Despite fundamental interests in learning quantum circuits, the existence of a computationally efficient algorithm for learning shallow quantum circuits remains an open question. Because shallow quantum circuits can generate distributions that are classically hard to sample from, existing learning algorithms do not apply. In this work, we present a polynomial-time classical algorithm for learning the description of any unknown $n$-qubit shallow quantum circuit $U$ (with arbitrary unknown architecture) within a small diamond distance using single-qubit measurement data on the output states of $U$. We also provide a polynomial-time classical algorithm for learning the description of any unknown $n$-qubit state $\lvert \psi \rangle = U \lvert 0^n \rangle$ prepared by a shallow quantum circuit $U$ (on a 2D lattice) within a small trace distance using single-qubit measurements on copies of $\lvert \psi \rangle$. Our approach uses a quantum circuit representation based on local inversions and a technique to combine these inversions. This circuit representation yields an optimization landscape that can be efficiently navigated and enables efficient learning of quantum circuits that are classically hard to simulate.

翻译：尽管学习量子电路具有基础性的研究价值，但高效计算浅层量子电路的学习算法是否存在仍是一个开放问题。由于浅层量子电路能生成经典计算难以采样的分布，现有学习算法在此场景下无法适用。本文提出一种多项式时间经典算法，该算法利用未知$n$量子比特浅层量子电路$U$（具有任意未知架构）输出态上的单比特测量数据，可在小钻石距离内学习其完整描述。此外，我们进一步给出一种多项式时间经典算法，利用$\lvert \psi \rangle$副本上的单比特测量，在迹距离约束下学习由浅层量子电路$U$（在二维晶格上）制备的任意$n$量子比特态$\lvert \psi \rangle = U \lvert 0^n \rangle$的完整描述。我们的方法基于局部逆变换的量子电路表示及组合技术，该表示方法构建了可高效导航的优化景观，从而实现对经典难模拟量子电路的高效学习。