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arXiv:2010.09075 · Physical Review Applied

Bayesian Quantum Multiphase Estimation Algorithm

Valentin Gebhart, Augusto Smerzi, Luca Pezzè

Published Jul 14, 2021• 38 citations
arXivHTMLPDF

Abstract

Quantum phase estimation (QPE) is the key subroutine of several quantum computing algorithms as well as a central ingredient in quantum computational chemistry and quantum simulation. While QPE strategies have focused on the estimation of a single phase, applications to the simultaneous estimation of several phases may bring substantial advantages; for instance, in the presence of spatial or temporal constraints. In this work, we study a Bayesian algorithm for the parallel (simultaneous) estimation of multiple arbitrary phases. The protocol gives access to correlations in the Bayesian multiphase distribution resulting in covariance matrix elements scaling as $O({N}_{T}^{\ensuremath{-}2})$, with respect to the total number of quantum resources ${N}_{T}$. The parallel estimation allows to surpass the sensitivity of sequential single-phase estimation strategies for optimal linear combinations of phases. Furthermore, the algorithm proves a certain noise resilience and can be implemented using single photons and standard optical elements in currently accessible experiments.

DOI
10.1103/physrevapplied.16.014035

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Quantum phase estimation (QPE) is the key subroutine of several quantum computing algorithms as well as a central ingredient in quantum computational chemistry and quantum simulation. While QPE strategies have focused on the estimation of a single phase, applications to the simultaneous estimation of several phases may bring substantial advantages; for instance, in the presence of spatial or temporal constraints. In this work, we study a Bayesian algorithm for the parallel (simultaneous) estimation of multiple arbitrary phases. The protocol gives access to correlations in the Bayesian multiphase distribution resulting in covariance matrix elements scaling as O(NT\ensuremath−2)O({N}_{T}^{\ensuremath{-}2})O(NT\ensuremath−2​), with respect to the total number of quantum resources NT{N}_{T}NT​. The parallel estimation allows to surpass the sensitivity of sequential single-phase estimation strategies for optimal linear combinations of phases. Furthermore, the algorithm proves a certain noise resilience and can be implemented using single photons and standard optical elements in currently accessible experiments.
Quantum phase estimation (QPE) is ubiquitous in quantum technologies. It is the key subroutine of several quantum computing algorithms as well as a central ingredient in quantum computational chemistry and quantum simulation. While QPE strategies have focused on the estimation of a single phase, one can envisage applications where the simultaneous estimation of multiple phases may bring substantial advantages, for instance in presence of spatial or temporal constraints. In this work, we study a Bayesian QPE algorithm for the parallel (simultaneous) estimation of multiple arbitrary phases. The algorithm allows the estimation of each phase, as well as any arbitrary linear combination, with Heisenberg-limited sensitivity with respect to the total number of quantum resources. The parallel estimation protocol gives access to correlations in the Bayesian multiphase distribution that allow to surpass the sensitivity of sequential single phase estimation strategies for optimal combinations of phases. Furthermore, the algorithm proves a certain noise resilience, and can be implemented using single photons and standard optical elements in currently accessible experiments.
Quantum phase estimation is fundamental to advancing quantum science and technology. While much of the research has concentrated on estimating a single phase, the simultaneous estimation of multiple phases can yield significantly enhanced sensitivities when using specially tailored input quantum states. This work reviews recent theoretical and experimental advancements in the parallel estimation of multiple arbitrary phases. We highlight strategies for constructing optimal measurement protocols and discuss the experimental platforms best suited for implementing these techniques.
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.
Quantum phase estimation (QPE) serves as a building block of many different quantum algorithms and finds important applications in computational chemistry problems. Despite the rapid development of quantum hardware, experimental demonstration of QPE for chemistry problems remains challenging due to its large circuit depth and the lack of quantum resources to protect the hardware from noise with fully fault-tolerant protocols. In the present work, we take a step towards fault-tolerant quantum computing by demonstrating a QPE algorithm on a Quantinuum trapped-ion computer. We employ a Bayesian approach to QPE and introduce a routine for optimal parameter selection, which we combine with a [[ n + 2 , n , 2 ]] quantum error detection code carefully tailored to the hardware capabilities. As a simple quantum chemistry example, we take a hydrogen molecule represented by a two-qubit Hamiltonian and estimate its ground state energy using our QPE protocol. In the experiment, we use the quantum circuits containing as many as 920 physical two-qubit gates to estimate the ground state energy within 6 × 10 − 3 hartree of the exact value. Published by the American Physical Society 2024
Quantum phase estimation (QPE) is one of the most important subroutines in quantum computing. In general applications, current QPE algorithms either suffer an exponential time overload or require a set of--notoriously quite fragile--GHZ states. These limitations have so far prevented the demonstration of QPE beyond proof of principles. Here we propose a new QPE algorithm based on a cascade of Gaussian spin states (GSSs) and a suitable adaptive measurement protocol. GSSs are renownedly resilient and have been created experimentally in a variety of platforms, from hundreds of ions up to millions of cold and ultracold neutral atoms. We show that our protocol achieves a QPE sensitivity overcoming previous schemes, including that obtained with GHZ states, scales linearly with time, and is robust against certain sources of noise and decoherence, such as detection noise, in particular. Our work paves the way toward an efficient implementation of the QPE, as well as applications of atomic squeezed states for quantum computation.