Spectral Minimax Direct Fidelity Estimation for Generic Target States

Abstract

Direct fidelity estimation benefits from tailoring measurements to a fixed target, but the operator-aware shadow importance sampling (OASIS) method optimizes an outcome-wise linear-program surrogate rather than the exact worst-case variance over physical states. We propose an exact spectral replacement for arbitrary target states under the same non-adaptive single-copy measurement model. Specifically, we characterize unbiased linear estimators by a single operator identity, determine the state-wise optimal sampling law for fixed reconstruction coefficients, and convert the exact minimax problem into a semidefinite program. The resulting offline design and online estimator are presented as an algorithm and implemented with local Pauli measurements. Numerical simulations under depolarizing noise demonstrate that our exact spectral optimization outperforms the OASIS surrogate in terms of estimation variance.

The paper asks how to optimize direct fidelity estimation for a fixed pure target under non-adaptive, single-copy measurements, replacing the outcome-wise surrogate used by operator-aware shadow importance sampling (OASIS) with the exact worst-case variance over physical states. It retains the same class of unbiased linear estimators and uses local Pauli measurements in its implementation. Writing the estimator output as $\alpha_{u,b}/q(u)$, unbiasedness is equivalent to $\sum_{u,b}\alpha_{u,b}P_{u,b}=O$. For the second-moment operator $M=\sum_{u,b}\alpha_{u,b}^2P_{u,b}/q(u)$, the central result is $\sup_\rho\operatorname{Var}_\rho(X)=\min_t\{\lambda_{\max}(M-2tO)+t^2\}$. This follows from a minimax interchange and spectral maximization over density operators. Perspective constraints then give an exact semidefinite program jointly optimizing sampling probabilities and reconstruction coefficients. The resulting sample mean is unbiased, with worst-case MSE $\Gamma^\star(O)/N$ and a Chebyshev guarantee when $N\geq\Gamma^\star(O)/(\delta\varepsilon^2)$. Simulations use one Haar-random target at each of $n=3,4,5,6$ qubits, with $\rho=0.9O+0.1I/2^n$ and 1,000 repetitions per configuration. At matched shot budgets, MSE falls from 3.67 to 3.49, 2.40 to 2.13, 2.22 to 1.52, and 1.49 to 1.01, respectively, in units of $10^{-4}$. The stated limitation is exponential offline complexity: the local-Pauli SDP has $6^n$ reconstruction coefficients and a $2^n$-dimensional matrix inequality. It is presented as a small-system benchmark for generic targets, not a scalable method or a replacement for specialized methods exploiting structured targets.