Optimal randomized measurements for a family of non-linear quantum properties
#Quantumlearning 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
#samplecomplexity 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 Tr(Oρ^2) for an arbitrary observable O - an essential quantity in
#quantumcomputing and
#manybodyphysics. ORM achieves this by decomposing the observable O into dichotomic observables and extracting the information of each eigenspace through randomized measurements with block-diagonal unitaries.
We establish an upper bound for ORM's sample complexity and prove its optimality for all Pauli observables, closing a gap in the literature. Furthermore, we develop simplified variants of ORM for local Pauli observables and introduce a braiding
#randomized measurement protocol for fidelity estimation, both of which significantly reduce
#circuitcomplexities in practical applications. Numerical experiments validate that ORM requires substantially fewer state samples to achieve the same precision compared to classical shadows.
Warm thanks to Zhenyu Du,
@YifanTang1999,
@AndreasElben, Ingo Roth, and
@liu_zhenhu77178 for the productive collaboration.
