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  • Open Access

Heisenberg-Limited Adaptive Gradient Estimation for Multiple Observables

Kaito Wada1,*, Naoki Yamamoto2,3,†, and Nobuyuki Yoshioka4,5,6,7,‡

  • 1Graduate School of Science and Technology, Keio University, 3-14-1 Hiyoshi, Kohoku, Yokohama, Kanagawa 223-8522, Japan
  • 2Department of Applied Physics and Physico-Informatics, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama, Kanagawa 223-8522, Japan
  • 3Quantum Computing Center, Keio University, Hiyoshi 3-14-1, Kohoku, Yokohama 223-8522, Japan
  • 4Department of Applied Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
  • 5Theoretical Quantum Physics Laboratory, RIKEN Cluster for Pioneering Research (CPR), Wako-shi, Saitama 351-0198, Japan
  • 6JST, PRESTO, 4-1-8 Honcho, Kawaguchi, Saitama 332-0012, Japan
  • 7International Center for Elementary Particle Physics, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan

  • *Contact author: wkai1013keio840@keio.jp
  • †Contact author: yamamoto@appi.keio.ac.jp
  • ‡Contact author: nyoshioka@ap.t.u-tokyo.ac.jp
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PRX Quantum 6, 020308 – Published 10 April, 2025

DOI: https://doi.org/10.1103/PRXQuantum.6.020308

Abstract

In quantum mechanics, measuring the expectation value of a general observable has an inherent statistical uncertainty that is quantified by variance or mean squared error of measurement outcome. While the uncertainty can be reduced by averaging several samples, the number of samples should be minimized when each sample is very costly. This is especially the case for fault-tolerant quantum computing that involves measurement of multiple observables of nontrivial states in large quantum systems that exceed the capabilities of classical computers. In this work, we provide an adaptive quantum algorithm for estimating the expectation values of M general observables within root mean squared error ε simultaneously, using O(ε−1MlogM) queries to a state preparation oracle of a target state. This remarkably achieves the scaling of Heisenberg limit 1/ε, a fundamental bound on the estimation precision in terms of mean squared error, together with the sublinear scaling of the number of observables M. The proposed method is an adaptive version of the quantum gradient-estimation algorithm and has a resource-efficient implementation due to its adaptiveness. Specifically, the space overhead in the proposed method is O(M), which is independent from the estimation precision ε unlike noniterative algorithms. In addition, our method can avoid the numerical instability problem for constructing quantum circuits in a large-scale task (e.g., ε≪1 in our case), which appears in the actual implementation of many algorithms relying on quantum signal processing techniques. Our method paves a new way to precisely understand and predict various physical properties in complicated quantum systems using quantum computers.

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Physics Subject Headings (PhySH)

Popular Summary

Quantum-enhanced measurements can vastly improve estimation efficiency compared to classical approaches. In particular, achieving the fundamental limit on precision, known as the Heisenberg limit, is vital for advanced quantum technologies. Such a quantum speedup is crucial in quantum computing, which measures expectation values of observables, since this task is essential but highly time consuming in various applications. We address the challenge of accurately estimating multiple observables from an expensive quantum circuit preparing a target state while minimizing overall resource use. Our adaptive quantum algorithm proves that it is possible to attain sublinear resource scaling in the number of observables and Heisenberg-limited precision in terms of mean squared error, a common measure of uncertainty.

Building on quantum metrology insights, we design an adaptive procedure by appropriately allocating quantum resources, combined with another quantum amplification for observable encoding in quantum computing. The resulting algorithm represents the first adaptive extension of the quantum gradient-estimation algorithm, offering quantum speedups in calculating multiple quantities across various domains. Crucially, the adaptivity allows us to reduce the additional space overhead to a level independent of the required precision and exponentially reduce classical computational resources for circuit synthesis compared to existing methods. Therefore, our method significantly lowers the barrier of practical implementation, making quantum-enhanced measurements of multiple observables more accessible.

Many target quantities require accurate and efficient estimation using quantum-enhanced measurements, both in quantum computing and in quantum metrology. We believe the developed method will provide a basic strategy for achieving this goal.

Article Text

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PRX Quantum 6, 020308– Published 10 April, 2025
  • Received 17 June 2024
  • Revised 28 January 2025
  • Accepted 12 March 2025
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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