An algorithm for the T-count

We consider quantum circuits composed of Clifford and T gates. In this context the T gate has a special status since it confers universal computation when added to the (classically simulable) Clifford gates. However it can be very expensive to implement fault-tolerantly. We therefore view this gate as a resource which should be used only when necessary. Given an n-qubit unitary U we are interested in computing a circuit that implements it using the minimum possible number of T gates (called the T -count of U). A related task is to decide if the T -count of U is less than or equal to m; we consider this problem as a function of N = 2n and m. We provide a classical algorithm which solves it using time and space both upper bounded as O(Nmpoly(m,N)). We implemented our algorithm and used it to show that any Clifford+T circuit for the Toffoli or the Fredkin gate requires at least 7 T gates. This implies that the known 7 T gate circuits for these gates are T -optimal. We also provide a simple expression for the T -count of single-qubit unitaries.