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📄 Abstract
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.
Key Contributions
This paper presents efficient algorithms for learning quantum states prepared with a limited number of non-Clifford gates (O(log n)). It offers two algorithms: one more efficient but requiring entangled measurements, and another using single-copy measurements at the cost of higher complexity. Additionally, it introduces an efficient property testing algorithm for stabilizer dimension.
Business Value
Advances the capabilities of quantum computing by enabling more efficient characterization and learning of quantum states, which is fundamental for developing quantum algorithms and applications.