Spectral analysis for Variational Quantum Algorithms: noise diagnostics, error mitigation, and efficient cost function recovery

We study the cost function of variational quantum algorithms (VQAs) from a spectral analysis perspective. This enables the use of signal processing tools for noisy and noiseless analysis. For different noise models, we quantify the additional, higher frequency modes in the output signal caused by device errors. We show that filtering these noise-induced modes effectively mitigates errors. We then employ spectral analysis and compressed sensing to identify settings where a noiseless VQA cost function can be recovered classically, and present theoretical and numerical evidence supporting the viability of similar sparse recovery techniques. As demonstration, we efficiently recover simulated Quantum Approximate Optimization Algorithm (QAOA) instances of large system size from a few samples, indicating that sparse recovery can enable a more efficient use of quantum resources in the optimisation of variational algorithms. Overall, we make the case that spectral analysis constitutes an exciting new tool for studying VQAs, and as such may be of interest to the broader community striving to bring near-term quantum computers up to their full potential.