Efficient Learning of Quantum States Prepared With Few Non-Clifford Gates

TL;DR

This paper presents two efficient algorithms for learning quantum states prepared with few non-Clifford gates, significantly improving the efficiency of quantum state tomography for such states.

Contribution

The paper introduces the first efficient algorithms for learning quantum states with limited non-Clifford gates, including a measurement-efficient method and a more general stabilizer dimension testing approach.

Findings

Algorithms run in polynomial time relative to system size and non-Clifford gates

Achieve trace distance accuracy with polynomially many copies of the state

Develop an efficient property testing algorithm for stabilizer dimension

Abstract

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 $|\psi\rangle$ prepared with at most $t$ non-Clifford gates, our algorithms use $\mathsf{poly}(n,2^t,1/\varepsilon)$ time and copies of $|\psi\rangle$ to learn $|\psi\rangle$ to trace distance at most $\varepsilon$. The first algorithm for this task is more efficient, but requires entangled measurements across two copies of $|\psi\rangle$. 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.