Learning quantum circuits of T -depth one

Abstract

In this paper, we study the problem of learning an unknown quantum circuit of a certain structure. If the unknown target is an n-qubit Clifford circuit, we devise an algorithm to reconstruct its circuit representation by using O(n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ) queries to it. It is unknown for decades how to handle circuits beyond the Clifford group for which the stabilizer formalism cannot be applied. Herein, we study quantum circuits of T -depth one on the computational basis. We show that their output states can be represented by a certain stabilizer pseudomixture. By analyzing the algebraic structure of the stabilizer pseudomixture, we can generate a hypothesis circuit that is equivalent to the unknown target T -depth one quantum circuit U on computational basis states, using Pauli and Bell measurements. If the number of T gates in U is of the order O(log n), our algorithm requires O(n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ) queries to U to produce its equivalent circuit representation on the computational basis in time O(n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> ). Using further additional O(4 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3n</sup> ) classical computations, we can derive an exact description of U for arbitrary input states. Our results greatly extend the previously known facts that stabilizer states can be efficiently identified based on the stabilizer formalism.The full manuscript can be found at [1].