We present a protocol that estimates the quantum fidelity, up to precision $\varepsilon$, between a known target state and unknown lab-prepared state with sublinear, $o(d^{0.9908}/\varepsilon^2)$, number of Pauli basis measurements.
We present a protocol that estimates the quantum fidelity, up to precision ε, between a known target state and unknown lab-prepared state with sublinear, o(d0.9908/ε2), number of Pauli basis measurements.
We propose an efficient protocol to estimate the fidelity of an n-qubit entangled measurement device, requiring only qubit state preparations and classical data post-processing. It works by measuring the eigenstates of Pauli operators, which are strategically selected according to their importance weights and collectively contributed by all measurement operators. We rigorously analyze the protocol's performance and demonstrate that its sample complexity is uniquely determined by the number of Pauli operators possessing non-zero expectation values with respect to the target measurement. Moreover, from a resource-theoretic perspective, we introduce the stabilizer Rényi entropy of quantum measurements as a precise metric to quantify the inherent difficulty of estimating measurement fidelity.
As quantum devices scale, quantifying how close an experimental state aligns with a target becomes both vital and challenging. Fidelity is the standard metric, but existing estimators either require full tomography or apply only to restricted state/measurement families. Huang, Preskill, and Soleimanifar (Nature Physics, 2025) introduced an efficient certification protocol for Haar-random states using only a polynomial number of non-adaptive, single-copy, local Pauli measurements. Here, we adopt the same data collection routine but recast it as a fidelity estimation protocol with rigorous performance guarantees and broaden its applicability. We analyze the bias in this estimator, linking its performance to the mixing time τ of a Markov chain induced by the target state, and resolve the three open questions posed by Huang, Preskill, and Soleimanifar (Nature Physics, 2025). Our analysis extends beyond Haar-random states to state t-designs, states prepared by low-depth random circuits, physically relevant states and families of mixed states. We introduce a k-generalized local escape property that identifies when the fidelity estimation protocol is both efficient and accurate, and design a practical empirical test to verify its applicability for arbitrary states. This work enables scalable benchmarking, error characterization, and tomography assistance, supports adaptive quantum algorithms in high dimensions, and clarifies fundamental limits of learning from local measurements.
Direct fidelity estimation is a protocol that estimates the fidelity between an experimental quantum state and a target pure state. By measuring the expectation values of Pauli operators selected through importance sampling, the method is exponentially faster than full quantum state tomography. We propose an enhanced direct fidelity estimation protocol that uses fewer copies of the experimental state by grouping Pauli operators before the sampling process. We derive analytical bounds on the measurement cost and estimator variance, showing improvements over the standard method. Numerical simulations validate our approach, demonstrating that for 8-qubit Haar-random states, our method achieves a one-third reduction in the required number of copies and reduces variance by an order of magnitude using only local measurements. These results underscore the potential of our protocol to enhance the efficiency of fidelity estimation in current quantum devices.
We study the number of copies needed to estimate the root Uhlmann fidelity between an unknown quantum state and a classically known reference, under collective measurements. If the unknown state has rank at most s, we give an estimator using O(s2/ε2) copies, uniformly in the ambient dimension and reference rank. The estimator applies a random-purification channel and a covariant pure-state measurement, then rescales the observed amplitude before evaluating a weighted nuclear norm. Combined with the lower bound established independently in our first version and in concurrent work of Wang, this determines the worst-case complexity under rank bounds r,s as min{r,s}2/ε2 up to logarithmic factors in the smaller rank.
The same estimator gives the upper bound O(r(trσ)2/ε2) for a rank-r reference σ. We combine spectral truncation with lower bounds obtained by embedding hard instances for the maximally mixed reference into spectral subspaces. For spectra λi∝i−α with fixed 1<α≤2, as ε↓0 with r≥Cαε−2/(α−1), the bounds determine the complexity as Θα(ε−4/(α−1)). In particular, inverse-square spectra have accuracy exponent four. The lower-bound construction uses exact moment matching and an explicitly computable Schur measure.
We consider a fundamental problem of \emph{mixedness testing}: Given n copies of an N-qubit state ρ, determine whether ρ=Id/d or ∥ρ−Id/d∥1≥ε with high probability, where d=2N. In particular, we focus on performing this task in the practical setting of single-qubit measurements, where measurements are prepared independently on each qubit. We provide a nearly complete picture of single-qubit mixedness tesing by showing n=Θ~(10N/ε2). To establish our lower bound, we introduce a measurement-dependent lower bound framework for adaptive single-copy state certification. For the upper bound, we present a randomized Pauli basis measurement protocol, which relies on a new primitive for computationally efficient uniformity testing of correlation-concentrated distributions on the Boolean hypercube. In conjunction, we provide lower and upper bounds for mixedness testing with fixed Pauli measurement protocols.
We describe a simple method for certifying that an experimental device prepares a desired quantum state ρ. Our method is applicable to any pure state ρ, and it provides an estimate of the fidelity between ρ and the actual (arbitrary) state in the lab, up to a constant additive error. The method requires measuring only a constant number of Pauli expectation values, selected at random according to an importance-weighting rule. Our method is faster than full tomography by a factor of d, the dimension of the state space, and extends easily and naturally to quantum channels.