Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2007.11629 · Quantum

Efficient Bayesian phase estimation using mixed priors

Ewout van den Berg

Published Jun 7, 2021• 7 citations
arXivHTMLPDF

Abstract

We describe an efficient implementation of Bayesian quantum phase estimation in the presence of noise and multiple eigenstates. The main contribution of this work is the dynamic switching between different representations of the phase distributions, namely truncated Fourier series and normal distributions. The Fourier-series representation has the advantage of being exact in many cases, but suffers from increasing complexity with each update of the prior. This necessitates truncation of the series, which eventually causes the distribution to become unstable. We derive bounds on the error in representing normal distributions with a truncated Fourier series, and use these to decide when to switch to the normal-distribution representation. This representation is much simpler, and was proposed in conjunction with rejection filtering for approximate Bayesian updates. We show that, in many cases, the update can be done exactly using analytic expressions, thereby greatly reducing the time complexity of the updates. Finally, when dealing with a superposition of several eigenstates, we need to estimate the relative weights. This can be formulated as a convex optimization problem, which we solve using a gradient-projection algorithm. By updating the weights at exponentially scaled iterations we greatly reduce the computational complexity without affecting the overall accuracy.

DOI
10.22331/q-2021-06-07-469

Related papers

Bayesian quantum phase estimation beyond the eigenstate approximation: sample complexity, Cramér–Rao bounds, and adaptive measurement design for non-eigenstate and mixed-state inputs on NISQ devices

Akoramurthy Balasubramaniam, B. Surendiran, Xiaochun Cheng +1 more

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

Frontiers in Quantum Science and Technology•Aug 11, 2026•0 citations

Efficient Bayesian Phase Estimation

Nathan Wiebe, Christopher Granade

We introduce a new method called rejection filtering that we use to perform adaptive Bayesian phase estimation. Our approach has several advantages: it is classically efficient, easy to implement, achieves Heisenberg limited scaling, resists depolarizing noise, tracks time-dependent eigenstates, recovers from failures, and can be run on a field programmable gate array. It also outperforms existing iterative phase estimation algorithms such as Kitaev's method.

arXiv:1508.00869•Physical Review Letters•Jun 30, 2016•196 citations

Using Random Walks for Iterative Phase Estimation

Cassandra Granade, Nathan Wiebe

In recent years there has been substantial development in algorithms for quantum phase estimation. In this work we provide a new approach to online Bayesian phase estimation that achieves Heisenberg limited scaling that requires exponentially less classical processing time with the desired error tolerance than existing Bayesian methods. This practically means that we can perform an update in microseconds on a CPU as opposed to milliseconds for existing particle filter methods. Our approach assumes that the prior distribution is Gaussian and exploits the fact, when optimal experiments are chosen, the mean of the prior distribution is given by the position of a random walker whose moves are dictated by the measurement outcomes. We then argue from arguments based on the Fisher information that our algorithm provides a near-optimal analysis of the data. This work shows that online Bayesian inference is practical, efficient and ready for deployment in modern FPGA driven adaptive experiments.

arXiv:2208.04526•Aug 9, 2022•4 citations

Bayesian quantum phase estimation with fixed photon states

Boyu Zhou, Saikat Guha, Christos N. Gagatsos

We consider a two-mode bosonic state with fixed photon number $$n \in \mathbb {N}$$ , whose upper and lower modes pick up a phase $$\phi $$ and $$-\phi $$ , respectively. We compute the optimal Fock coefficients of the input state, such that the mean square error (MSE) for estimating $$\phi $$ is minimized, while the minimum MSE is always attainable by a measurement. Our setting is Bayesian, i.e., we consider $$\phi $$ to be a random variable that follows a prior probability distribution function (PDF). Initially, we consider the flat prior PDF and we discuss the well-known fact that the MSE is not an informative tool for estimating a phase when the variance of the prior PDF is large. Therefore, we move on to study truncated versions of the flat prior in both single-shot and adaptive approaches. For our adaptive technique, we consider $$n=1$$ and truncated prior PDFs. Each subsequent step utilizes as prior PDF the posterior probability of the previous step, and at the same time we update the optimal state and optimal measurement.

arXiv:2308.01293•Quantum Information Processing•Oct 28, 2024•3 citations

Efficient Bayesian phase estimation via entropy-based sampling

Yuxiang Qiu, Min Zhuang, Jiahao Huang +1 more

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.

Quantum Science and Technology•May 31, 2022•7 citations

Bayesian Phase Estimation via Active Learning

Yuxiang Qiu, Zhuang, Min, Jiahao Huang +1 more

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.

arXiv:2107.00196•Jul 1, 2021•0 citations
We describe an efficient implementation of Bayesian quantum phase estimation in the presence of noise and multiple eigenstates. The main contribution of this work is the dynamic switching between different representations of the phase distributions, namely truncated Fourier series and normal distributions. The Fourier-series representation has the advantage of being exact in many cases, but suffers from increasing complexity with each update of the prior. This necessitates truncation of the series, which eventually causes the distribution to become unstable. We derive bounds on the error in representing normal distributions with a truncated Fourier series, and use these to decide when to switch to the normal-distribution representation. This representation is much simpler, and was proposed in conjunction with rejection filtering for approximate Bayesian updates. We show that, in many cases, the update can be done exactly using analytic expressions, thereby greatly reducing the time complexity of the updates. Finally, when dealing with a superposition of several eigenstates, we need to estimate the relative weights. This can be formulated as a convex optimization problem, which we solve using a gradient-projection algorithm. By updating the weights at exponentially scaled iterations we greatly reduce the computational complexity without affecting the overall accuracy.
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
In recent years there has been substantial development in algorithms for quantum phase estimation. In this work we provide a new approach to online Bayesian phase estimation that achieves Heisenberg limited scaling that requires exponentially less classical processing time with the desired error tolerance than existing Bayesian methods. This practically means that we can perform an update in microseconds on a CPU as opposed to milliseconds for existing particle filter methods. Our approach assumes that the prior distribution is Gaussian and exploits the fact, when optimal experiments are chosen, the mean of the prior distribution is given by the position of a random walker whose moves are dictated by the measurement outcomes. We then argue from arguments based on the Fisher information that our algorithm provides a near-optimal analysis of the data. This work shows that online Bayesian inference is practical, efficient and ready for deployment in modern FPGA driven adaptive experiments.
We consider a two-mode bosonic state with fixed photon number n∈Nn \in \mathbb {N}n∈N , whose upper and lower modes pick up a phase ϕ\phi ϕ and −ϕ-\phi −ϕ , respectively. We compute the optimal Fock coefficients of the input state, such that the mean square error (MSE) for estimating ϕ\phi ϕ is minimized, while the minimum MSE is always attainable by a measurement. Our setting is Bayesian, i.e., we consider ϕ\phi ϕ to be a random variable that follows a prior probability distribution function (PDF). Initially, we consider the flat prior PDF and we discuss the well-known fact that the MSE is not an informative tool for estimating a phase when the variance of the prior PDF is large. Therefore, we move on to study truncated versions of the flat prior in both single-shot and adaptive approaches. For our adaptive technique, we consider n=1n=1n=1 and truncated prior PDFs. Each subsequent step utilizes as prior PDF the posterior probability of the previous step, and at the same time we update the optimal state and optimal measurement.
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.
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.