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Variational quantum algorithms (VQAs) offer the most promising path toobtaining quantum advantages via noisy intermediate-scale quantum (NISQ)processors. Such systems leverage classical optimization to tune the parametersof a parameterized quantum circuit (PQC). The goal is minimizing a costfunction that depends on measurement outputs obtained from the PQC.Optimization is typically implemented via stochastic gradient descent (SGD). OnNISQ computers, gate noise due to imperfections and decoherence affects thestochastic gradient estimates by introducing a bias. Quantum error mitigation(QEM) techniques can reduce the estimation bias without requiring any increasein the number of qubits, but they in turn cause an increase in the variance ofthe gradient estimates. This work studies the impact of quantum gate noise onthe convergence of SGD for the variational eigensolver (VQE), a fundamentalinstance of VQAs. The main goal is ascertaining conditions under which QEM canenhance the performance of SGD for VQEs. It is shown that quantum gate noiseinduces a non-zero error-floor on the convergence error of SGD (evaluated withrespect to a reference noiseless PQC), which depends on the number of noisygates, the strength of the noise, as well as the eigenspectrum of theobservable being measured and minimized. In contrast, with QEM, any arbitrarilysmall error can be obtained. Furthermore, for error levels attainable with orwithout QEM, QEM can reduce the number of required iterations, but only as longas the quantum noise level is sufficiently small, and a sufficiently largenumber of measurements is allowed at each SGD iteration. Numerical examples fora max-cut problem corroborate the main theoretical findings.

Error Mitigation-Aided Optimization of Parameterized Quantum Circuits: Convergence Analysis