Divide-and-conquer verification method for noisy intermediate-scale quantum computation

Several noisy intermediate-scale quantum computations can be regarded as logarithmic-depth quantum circuits on a sparse quantum computing chip, where two-qubit gates can be directly applied on only some pairs of qubits. In this paper, we propose a method to efficiently verify such noisy intermediate-scale quantum computation. To this end, we first characterize small-scale quantum operations with respect to the diamond norm. Then by using these characterized quantum operations, we estimate the fidelity psi(t)|(p) over cap (out)| psi(t) between an actual n-qubit output state (p) over cap (out) obtained from the noisy intermediate-scale quantum computation and the ideal output state (i.e., the target state) |psi(t). Although the direct fidelity estimation method requires O(2(n)) copies of (p) over cap (out) on average, our method requires only O(D(3)2(12D)) copies even in the worst case, where D is the denseness of |psi(t). For logarithmic-depth quantum circuits on a sparse chip, D is at most O(log n), and thus O(D(3)2(12D)) is a polynomial in n. By using the IBM Manila 5-qubit chip, we also perform a proof-of-principle experiment to observe the practical performance of our method.