Optimal randomized measurements for a family of non-linear quantum properties

TL;DR

This paper introduces the ORM protocol for efficiently estimating non-linear quantum properties like Tr(Oρ²), demonstrating optimal sample complexity and practical advantages over classical shadows.

Contribution

It presents the observable-driven randomized measurement (ORM) protocol, achieving optimal sample complexity for broad classes of non-linear properties in quantum systems.

Findings

ORM requires fewer samples than classical shadows for the same precision.

ORM is optimal for observables with large trace-norm, including Pauli and local observables.

Numerical experiments confirm ORM's efficiency and practical implementation advantages.

Abstract

Quantum learning encounters fundamental challenges when estimating non-linear 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 non-linear properties without sacrificing optimality remains an open problem. In this work, we introduce the observable-driven randomized measurement (ORM) protocol enabling the estimation of ${\rm Tr}(O\rho^2)$ 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 non-linear observables, reducing circuit complexities in practical applications.