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arXiv:2208.04526

Using Random Walks for Iterative Phase Estimation

Cassandra Granade, Nathan Wiebe

Published Aug 9, 2022• 4 citations
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Abstract

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.

DOI
10.48550/arxiv.2208.04526

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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.
Abstract Optimal phase estimation protocols require complex state preparation and readout schemes, generally unavailable or unscalable in many quantum platforms. We develop a scheme that achieves near-optimal precision up to a constant overhead for Bayesian phase estimation, using simple digital quantum circuits with depths scaling logarithmically with the number of qubits. This is done by approximating the optimal initial states with products of Greenberger-Horne-Zeilinger states for Gaussian prior phase distributions with arbitrary widths. We study various protocols that employ this class of states with different levels of measurement and post-processing complexities, and obtain improvement compared to previously proposed schemes. We then use our scheme to address phase slip errors and laser noise, which impose a major limitation in Bayesian phase estimation and atomic clocks. Based on our scheme, we develop an efficient protocol to suppress this noise that outperforms existing methods.
Quantum-phase-estimation algorithms are critical subroutines in many applications for quantum computers and in quantum-metrology protocols. These algorithms estimate the unknown strength of a unitary evolution. By using coherence or entanglement to sample the unitary N tot times, the variance of the estimates can scale as O ( 1 / N tot 2 ) , compared to the best “classical” strategy with O ( 1 / N tot ) . The original algorithm for quantum phase estimation cannot be implemented on near-term hardware as it requires large-scale entangled probes and fault-tolerant quantum computing. Therefore, alternative algorithms have been introduced that rely on coherence and statistical inference. These algorithms produce quantum-boosted phase estimates without interprobe entanglement. This family of phase-estimation algorithms have, until now, never exhibited the possibility of achieving optimal scaling O ( 1 / N tot 2 ) . Moreover, previous works have not considered the effect of noise on these algorithms. Here, we present a coherence-based phase-estimation algorithm which can achieve the optimal quadratic scaling in the mean absolute error and the mean squared error. In the presence of noise, our algorithm produces errors that approach the theoretical lower bound. The optimality of our algorithm stems from its adaptive nature: Each step is determined, iteratively, using a Bayesian protocol that analizes the results of previous steps. Published by the American Physical Society 2024
Quantum-phase-estimation algorithms are critical subroutines in many applications for quantum computers and in quantum-metrology protocols. These algorithms estimate the unknown strength of a unitary evolution. By using coherence or entanglement to sample the unitary NtotN_{\mathrm{tot}}Ntot​ times, the variance of the estimates can scale as O(1/Ntot2)O(1/{N^2_{\mathrm{tot}}})O(1/Ntot2​), compared to the best ``classical'' strategy with O(1/Ntot)O(1/{N_{\mathrm{tot}}})O(1/Ntot​). The original algorithm for quantum phase estimation cannot be implemented on near-term hardware as it requires large-scale entangled probes and fault-tolerant quantum computing. Therefore, alternative algorithms have been introduced that rely on coherence and statistical inference. These algorithms produce quantum-boosted phase estimates without inter-probe entanglement. This family of phase-estimation algorithms have, until now, never exhibited the possibility of achieving optimal scaling O(1/Ntot2)O(1/{N^2_{\mathrm{tot}}})O(1/Ntot2​). Moreover, previous works have not considered the effect of noise on these algorithms. Here, we present a coherence-based phase-estimation algorithm which can achieve the optimal quadratic scaling in the mean absolute error and the mean squared error. In the presence of noise, our algorithm produces errors that approach the theoretical lower bound. The optimality of our algorithm stems from its adaptive nature: Each step is determined, iteratively, using a Bayesian protocol that analyses the results of previous steps.
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.
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.