Learning quantum circuits of some $T$ gates

In this paper, we study the problem of learning quantum circuits 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 $O(n^2)$ 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 given the all-zero state as an input. We show that their output states can be represented by some stabilizer pseudomixtures. By analyzing the algebraic structure of the stabilizer pseudomixture, we can reconstruct the output state of an unknown $T$-depth one quantum circuit on input $|0^n\rangle$ from the outcomes of Pauli measurements and Bell measurements. If the number of $T$ gates is of the order $O(\log n)$, our algorithm requires $O(n^2)$ queries. Our results greatly extend the previous known facts that stabilizer states can be efficiently identified based on the stabilizer formalism. Hence, the proposed expanded stabilizer formalism and our analysis might pave the way towards learning quantum circuits beyond the Clifford structure.