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arXiv:2107.00196

Bayesian Phase Estimation via Active Learning

Yuxiang Qiu, Zhuang, Min, Jiahao Huang, Chaohong Lee

Published Jul 1, 2021• 0 citations
arXivHTMLPDF

Abstract

Bayesian estimation approaches, which are capable of combining the information of experimental data from different likelihood functions to achieve high precisions, have been widely used in phase estimation via introducing a controllable auxiliary phase. Here, we present a non-adaptive Bayesian phase estimation (BPE) algorithms with an ingenious update rule of the auxiliary phase designed via active learning. Unlike adaptive BPE algorithms, the auxiliary phase in our algorithm is determined by a pre-established update rule with simple statistical analysis of a small batch of data, instead of complex calculations in every update trails. As the number of measurements for a same amount of Bayesian updates is significantly reduced via active learning, our algorithm can work as efficient as adaptive ones and shares the advantages (such as wide dynamic range and perfect noise robustness) of non-adaptive ones. Our algorithm is of promising applications in various practical quantum sensors such as atomic clocks and quantum magnetometers.

DOI
10.48550/arxiv.2107.00196

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Bayesian estimation approaches, which are capable of combining the information of experimental data from different likelihood functions to achieve high precisions, have been widely used in phase estimation via introducing a controllable auxiliary phase. Here, we present a non-adaptive Bayesian phase estimation (BPE) algorithms with an ingenious update rule of the auxiliary phase designed via active learning. Unlike adaptive BPE algorithms, the auxiliary phase in our algorithm is determined by a pre-established update rule with simple statistical analysis of a small batch of data, instead of complex calculations in every update trails. As the number of measurements for a same amount of Bayesian updates is significantly reduced via active learning, our algorithm can work as efficient as adaptive ones and shares the advantages (such as wide dynamic range and perfect noise robustness) of non-adaptive ones. Our algorithm is of promising applications in various practical quantum sensors such as atomic clocks and quantum magnetometers.
Abstract Bayesian estimation approaches, which are capable of combining the information of experimental data from different likelihood functions to achieve high precisions, have been widely used in phase estimation via introducing a controllable auxiliary phase. Here, we present a Bayesian phase estimation (BPE) algorithm with an ingenious update rule of the auxiliary phase designed via entropy-based sampling. Unlike other adaptive BPE algorithms, the auxiliary phase in our algorithm is determined only once in a pre-estimation step. With simple statistical analysis on a small batch of data, an iteration rule for the auxiliary phase is pre-established and used in all afterward updates, instead of complex calculations in every update trails. During this pre-estimation process the most informative data can be selected, which guides one to perform the BPE with much less measurement times. As the measurement times for the same amount of Bayesian updates is significantly reduced, our algorithm via entropy-based sampling can work as efficient as other adaptive BPE algorithms and shares the advantages (such as wide dynamic range and perfect noise robustness) of non-adaptive BPE algorithms. Our algorithm is of promising applications in various practical quantum sensors such as atomic clocks and quantum magnetometers.
Phase estimation is known to be a robust method for single-qubit gate calibration in quantum computers, while Bayesian estimation is widely used in devising optimal methods for learning in quantum systems. We present a Bayesian method that adaptively chooses a control phase and the time of coherent evolution. The uniqueness of the approach is to use analytics to implement an efficient numerical algorithm that offers exact calculation of the Bayesian probability and associated utility functions, even in the presence of noise. We determine the utility of control parameter values using functions of the prior probability of the phase that quantify expected knowledge gain in terms of either expected narrowing of the posterior or expected information gain. We find that by maximizing the of expected gain we obtain phase estimates having standard deviation a factor of 1.43 larger than the Heisenberg limit using a classical sequential strategy for noise-free experiments. In the presence of dephasing, we find near-optimal performance with respect to the Cramér-Rao bound. We also demonstrate some robustness of the estimates to noise that is not accounted for in the model of the estimator, making the methods suitable for calibrating operations in quantum computers. The rate of gain can be adjusted to account for times in the experiment other than the coherent evolution of the unknown phase, such as times required for state preparation or readout. The combined features of the method make it well suited to single-qubit gate calibration in trapped-ion or neutral atom experiments and promises a wide range of other applications by offering near-optimal phase estimation that straightforwardly accounts for experimental imperfections.
Background Quantum phase estimation (QPE) is a foundational subroutine in quantum algorithms that ranges from Shor’s factoring scheme to variational quantum eigensolvers and quantum metrology protocols. Classical post-processing replacements for the inverse quantum Fourier transform (iQFT) —specifically Kitaev-style iterative estimation and Bayesian update schemes—substantially reduce circuit depth, making them practical for noisy intermediate-scale quantum (NISQ) hardware. Despite this promise, the statistical performance of these methods under the conditions that actually arise in practice—non-eigenstate inputs characterised by eigenstate overlap p ≡ O = | ψ | ϕ | 2 < 1 and mixed-state purity γ ≡ P = Tr ( ρ 2 ) < 1 —remains uncharacterised in the literature. Methods We develop a rigorous information-theoretic framework for Bayesian QPE (BQPE) beyond the eigenstate assumption. Starting from the binary measurement likelihood attenuated by the joint factor O P , we derive exact classical Fisher information expressions and apply the Cramér–Rao inequality to obtain tight variance lower bounds. We then construct an adaptive measurement protocol based on von Mises posterior updates, prove its asymptotic efficiency in the frequentist sense, and establish matching upper and lower bounds on sample complexity. All theoretical results are validated against 2,000 -trial Monte Carlo simulations and cross-checked against published nitrogen-vacancy (NV) centre and photonic experimental data. Results We prove that the fundamental variance floor for any unbiased frequentist estimator is ( 1 − p γ ) / ( N p γ n ̄ 2 ) 1 / ( N ( p γ ) 2 n ̄ 2 ) , where N
Abstract Realisation of experiments even on small and medium-scale quantum computers requires an optimisation of several parameters to achieve high-fidelity operations. As the size of the quantum register increases, the characterisation of quantum states becomes more difficult since the number of parameters to be measured grows as well and finding efficient observables in order to estimate the parameters of the model becomes a crucial task. Here we propose a method relying on application of Bayesian inference that can be used to determine systematic, unknown phase shifts of multi-qubit states. This method offers important advantages as compared to Ramsey-type protocols. First, application of Bayesian inference allows the selection of an adaptive basis for the measurements which yields the optimal amount of information about the phase shifts of the state. Secondly, this method can process the outcomes of different observables at the same time. This leads to a substantial decrease in the resources needed for the estimation of phases, speeding up the state characterisation and optimisation in experimental implementations. The proposed Bayesian inference method can be applied in various physical platforms that are currently used as quantum processors.
Abstract Bayesian Phase Estimation (BPE) is a powerful tool for quantum computing and metrology, yet its precision is inevitably compromised by noise in intermediate-scale quantum (NISQ) devices. In this work, we systematically analyze and simulate photonic BPE performance under depolarizing, amplitude damping, and bit flipping quantum noise channels. We identify a critical trade-off between noise-induced errors and resource consumption, leading to a proposed adaptive photon-number strategy. By dynamically selecting the optimal photon number in each iteration, our scheme significantly improves the estimation precision under equivalent resource constraints. This resource-efficient approach offers a practical guideline for implementing robust quantum phase estimation in noisy environments.
Quantum computers can perform full configuration interaction (full-CI) calculations by utilising the quantum phase estimation (QPE) algorithms including Bayesian phase estimation (BPE) and iterative quantum phase estimation (IQPE). In these quantum algorithms, the time evolution of wave functions for atoms and molecules is simulated conditionally with an ancillary qubit as the control, which make implementation to real quantum devices difficult. Also, most of the problems in chemistry discuss energy differences between two electronic states rather than total energies themselves, and thus direct calculations of energy gaps are promising for future applications of quantum computers to real chemistry problems. In the race of finding efficient quantum algorithms to solve quantum chemistry problems, we test a Bayesian phase difference estimation (BPDE) algorithm, which is a general algorithm to calculate the difference of two eigenphases of unitary operators in the several cases of the direct calculations of energy gaps between two electronic states on quantum computers, including vertical ionisation energies, singlet-triplet energy gaps, and vertical excitation energies. In the BPDE algorithm, state preparation is carried out conditionally on the ancillary qubit, and the time evolution of the wave functions in superposition of two electronic states are executed unconditionally. Based on our test, we conclude that BPDE is capable of computing the energy gap with an accuracy similar to BPE without controlled-time evolution simulations and with the smaller number of iterations in Bayesian optimisations.