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arXiv:2307.16083 · Physical Review X

Tackling Sampling Noise in Physical Systems for Machine Learning Applications: Fundamental Limits and Eigentasks

Fangjun Hu, Gerasimos Angelatos, Saeed A. Khan, Marti Vives, Hakan E. Türeci, Leon Bello, Graham E. Rowlands, Guilhem J. Ribeill, Hakan E. Türeci

Published Oct 30, 2023• 22 citations
arXivHTMLPDF

Abstract

The expressive capacity of physical systems employed for learning is limited by the unavoidable presence of noise in their extracted outputs.Though present in physical systems across both the classical and quantum regimes, the precise impact of noise on learning remains poorly understood.Focusing on supervised learning, we present a mathematical framework for evaluating the resolvable expressive capacity (REC) of general physical systems under finite sampling noise and provide a methodology for extracting its extrema, the eigentasks.Eigentasks are a native set of functions that a given physical system can approximate with minimal error.We show that the REC of a quantum system is limited by the fundamental theory of quantum measurement and obtain a tight upper bound for the REC of any finitely sampled physical system.We then provide empirical evidence that extracting low-noise eigentasks can lead to improved performance for machine learning tasks such as classification, displaying robustness to overfitting.We present analyses suggesting that correlations in the measured quantum system enhance learning capacity by reducing noise in eigentasks.The applicability of these results in practice is demonstrated with experiments on superconducting quantum processors.Our findings have broad implications for quantum machine learning and sensing applications.

DOI
10.1103/physrevx.13.041020

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The expressive capacity of physical systems employed for learning is limited by the unavoidable presence of noise in their extracted outputs.Though present in physical systems across both the classical and quantum regimes, the precise impact of noise on learning remains poorly understood.Focusing on supervised learning, we present a mathematical framework for evaluating the resolvable expressive capacity (REC) of general physical systems under finite sampling noise and provide a methodology for extracting its extrema, the eigentasks.Eigentasks are a native set of functions that a given physical system can approximate with minimal error.We show that the REC of a quantum system is limited by the fundamental theory of quantum measurement and obtain a tight upper bound for the REC of any finitely sampled physical system.We then provide empirical evidence that extracting low-noise eigentasks can lead to improved performance for machine learning tasks such as classification, displaying robustness to overfitting.We present analyses suggesting that correlations in the measured quantum system enhance learning capacity by reducing noise in eigentasks.The applicability of these results in practice is demonstrated with experiments on superconducting quantum processors.Our findings have broad implications for quantum machine learning and sensing applications.
The expressive capacity of quantum systems for machine learning is limited by quantum sampling noise incurred during measurement. Although it is generally believed that noise limits the resolvable capacity of quantum systems, the precise impact of noise on learning is not yet fully understood. We present a mathematical framework for evaluating the available expressive capacity of general quantum systems from a finite number of measurements, and provide a methodology for extracting the extrema of this capacity, its eigentasks. Eigentasks are a native set of functions that a given quantum system can approximate with minimal error. We show that extracting low-noise eigentasks leads to improved performance for machine learning tasks such as classification, displaying robustness to overfitting. We obtain a tight bound on the expressive capacity, and present analyses suggesting that correlations in the measured quantum system enhance learning capacity by reducing noise in eigentasks. These results are supported by experiments on superconducting quantum processors. Our findings have broad implications for quantum machine learning and sensing applications.
Tackling output sampling noise due to finite shots of quantum measurement is an unavoidable challenge when extracting information in machine learning with physical systems. A technique called Eigentask Learning was developed recently as a framework for learning with infinite input training data in the presence of output sampling noise. In the work of Eigentask Learning, numerical evidence was presented that extracting low-noise contributions of features can practically improve performance for machine learning tasks, displaying robustness to overfitting and increasing generalization accuracy. However, it remains unsolved to quantitatively characterize generalization errors in situations where the training dataset is finite, while output sampling noise still exists. In this study, we use methodologies from statistical mechanics to calculate the training and generalization errors of a generic quantum machine learning system when the input training dataset and output measurement sampling shots are both finite. Our analytical findings, supported by numerical validation, offer solid justification that Eigentask Learning provides optimal learning in the sense of minimizing generalization errors.
We introduce an end-to-end optimization strategy for quantum machine learning that directly targets performance under finite measurement resources, where learning objectives are defined directly at the level of task performance. The method is applied on a Bayesian quantum metrology task since it provides a natural testbed with known fundamental limits and scaling with system size. The sampling-aware hybrid algorithm achieves a single-shot risk within 1 dB of the -20 dB Bayesian limit using 32 qubits. We extend the Bayesian framework from parameter estimation to global function inference, where the task is to infer a target function of the sensor input drawn from an arbitrary prior, and we demonstrate a clear computational-sensing advantage for direct functional inference over indirect reconstruction. We relate the corresponding Bayesian risk to the Capacity metric and argue that the Resolvable Expressive Capacity provides a natural measure of the space of functions accessible in a single shot. The resulting eigentask analysis identifies noise-robust feature combinations that yield compact estimators with improved accuracy and reduced optimization cost in resource-limited or real-time on-device settings. Going beyond the irreducible statistical uncertainty associated with finite measurement records, we study implementation-induced distortions of the sensor response focusing on representative imperfections arising from readout errors and static coherent gate disorder. We show that when these imperfections are stationary across training and inference, end-to-end optimization can adapt the quantum circuit and classical estimator to the implemented hardware response, preserving high inference performance over a broad range of imperfection strengths.
The development of practical quantum processors relies on the ability to control and predict their functioning despite the presence of noise. This is particularly challenging for temporarily correlated noise. Here we propose a physics-inspired supervised machine learning approach to efficiently and accurately predict the functioning of quantum processors in the presence of correlated noise, which only requires data from randomized benchmarking experiments. To demonstrate the efficacy of our technique, we analyze the data from a superconducting quantum processor with tunable correlated noise. We produce training data by evolving the system for a number of time steps, and with this, we fully quantify the correlated noise and accurately predict the dynamics of the system for times beyond the training data. This approach shows a path towards efficient and effective learning of noisy quantum dynamics and optimally control quantum processors over long and complex computations even in the presence of correlated noise. Combining standard quantum benchmarking techniques with a physics-inspired supervised machine learning algorithm, the authors efficiently and accurately predict the functioning of a superconducting qubit in the presence of complex noise. The proposed approach opens a path towards a better control of current quantum devices.
Quantum computing has emerged as a powerful computational paradigm capable of solving problems beyond the reach of classical computers. However, today’s quantum computers are noisy, posing challenges to obtaining accurate results. Here, we explore the impact of noise on quantum computing, focusing on the challenges in sampling bit strings from noisy quantum computers and the implications for optimization and machine learning. We formally quantify the sampling overhead to extract good samples from noisy quantum computers and relate it to the layer fidelity, a metric to determine the performance of noisy quantum processors. Further, we show how this allows us to use the conditional value at risk of noisy samples to determine provable bounds on noise-free expectation values. We discuss how to leverage these bounds for different algorithms and demonstrate our findings through experiments on real quantum computers involving up to 127 qubits. The results show strong alignment with theoretical predictions. In this study, the authors investigate the impact of noise on quantum computing with a focus on the challenges in sampling bit strings from noisy quantum computers, which has implications for optimization and machine learning.
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.