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arXiv:2108.06978 · EPJ Quantum Technology

Benchmarking machine learning algorithms for adaptive quantum phase estimation with noisy intermediate-scale quantum sensors

Nelson Filipe Costa, Yasser Omar, Aidar Sultanov, Gheorghe Sorin Paraoanu

Published Jun 3, 2021• 20 citations
arXivHTMLPDF

Abstract

Abstract Quantum phase estimation is a paradigmatic problem in quantum sensing and metrology. Here we show that adaptive methods based on classical machine learning algorithms can be used to enhance the precision of quantum phase estimation when noisy non-entangled qubits are used as sensors. We employ the Differential Evolution (DE) and Particle Swarm Optimization (PSO) algorithms to this task and we identify the optimal feedback policies which minimize the Holevo variance. We benchmark these schemes with respect to scenarios that include Gaussian and Random Telegraph fluctuations as well as reduced Ramsey-fringe visibility due to decoherence. We discuss their robustness against noise in connection with real experimental setups such as Mach–Zehnder interferometry with optical photons and Ramsey interferometry in trapped ions, superconducting qubits and nitrogen-vacancy (NV) centers in diamond.

DOI
10.1140/epjqt/s40507-021-00105-y

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Abstract Quantum phase estimation is a paradigmatic problem in quantum sensing and metrology. Here we show that adaptive methods based on classical machine learning algorithms can be used to enhance the precision of quantum phase estimation when noisy non-entangled qubits are used as sensors. We employ the Differential Evolution (DE) and Particle Swarm Optimization (PSO) algorithms to this task and we identify the optimal feedback policies which minimize the Holevo variance. We benchmark these schemes with respect to scenarios that include Gaussian and Random Telegraph fluctuations as well as reduced Ramsey-fringe visibility due to decoherence. We discuss their robustness against noise in connection with real experimental setups such as Mach–Zehnder interferometry with optical photons and Ramsey interferometry in trapped ions, superconducting qubits and nitrogen-vacancy (NV) centers in diamond.
Quantum metrology is one of the most relevant applications of quantum information theory to quantum technologies.Here, quantum probes are exploited to overcome classical bounds in the estimation of unknown parameters.In this context, phase estimation, where the unknown parameter is a phase shift between two modes of a quantum system, is a fundamental problem.In practical and realistic applications, it is necessary to devise methods to optimally estimate an unknown phase shift by using a limited number of probes.Here we introduce and experimentally demonstrate a machine learning-based approach for the adaptive estimation of a phase shift in a Mach-Zehnder interferometer, tailored for optimal performances with limited resources.The employed technique is a genetic algorithm used to devise the optimal feedback phases employed during the estimation in an offline fashion.The results show the capability to retrieve the true value of the phase by using few photons, and to reach the sensitivity bounds in such small probe regime.We finally investigate the robustness of the protocol with respect to common experimental errors, showing that the protocol can be adapted to a noisy scenario.Such approach promises to be a useful tool for more complex and general tasks where optimization of feedback parameters is required.
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.
Adaptive quantum machine learning (QML) represents an exciting frontier at the intersection of quantum computing and classical bio-inspired optimization. The constraints of noisy intermediate-scale quantum (NISQ) devices—limited qubit counts, significant decoherence, and measurement uncertainties—pose formidable challenges to training parameterized quantum circuits (PQCs) using traditional gradient-based methods. To overcome these hurdles, we propose a novel Adaptive PSO–QML framework that integrates a particle swarm optimization (PSO)–inspired algorithm with dynamic hyperparameter adjustment, enabling real-time tuning of PQC parameters based on performance feedback. Unlike static optimizers, our approach continuously adapts its inertia weight and cognitive/social coefficients to balance exploration and exploitation, responding to stagnation or rapid improvement in measured fitness. We evaluate this hybrid algorithm on two benchmark classification tasks: the classical Iris dataset encoded into a 2-qubit circuit and a quantum-encoded subset of the MNIST dataset on 4 qubits. Our experiments, conducted on the Qiskit Aer simulator under realistic noise models (depolarizing, amplitude damping, and mixed), demonstrate that Adaptive PSO–QML achieves an average accuracy improvement of 12% over fixed-parameter PSO and 15% over Adam-based gradient descent, with p-values below 0.01 across paired t-tests. Statistical analysis summarized in Table 1 confirms the significance of these gains. Detailed simulation research further reveals that the adaptive mechanism not only accelerates convergence by approximately 30%—reducing total circuit evaluations—but also exhibits robust performance across varying circuit depths (1–3 entangling layers) and noise intensities. Our findings suggest that bio-inspired adaptation is a powerful tool for mitigating barren plateaus and noise effects in NISQ-era QML, paving the way for scalable quantum advantage in near-term hardware.
We devise powerful algorithms based on differential evolution for adaptive many-particle quantum metrology. Our new approach delivers adaptive quantum metrology policies for feedback control that are orders-of-magnitude more efficient and surpass the few-dozen-particle limitation arising in methods based on particle-swarm optimization. We apply our method to the binary-decision-tree model for quantum-enhanced phase estimation as well as to a new problem: a decision tree for adaptive estimation of the unknown bias of a quantum coin in a quantum walk and show how this latter case can be realized experimentally.
In an idealistic setting, quantum metrology protocols allow to sense physical parameters with mean squared error that scales as 1/N21/N^21/N2 with the number of particles involved---substantially surpassing the 1/N1/N1/N-scaling characteristic to classical statistics. A natural question arises, whether such an impressive enhancement persists when one takes into account the decoherence effects that are unavoidable in any real-life implementation. In this thesis, we resolve a major part of this issue by describing general techniques that allow to quantify the attainable precision in metrological schemes in the presence of uncorrelated noise. We show that the abstract geometrical structure of a quantum channel describing the noisy evolution of a single particle dictates then critical bounds on the ultimate quantum enhancement. Our results prove that an infinitesimal amount of noise is enough to restrict the precision to scale classically in the asymptotic NNN limit, and thus constrain the maximal improvement to a constant factor. Although for low numbers of particles the decoherence may be ignored, for large NNN the presence of noise heavily alters the form of both optimal states and measurements attaining the ultimate resolution. However, the established bounds are then typically achievable with use of techniques natural to current experiments. In this work, we thoroughly introduce the necessary concepts and mathematical tools lying behind metrological tasks, including both frequentist and Bayesian estimation theory frameworks. We provide examples of applications of the methods presented to typical qubit noise models, yet we also discuss in detail the phase estimation tasks in Mach-Zehnder interferometry both in the classical and quantum setting---with particular emphasis given to photonic losses while analysing the impact of decoherence.