Martin Head-Gordon. Simulated quantum computation of molecular energies. Science, 309 (5741): 1704–1707, 2005. 10.1126/​science.1113479.
https:/​/​doi.org/​10.1126/​science.1113479

[3] Dominic W. Berry, Brendon L. Higgins, Stephen D. Bartlett, Morgan W. Mitchell, Geoff J. Pryde, and Howard M. Wiseman. How to perform the most accurate possible phase measurements. Physical Review A, 80 (5): 052114, 2009. 10.1103/​PhysRevA.80.052114.
https:/​/​doi.org/​10.1103/​PhysRevA.80.052114

[4] Dimitri P. Bertsekas. Nonlinear Programming. Athena, 2nd edition, 1999.

[5] Wlodzimierz Bryc. The Normal Distribution: Characterizations with Applications. Springer-Verlag, 1995. 10.1007/​978-1-4612-2560-7.
https:/​/​doi.org/​10.1007/​978-1-4612-2560-7

[6] Richard Cleve, Arthur Ekert, Chiara Macchiavello, and Michele Mosca. Quantum algorithms revisited. Proceedings of the Royal Society A, 454 (1969): 339–354, 1998. 10.1098/​rspa.1998.0164.
https:/​/​doi.org/​10.1098/​rspa.1998.0164

[7] Laurent Condat. Fast projection onto the simplex and the $\ell_1$ ball. Mathematical Programming, Series A, 158 (1–2): 575–585, 2016. 10.1007/​s10107-015-0946-6.
https:/​/​doi.org/​10.1007/​s10107-015-0946-6

[8] Loukas Grafakos. Classical Fourier analysis. Springer, 3rd edition, 2008. 10.1007/​978-1-4939-1194-3.
https:/​/​doi.org/​10.1007/​978-1-4939-1194-3

[9] Alexander J. F. Hayes and Dominic W. Berry. Swarm optimization for adaptive phase measurement with low visibility. Physical Review A, 89 (1): 013838, 2014. 10.1103/​PhysRevA.89.013838.
https:/​/​doi.org/​10.1103/​PhysRevA.89.013838

[10] Shelby Kimmel, Guang Hao Low, and Theodore J. Yoder. Robust calibration of a universal single-qubit gate set via robust phase estimation. Physical Review A, 92 (6): 062315, 2015. 10.1103/​PhysRevA.92.062315.
https:/​/​doi.org/​10.1103/​PhysRevA.92.062315

[11] Alexei Yu. Kitaev. Quantum measurements and the Abelian stabilizer problem. arXiv preprint quant-ph/​9511026, 1995. URL https:/​/​arxiv.org/​abs/​quant-ph/​9511026. (See also Electronic Colloquium on Computational Complexity, TR96-003, 1996).
arXiv:quant-ph/9511026

[12] Jorge Nocedal and Stephen J. Wright. Numerical Optimization. Springer series in operations research and financial engineering. Springer, second edition, 2006. 10.1007/​978-0-387-40065-5.
https:/​/​doi.org/​10.1007/​978-0-387-40065-5

[13] Thomas E. O'Brien, Brian Tarasinski, and Barbara M. Terhal. Quantum phase estimation of multiple eigenvalues for small-scale (noisy) experiments. New Journal of Physics, 21: 023022, 2019. 10.1088/​1367-2630/​aafb8e.
https:/​/​doi.org/​10.1088/​1367-2630/​aafb8e

[14] Alessandro Roggero. Spectral-density estimation with the Gaussian integral transform. Physical Review A, 102: 022409, Aug 2020. 10.1103/​PhysRevA.102.022409.
https:/​/​doi.org/​10.1103/​PhysRevA.102.022409

[15] Peter W. Shor. Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM Journal on Computing, 26 (5): 1484–1509, 1997. 10.1137/​S0097539795293172.
https:/​/​doi.org/​10.1137/​S0097539795293172

[16] Rolando D. Somma. Quantum eigenvalue estimation via time series analysis. New Journal of Physics, 21: 123025, 2019. 10.1088/​1367-2630/​ab5c60.
https:/​/​doi.org/​10.1088/​1367-2630/​ab5c60

[17] Krysta M. Svore, Matthew B. Hastings, and Michael Freedman. Faster phase estimation. Quantum Information & Computation, 14 (3–4): 306–328, March 2014.

[18] Kristan Temme, Tobias J. Osborne, Karl G. Vollbrecht, David Poulin, and Frank Verstraete. Quantum Metropolis sampling. Nature, 471: 87–90, 2011. 10.1038/​nature09770.
https:/​/​doi.org/​10.1038/​nature09770

[19] Nathan Wiebe and Chris Granade. Efficient Bayesian phase estimation. Physical Review Letters, 117: 010503, 2016. 10.1103/​PhysRevLett.117.010503.
https:/​/​doi.org/​10.1103/​PhysRevLett.117.010503

Cited by

[1] K. Craigie, E. M. Gauger, Y. Altmann, and C. Bonato, "Resource-efficient adaptive Bayesian tracking of magnetic fields with a quantum sensor", Journal of Physics Condensed Matter 33 19, 195801 (2021).

[2] Alicja Dutkiewicz, Barbara M. Terhal, and Thomas E. O'Brien, "Heisenberg-limited quantum phase estimation of multiple eigenvalues with up to two control qubits", arXiv:2107.04605.

The above citations are from SAO/NASA ADS (last updated successfully 2021-09-23 07:59:12). The list may be incomplete as not all publishers provide suitable and complete citation data.

On Crossref's cited-by service no data on citing works was found (last attempt 2021-09-23 07:59:10).