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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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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Cited by

[1] Nelson Filipe Costa, Yasser Omar, Aidar Sultanov, and Gheorghe Sorin Paraoanu, "Benchmarking machine learning algorithms for adaptive quantum phase estimation with noisy intermediate-scale quantum sensors", EPJ Quantum Technology 8 1, 16 (2021).

[2] Leonardo Alchieri, Davide Badalotti, Pietro Bonardi, and Simone Bianco, "An introduction to quantum machine learning: from quantum logic to quantum deep learning", Quantum Machine Intelligence 3 2, 28 (2021).

[3] M. A. Rodríguez-García, M. T. DiMario, P. Barberis-Blostein, and F. E. Becerra, "Determination of the asymptotic limits of adaptive photon counting measurements for coherent-state optical phase estimation", npj Quantum Information 8 1, 94 (2022).

[4] Joseph G. Smith, Crispin H. W. Barnes, and David R. M. Arvidsson-Shukur, "Adaptive Bayesian quantum algorithm for phase estimation", Physical Review A 109 4, 042412 (2024).

[5] Nelson Filipe Costa, Yasser Omar, Aidar Sultanov, and Gheorghe Sorin Paraoanu, "Benchmarking Machine Learning Algorithms for Adaptive Quantum Phase Estimation with Noisy Intermediate-Scale Quantum Sensors", arXiv:2108.06978, (2021).

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