Efficient Learning of Quantum States Prepared With Few Non-Clifford Gates
This addresses the challenge of quantum state learning for near-Clifford states, which is incremental but important for quantum computing applications.
The paper tackles the problem of efficiently learning quantum states prepared with few non-Clifford gates, achieving algorithms that use polynomial time and copies to learn such states to trace distance at most ε, with complexities scaling as poly(n, 2^t, 1/ε) for t non-Clifford gates.
We give a pair of algorithms that efficiently learn a quantum state prepared by Clifford gates and $O(\log n)$ non-Clifford gates. Specifically, for an $n$-qubit state $|ψ\rangle$ prepared with at most $t$ non-Clifford gates, our algorithms use $\mathsf{poly}(n,2^t,1/\varepsilon)$ time and copies of $|ψ\rangle$ to learn $|ψ\rangle$ to trace distance at most $\varepsilon$. The first algorithm for this task is more efficient, but requires entangled measurements across two copies of $|ψ\rangle$. The second algorithm uses only single-copy measurements at the cost of polynomial factors in runtime and sample complexity. Our algorithms more generally learn any state with sufficiently large stabilizer dimension, where a quantum state has stabilizer dimension $k$ if it is stabilized by an abelian group of $2^k$ Pauli operators. We also develop an efficient property testing algorithm for stabilizer dimension, which may be of independent interest.