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  • Open Access

Optimal Randomized Measurements for a Family of Nonlinear Quantum Properties

Zhenyu Du1,*, Yifan Tang2,*, Andreas Elben3,4, Ingo Roth5, Jens Eisert2, and Zhenhuan Liu1,†

  • *Z.D. and Y.T. contributed equally to this work.
  • †Contact author: qubithuan@gmail.com
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PRX Quantum 7, 010360 – Published 26 March, 2026

DOI: https://doi.org/10.1103/4rkx-xkwy

Abstract

Quantum learning encounters fundamental challenges when estimating nonlinear properties, owing to the inherent linearity of quantum mechanics. Although recent advances in single-copy randomized measurement protocols have achieved optimal sample complexity for specific tasks like state purity estimation, generalizing these protocols to estimate broader classes of nonlinear properties without sacrificing optimality remains an open problem. In this work, we introduce the observable-driven randomized measurement (ORM) protocol enabling the estimation of Tr(Oρ2) for an arbitrary observable O—an essential quantity in quantum computing and many-body physics. We establish an upper bound for ORM’s sample complexity and show its optimality for observables with a large trace-norm, including Pauli and local observables, closing a gap in the literature. For these observables, ORM admits an efficient implementation with Clifford circuits. Numerical experiments validate that ORM requires substantially fewer state samples to achieve the same precision compared to classical shadows. Additionally, we introduce a braiding randomized measurement protocol for multiple low-rank nonlinear observables, reducing circuit complexities in practical applications.

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Popular Summary

Quantum learning theory offers a powerful lens through which to infer the properties of quantum systems from data. It lies at the heart of quantum readout, benchmarking, and emerging strategies for error mitigation in quantum technologies. Beyond its practical relevance, it poses deep foundational questions probing the very limits of how quantum properties can be efficiently and accurately measured. Yet, perhaps most critically, it stands as a technological imperative: as quantum technologies evolve, their progress appears inextricably linked to the development of equally sophisticated quantum learning techniques.

In this work, we explore a pressing and high-impact challenge at the intersection of physics, computer science, and information theory: how to efficiently learn complex, nonlinear features of quantum systems using minimal quantum resources. We introduce an observable-driven randomized measurement protocol that reaches the optimal sample complexity while requiring only single-copy measurements and a constant number of measurement settings. Tailored for near-term experimental platforms, our approach is both hardware-efficient and scalable, enabling simultaneous estimation of multiple observables through purely local operations. These contributions help close a critical gap in quantum learning, equipping current-generation quantum devices with practical, high-performance tools for tasks ranging from error mitigation and virtual cooling to entropy estimation in quantum computing, simulation, and many-body physics.

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PRX Quantum 7, 010360– Published 26 March, 2026
  • Received 24 June 2025
  • Accepted 12 December 2025
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