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

• URL: http://arxiv.org/abs/2305.13409v4
• Date: Fri, 5 Apr 2024 02:26:02 GMT
• Title: Efficient Learning of Quantum States Prepared With Few Non-Clifford Gates
• Authors: Sabee Grewal, Vishnu Iyer, William Kretschmer, Daniel Liang,
• Abstract summary: 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 $|psirangle$ prepared with at most $t$ non-Clifford gates, our algorithms use $mathsfpoly(n,2t,1/varepsilon)$ time and copies of $|psirangle$ to learn $|psirangle$ to trace distance at most gates.
• Score: 0.43123403062068827
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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