Tolerant Testing of Stabilizer States with Mixed State Inputs
Vishnu Iyer, Daniel Liang
TL;DR
The paper addresses tolerant testing of stabilizer states when the input is a mixed quantum state. It builds a GNW-based testing framework that uses Bell difference sampling and an ancilla-free SWAP-test estimator to approximate the symplectic-Fourier–domain quantities $\widehat{p}_\rho(x)$ and a derived bias $\eta=4^n\sum_x q_\rho(x)\widehat{p}_\rho(x)$, achieving $\poly(1/\varepsilon_1)$ copies and $O(n\,\mathrm{poly}(1/\varepsilon_1))$ time with success probability $>2/3$ for mixed inputs. Completeness results show the test remains faithful when the state has high stabilizer fidelity, while soundness extends prior pure-state proofs by a sequence of steps (approximate subspace -> affine subspace -> proper subspace -> isotropic subspace) to bound the stabilizer fidelity from test outcomes. The work preserves GNW-like resource properties (6 copies per iteration, linear time, Clifford-only operations) and matches known bounds in the close regime, contributing a practical, noise-tolerant tool for stabilizer-state testing in realistic, nonideal quantum systems, while highlighting avenues for improving constants with newer techniques.
Abstract
We study the problem of tolerant testing of stabilizer states. In particular, we give the first such algorithm that accepts mixed state inputs. Formally, given a mixed state $ρ$ that either has fidelity at least $\varepsilon_1$ with some stabilizer pure state or fidelity at most $\varepsilon_2$ with all such states, where $\varepsilon_2 \leq \varepsilon_1^{O(1)}$, our algorithm distinguishes the two cases with sample complexity $\text{poly}(1/\varepsilon_1)$ and time complexity $O(n \cdot \text{poly}(1/\varepsilon_1))$.