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.
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.
Abstract The estimation of quantum phase differences plays an important role in quantum simulation and quantum computation, yet existing quantum phase estimation algorithms face critical limitations in noisy intermediate-scale quantum (NISQ) devices due to their excessive depth and circuit complexity. We demonstrate a high-precision phase difference estimation protocol based on the Bayesian phase difference estimation algorithm and single-photon projective measurement. The iterative framework of the algorithm, combined with the independence from controlled unitary operations, inherently mitigates circuit depth and complexity limitations. Through an experimental realization on the photonic system, we demonstrate high-precision estimation of diverse phase differences, showing root-mean-square errors (RMSE) below the standard quantum limit O ( 1 / N ) and reaching the Heisenberg scaling O ( 1 / N ) after a certain number of iterations. Our scheme provides a critical advantage in quantum resource-constrained scenarios, and advances practical implementations of quantum information tasks under realistic hardware constraints.
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.
ABSTRACT Quantum Phase Estimation (QPE) is a cornerstone algorithm in quantum computing, with applications ranging from integer factorization to quantum chemistry simulations. However, the resource demands of standard QPE, which require a large number of coherent qubits and deep circuits, pose significant challenges for current Noisy Intermediate Scale Quantum (NISQ) devices. In this work, we introduce the Adaptive Windowed Quantum Phase Estimation (AWQPE) algorithm, a novel method designed to address the limitations of standard QPE. AWQPE utilizes small, independent blocks of control qubits to estimate multiple phase bits simultaneously within a “window,” thereby significantly reducing the number of coherent qubits required to achieve a desired precision. These independent blocks are amenable to parallelization and, when combined with a robust least‐significant‐bit (LSB) to most‐significant‐bit (MSB) ambiguity resolution mechanism, enhance the algorithm's accuracy while mitigating the risk of error propagation. Our numerical simulations demonstrate AWQPE's accuracy and improved robustness relative to standard QPE under representative noise models, showcasing a favorable balance between resource efficiency and computational speed. This makes AWQPE particularly well‐suited for near‐term quantum platforms.
Quantum Amplitude Estimation (QAE) is a key primitive in quantum computing, but its standard implementation using Quantum Phase Estimation is resource-intensive, requiring a large number of coherent qubits in a single circuit block to achieve high precision. This presents a significant challenge for near-term Noisy Intermediate-Scale Quantum (NISQ) devices. To address this, we introduce the Adaptive Windowed Quantum Amplitude Estimation (AWQAE) framework, a modular, scalable and adaptive approach that decouples estimation precision from the number of physical qubits required in a single circuit. AWQAE operates by iteratively estimating the phase bits in small, fixed-size chunks, using a number of smaller, independent quantum circuits, which are amenable to parallel processing. A key technical contribution of this work is introduction of a phase resolution circuit and an ancilla-guided mechanism that enables accurate chunk assignment and eigenphase reconstruction in the presence of multiple eigenstates. This design is inherently NISQ-friendly, by lowering circuit depth and qubit count per block to reduce decoherence and noise effects. A key component of our approach is a robust classical post-processing algorithm that resolves measurement ambiguities that arise during the iterative process. This post-processing routine uses a least-significant-bit (LSB)-to-most-significant-bit (MSB) correction to reconstruct the full, high-precision phase estimate, ensuring accuracy. By combining a modular quantum-classical loop with an ambiguity-aware reconstruction method, AWQAE offers a powerful and flexible solution for performing high-precision QAE on resource-constrained quantum hardware. Our approach demonstrates enhanced scalability, and adaptability, making it a promising candidate for practical applications of QAE in the NISQ era.
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.
We aim to construct tests for evaluating whether policies for adaptive quantum-enhanced metrology are robust against unknown phase noise and are tractable to construct and to execute. Specifically, one of our tests determines scaling of phase-estimate precision with respect to photon number; the other two tests concern resource complexity with respect to the training time required to construct the policy and the execution time for the policy so constructed. The robustness test is performed on quantum-enhanced adaptive phase estimation by simulating the scheme under four phase-noise models corresponding to normal-distribution noise, random-telegraph noise, skew-normal-distribution noise, and log-normal-distribution noise. Control policies are devised either by an evolutionary algorithm under the same noisy conditions, albeit ignorant of its properties, or a Bayesian-based feedback method that assumes no noise. We have introduced an approach to evaluating quantum-control policies for metrology that relies on testing against unusual phase-noise models and accepting that the policies are robust only if scaling of phase-estimate precision beats the standard quantum limit and policy-design resource complexity, in the presence of phase noise, is polynomial in the number of photons.