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

Abstract

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 ⟨ψt|ρ^out|ψt⟩ between an actual n-qubit output state ρ^out obtained from the noisy intermediate-scale quantum computation and the ideal output state (i.e., the target state) |ψt⟩. Although the direct fidelity estimation method requires O(2n) copies of ρ^out on average, our method requires only O(D3212D) copies even in the worst case, where D is the denseness of |ψt⟩. For logarithmic-depth quantum circuits on a sparse chip, D is at most O(log⁡n), and thus O(D3212D) 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.