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

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 $\mathrm{Tr}(O{\rho }^2)$ 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.

17+25 pages, 5 figures

References in corpus (32)

• Supplementary information for "Quantum supremacy using a programmable superconducting processor"
• Randomized Benchmarking of Quantum Gates
• Challenges and Opportunities in Quantum Machine Learning
• Quantum Error Mitigation
• Quantum advantage in learning from experiments
• Direct Fidelity Estimation from Few Pauli Measurements
• The randomized measurement toolbox
• Evenly distributed unitaries: on the structure of unitary designs
• Fundamental limits of quantum error mitigation
• Quantum criticality under decoherence or weak measurement
• Computational advantage of quantum random sampling
• Introduction to Haar Measure Tools in Quantum Information: A Beginner's Tutorial
• Exact emergent quantum state designs from quantum chaotic dynamics
• Nishimori's cat: stable long-range entanglement from finite-depth unitaries and weak measurements
• Strong-to-Weak Spontaneous Symmetry Breaking in Mixed Quantum States
• Shadow Distillation: Quantum Error Mitigation with Classical Shadows for Near-Term Quantum Processors
• Symmetry protected topological phases under decoherence
• Shallow shadows: Expectation estimation using low-depth random Clifford circuits
• Universal Sampling Lower Bounds for Quantum Error Mitigation
• Universal cost bound of quantum error mitigation based on quantum estimation theory
• Measuring Arbitrary Physical Properties in Analog Quantum Simulation
• Detecting entanglement in quantum many-body systems via permutation moments
• Analysing quantum systems with randomised measurements
• Thrifty shadow estimation: re-using quantum circuits and bounding tails
• Distributed quantum inner product estimation
• Performance analysis of multi-shot shadow estimation
• Robust estimation of the Quantum Fisher Information on a quantum processor
• Certifying Quantum Temporal Correlation via Randomized Measurements: Theory and Experiment
• Efficient distributed inner product estimation via Pauli sampling
• Symmetric Clifford twirling for cost-optimal quantum error mitigation in early FTQC regime
• Efficiently measuring $d$-wave pairing and beyond in quantum gas microscopes
• Quantum subspace verification for error correction codes