Single-copy stabilizer learning: average case and worst case

TL;DR

This paper investigates the efficiency of single-copy stabilizer learning in quantum systems, demonstrating that average-case scenarios are more efficient than worst-case, with implications for quantum advantage in symmetry identification.

Contribution

It introduces a new understanding of single-copy stabilizer learning, showing logarithmic-depth circuits suffice on average and establishing exponential sample complexity in the worst case.

Findings

Logarithmic-depth local Clifford circuits can learn most stabilizer groups efficiently.

Numerical simulations support the average-case efficiency up to 100 qubits.

Worst-case adaptive schemes require exponentially many samples in t.

Abstract

We study single-copy stabilizer learning, the problem of identifying a stabilizer group of dimension $n-t$ from an $n$-qubit quantum state $\rho$. We obtain two complementary results. First, in the average case, logarithmic-depth local Clifford circuits suffice to efficiently learn almost all stabilizer groups with $t=O(\log n)$, instead of the linear-depth measurements required in previous approaches. We support this result with numerical simulations for systems of up to 100 qubits. Second, we show that, in the worst case, any adaptive single-copy measurement scheme requires a number of samples that scales exponentially in $t$. Together with existing results on two-copy learning, our findings suggest that, for large $t$, identifying Pauli symmetries of a quantum system exhibits a quantum advantage in the learning setting.