Variational quantum algorithms (VQAs) offer the most promising path to obtaining quantum advantages via noisy intermediate-scale quantum (NISQ) processors. Such systems leverage classical optimization to tune the parameters of a parameterized quantum circuit (PQC). The goal is minimizing a cost function that depends on measurement outputs obtained from the PQC. Optimization is typically implemented via stochastic gradient descent (SGD). On NISQ computers, gate noise due to imperfections and decoherence affects the stochastic gradient estimates by introducing a bias. Quantum error mitigation (QEM) techniques can reduce the estimation bias without requiring any increase in the number of qubits, but they in turn cause an increase in the variance of the gradient estimates. This work studies the impact of quantum gate noise on the convergence of SGD for the variational eigensolver (VQE), a fundamental instance of VQAs. The main goal is ascertaining conditions under which QEM can enhance the performance of SGD for VQEs. It is shown that quantum gate noise induces a non-zero error-floor on the convergence error of SGD (evaluated with respect to a reference noiseless PQC), which depends on the number of noisy gates, the strength of the noise, as well as the eigenspectrum of the observable being measured and minimized. In contrast, with QEM, any arbitrarily small error can be obtained. Furthermore, for error levels attainable with or without QEM, QEM can reduce the number of required iterations, but only as long as the quantum noise level is sufficiently small, and a sufficiently large number of measurements is allowed at each SGD iteration. Numerical examples for a max-cut problem corroborate the main theoretical findings.
翻译:快速量子算法(VQAs)提供了最有希望的途径,通过杂音中间级量量处理器(NISQQ)获得量子优势。这种系统可以利用经典优化来调节参数化量子电路(PQC)的参数。目标是最大限度地减少一个取决于从PQC获得的测量产出的成本函数。优化一般是通过随机梯度梯度下降(SGD)来实施的。在 NISQ计算机上,由于不完善和不一致性造成的大门噪音通过引入偏差来影响随机偏差的梯度估计。量级差缓解(QEM)技术可以减少估计偏差,而不需要增加qubits的量子线性测量,但反过来又导致梯度的偏差增加。 量门级噪声对SGD(VQE)的趋同作用影响很大。 量级峰值的精确度可以降低 VQED的性能,但只有QEMD(QEM)的性能提高性能提高性能。 量门的量级噪声比可以降低, QQNQQQQQQ的低的值比值比值比值比值比值越低, QQQQQQQQ的值的值的值能的值能能水平的值比值比值比值越低, QQQQQQQQ的值值值值值值值值值值值值值值值值值值值能的值值值值值值比值比值比值比值越高, QQ值越低, Q值值值值值值值值值值值越低值值越低, Q值越低, Q值越低值越低, Q值越低, Q值越高越高越低, Q值越低值越高越高越高越高越高越高越高越高越高越高越高越高越高越高越高越高越低值值值值值越高越低,越高越高越高越高越低值值值值越高越低,越高越高越高越低,越低,越高越高越高越高越高越高,越低,越高,越高,越高,越低,越高