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arXiv:2403.20131

QestOptPOVM: An iterative algorithm to find optimal measurements for quantum parameter estimation

Jianchao Zhang, Suzuki, Jun

Published Mar 29, 2024• 1 citation
arXivHTMLPDF

Abstract

Quantum parameter estimation holds significant promise for achieving high precision through the utilization of the most informative measurements. While various lower bounds have been developed to assess the best accuracy for estimates, they are not tight, nor provide a construction of the optimal measurement in general. Thus, determining the explicit forms of optimal measurements has been challenging due to the non-trivial optimization. In this study, we introduce an algorithm, termed QestOptPOVM, designed to directly identify optimal positive operator-valued measure (POVM) using the steepest descent method. Through rigorous testing on several examples for multiple copies of qubit states (up to six copies), we demonstrate the efficiency and accuracy of our proposed algorithm. Moreover, a comparative analysis between numerical results and established lower bounds serves to validate the tightness of the Nagaoka-Hayashi bound in finite-sample quantum metrology for our examples. Concurrently, our algorithm functions as a tool for elucidating the explicit forms of optimal POVMs, thereby enhancing our understanding of quantum parameter estimation methodologies.

DOI
10.48550/arxiv.2403.20131

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Quantum parameter estimation holds significant promise for achieving high precision through the utilization of the most informative measurements. While various lower bounds have been developed to assess the best accuracy for estimates, they are not tight, nor provide a construction of the optimal measurement in general. Thus, determining the explicit forms of optimal measurements has been challenging due to the non-trivial optimization. In this study, we introduce an algorithm, termed QestOptPOVM, designed to directly identify optimal positive operator-valued measure (POVM) using the steepest descent method. Through rigorous testing on several examples for multiple copies of qubit states (up to six copies), we demonstrate the efficiency and accuracy of our proposed algorithm. Moreover, a comparative analysis between numerical results and established lower bounds serves to validate the tightness of the Nagaoka-Hayashi bound in finite-sample quantum metrology for our examples. Concurrently, our algorithm functions as a tool for elucidating the explicit forms of optimal POVMs, thereby enhancing our understanding of quantum parameter estimation methodologies.
Quantum parameter estimation promises a high-precision measurement in theory; however, how to design the optimal scheme in a specific scenario, especially under a practical condition, is still a serious problem that needs to be solved case by case due to the existence of multiple mathematical bounds and optimization methods. Depending on the scenario considered, different bounds may be more or less suitable, both in terms of computational complexity and the tightness of the bound itself. At the same time, the metrological schemes provided by different optimization methods need to be tested against realization complexity, robustness, etc. Hence, a comprehensive toolkit containing various bounds and optimization methods is essential for the scheme design in quantum metrology. To fill this vacancy, here we present a Python-Julia-based open-source toolkit for quantum parameter estimation, which includes many well-used mathematical bounds and optimization methods. Utilizing this toolkit, all procedures in the scheme design, such as the optimizations of the probe state, control and measurement, can be readily and efficiently performed.
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Abstract In this paper we design the following two-step scheme to estimate the model parameter ω 0 of the quantum system: first we utilize the Fisher information with respect to an intermediate variable v = cos ( ω 0 t ) to determine an optimal initial state and to seek optimal parameters of the POVM measurement operators; second we explore how to estimate ω 0 from v by choosing t when a priori information knowledge of ω 0 is available. Our optimal initial state can achieve the maximum quantum Fisher information. The formulation of the optimal time t is obtained and the complete algorithm for parameter estimation is presented. We further explore how the lower bound of the estimation deviation depends on the a priori information of the model .
Determining when the multiparameter quantum Cramér--Rao bound (QCRB) is saturable with experimentally relevant single-copy measurements is a central open problem in quantum metrology. Here we establish an equivalence between QCRB saturation and the simultaneous hollowization of a set of traceless operators associated with the estimation model, i.e., the existence of complete (generally nonorthogonal) bases in which all corresponding diagonal matrix elements vanish. This formulation yields a geometric characterization: optimal rank-one measurement vectors are confined to a subspace orthogonal to a state-determined Hermitian span. This provides a direct criterion to construct optimal Positive Operator-Valued Measures(POVMs). We then identify conditions under which the partial commutativity condition proposed in [Phys. Rev. A 100, 032104(2019)] becomes necessary and sufficient for the saturation of the QCRB, demonstrate that this condition is not always sufficient, and prove the counter-intuitive uselessness of informationally-complete POVMs.
Optimization problems in disciplines such as machine learning are commonly solved with iterative methods. Gradient descent algorithms find local minima by moving along the direction of steepest descent while Newton's method takes into account curvature information and thereby often improves convergence. Here, we develop quantum versions of these iterative optimization algorithms and apply them to polynomial optimization with a unit norm constraint. In each step, multiple copies of the current candidate are used to improve the candidate using quantum phase estimation, an adapted quantum state exponentiation scheme, as well as quantum matrix multiplications and inversions. The required operations perform polylogarithmically in the dimension of the solution vector and exponentially in the number of iterations. Therefore, the quantum algorithm can be useful for high-dimensional problems where a small number of iterations is sufficient.
In multiparameter quantum estimation, incompatible measurements can impose trade-offs among estimation errors. Understanding the precision attainable with a finite number of jointly measured copies is therefore important, but direct optimization becomes costly as the tensor product space grows exponentially. For independent and identically distributed (i.i.d.) copies, the quantum statistical model is invariant under permutations of the copies. We exploit this symmetry through Schur--Weyl duality to remove permutation multiplicity spaces while using their dimensions as weights. This gives a reduced statistical model equivalent to the original under arbitrary collective measurements. On this reduced model, we derive an exact conic formulation of the tight Cramér--Rao bound and a semidefinite program for its Nagaoka--Hayashi (NH) relaxation. For fixed single-copy dimension, the reduced operator representation grows polynomially rather than exponentially with the number of copies. To approximate the tight bound, we use an exact quadratic identity to correct finite-ray dual solutions and obtain rigorous upper and lower bounds. For fixed copy number and parameter dimension, sufficiently fine meshes admit feasible primal points, and dual solutions with bounded objective gaps have uniformly bounded compact-search radii. These results yield quadratic convergence in the projective covering radius. Numerical examples illustrate the quadratic scaling of the mesh correction term and the computational reach of the reduced NH formulation.