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 $\langle\psi_t|\hat{\rho}_{\rm out}|\psi_t\rangle$ between an actual $n$-qubit output state $\hat{\rho}_{\rm out}$ obtained from the noisy intermediate-scale quantum computation and the ideal output state (i.e., the target state) $|\psi_t\rangle$. Although the direct fidelity estimation method requires $O(2^n)$ copies of $\hat{\rho}_{\rm out}$ on average, our method requires only $O(D^32^{12D})$ copies even in the worst case, where $D$ is the denseness of $|\psi_t\rangle$. For logarithmic-depth quantum circuits on a sparse chip, $D$ is at most $O(\log{n})$, and thus $O(D^32^{12D})$ 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.

中文翻译：

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噪声中等规模量子计算的分治验证方法

几个嘈杂的中尺度量子计算可以看作是稀疏量子计算芯片上的对数深度量子电路，其中两个量子比特门只能直接应用于一些量子比特对。在本文中，我们提出了一种有效验证这种嘈杂的中尺度量子计算的方法。为此，我们首先根据钻石范数来描述小规模的量子操作。然后通过使用这些特征化的量子操作，我们估计了实际 $n$-qubit 输出状态 $\hat{ 之间的保真度 $\langle\psi_t|\hat{\rho}_{\rm out}|\psi_t\rangle$ \rho}_{\rm out}$ 从有噪声的中尺度量子计算和理想输出状态（即目标状态）$|\psi_t\rangle$ 中获得。虽然直接保真估计方法平均需要 $O(2^n)$ 个 $\hat{\rho}_{\rm out}$ 副本，但我们的方法只需要 $O(D^32^{12D})$即使在最坏的情况下也会复制，其中 $D$ 是 $|\psi_t\rangle$ 的密度。对于稀疏芯片上的对数深度量子电路，$D$ 最多为 $O(\log{n})$，因此 $O(D^32^{12D})$ 是 $n$ 中的多项式。通过使用 IBM Manila 5-qubit 芯片，我们还进行了原理验证实验，以观察我们方法的实际性能。