Efficient learning of $t$-doped stabilizer states with single-copy measurements

Quantum 8, 1250 (2024).

https://doi.org/10.22331/q-2024-02-12-1250

One of the primary objectives in the field of quantum state learning is to develop algorithms that are time-efficient for learning states generated from quantum circuits. Earlier investigations have demonstrated time-efficient algorithms for states generated from Clifford circuits with at most $log(n)$ non-Clifford gates. However, these algorithms necessitate multi-copy measurements, posing implementation challenges in the near term due to the requisite quantum memory. On the contrary, using solely single-qubit measurements in the computational basis is insufficient in learning even the output distribution of a Clifford circuit with one additional $T$ gate under reasonable post-quantum cryptographic assumptions. In this work, we introduce an efficient quantum algorithm that employs only nonadaptive single-copy measurement to learn states produced by Clifford circuits with a maximum of $O(log n)$ non-Clifford gates, filling a gap between the previous positive and negative results.