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Stochastic parameter-shift rule for quantum metrology with general Hamiltonians

Le Bin Ho

Published Jun 14, 2022• 0 citations
Publisher

Abstract

Abstract Recently, quantum metrology with multiplicative Hamiltonians has been proposed in the variational quantum algorithms, from which the estimation precision can be adaptively optimized via the variational circuits. Systems with general Hamiltonians, however, still lack these variational schemes. This work introduces a quantum-circuit-based approach for studying quantum metrology with general Hamiltonians. We present a stochastic parameter-shift rule for the derivatives of the evolved quantum state under the parameterized gates in the circuit, whereby the quantum Fisher information can be obtained. Here the parameters are those we wish to estimate. We find that our scheme can be executed in universal quantum computers under the family of parameterized gates. Moreover, in the examples of the magnetic field estimation, we show the consistency between the results obtained from the stochastic parameter-shift rule and the exact results, while the results obtained from the standard parameter-shift rule slightly deviate from the exact ones. Our work sheds new light on studying quantum metrology with general Hamiltonians using quantum circuit algorithms.

DOI
10.21203/rs.3.rs-1705991/v1

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Quantum metrology enhances the sensitivity of parameter estimation using the distinctive resources of quantum mechanics such as entanglement. It has been shown that the precision of estimating an overall multiplicative factor of a Hamiltonian can be increased to exceed the classical limit, yet little is known about estimating a general Hamiltonian parameter. In this paper, we study this problem in detail. We find that the scaling of the estimation precision with the number of systems can always be optimized to the Heisenberg limit, while the time scaling can be quite different from that of estimating an overall multiplicative factor. We derive the generator of local parameter translation on the unitary evolution operator of the Hamiltonian, and use it to evaluate the estimation precision of the parameter and establish a general upper bound on the quantum Fisher information. The results indicate that the quantum Fisher information generally can be divided into two parts: one is quadratic in time, while the other oscillates with time. When the eigenvalues of the Hamiltonian do not depend on the parameter, the quadratic term vanishes, and the quantum Fisher information will be bounded in this case. To illustrate the results, we give an example of estimating a parameter of a magnetic field by measuring a spin-12\frac{1}{2}21​ particle and compare the results for estimating the amplitude and the direction of the magnetic field.
Abstract Variational quantum metrology represents a powerful tool to optimize estimation strategies, resulting particularly beneficial for multiparameter estimation problems that often suffer from limitations due to the curse of dimensionality and computational complexity. To overcome these challenges, we develop a variational approach able to efficiently optimize a quantum multiphase sensor. Leveraging the reconfigurability of an integrated photonic device, we implement a hybrid quantum-classical feedback loop able to enhance the estimation performances. The quantum circuit evaluations are used to compute the system partial derivatives by applying the parameter-shift rule, and thus reconstruct experimentally the Fisher information matrix. This in turn is adopted as the cost function of a classical learning algorithm run to optimize the measurement settings. Our experimental results showcase significant improvements in estimation accuracy and noise robustness, highlighting the potential of variational techniques for practical applications in quantum sensing and more generally in quantum information processing using photonic circuits.
Hybrid quantum-classical optimization algorithms represent one of the most promising application for near-term quantum computers. In these algorithms the goal is to optimize an observable quantity with respect to some classical parameters, using feedback from measurements performed on the quantum device. Here we study the problem of estimating the gradient of the function to be optimized directly from quantum measurements, generalizing and simplifying some approaches present in the literature, such as the so-called parameter-shift rule. We derive a mathematically exact formula that provides a stochastic algorithm for estimating the gradient of any multi-qubit parametric quantum evolution, without the introduction of ancillary qubits or the use of Hamiltonian simulation techniques. The gradient measurement is possible when the underlying device can realize all Pauli rotations in the expansion of the Hamiltonian whose coefficients depend on the parameter. Our algorithm continues to work, although with some approximations, even when all the available quantum gates are noisy, for instance due to the coupling between the quantum device and an unknown environment.
Abstract Quantum metrology has been studied for a wide range of systems with time-independent Hamiltonians. For systems with time-dependent Hamiltonians, however, due to the complexity of dynamics, little has been known about quantum metrology. Here we investigate quantum metrology with time-dependent Hamiltonians to bridge this gap. We obtain the optimal quantum Fisher information for parameters in time-dependent Hamiltonians, and show proper Hamiltonian control is generally necessary to optimize the Fisher information. We derive the optimal Hamiltonian control, which is generally adaptive, and the measurement scheme to attain the optimal Fisher information. In a minimal example of a qubit in a rotating magnetic field, we find a surprising result that the fundamental limit of T2 time scaling of quantum Fisher information can be broken with time-dependent Hamiltonians, which reaches T4 in estimating the rotation frequency of the field. We conclude by considering level crossings in the derivatives of the Hamiltonians, and point out additional control is necessary for that case.