Limitations of variational quantum algorithms: a quantum optimal transport approach

TL;DR

This paper establishes tight theoretical bounds on the performance of variational quantum algorithms in noisy and noiseless regimes, demonstrating fundamental limitations for NISQ devices in solving combinatorial optimization problems.

Contribution

It introduces novel quantum entropic and concentration inequalities based on optimal mass transport theory to analyze the limitations of variational quantum algorithms.

Findings

Noisy quantum circuits at depth L=O(p^{-1}) are unlikely to outperform classical algorithms.

Classical algorithms outperform noisy quantum circuits at constant depth in optimization tasks.

New theoretical tools from quantum optimal transport are developed for analyzing quantum algorithm limitations.

Abstract

The impressive progress in quantum hardware in the last years has raised the interest of the quantum computing community in harvesting the computational power of such devices. However, in the absence of error correction, these devices can only reliably implement very shallow circuits or comparatively deeper circuits at the expense of a nontrivial density of errors. In this work, we obtain extremely tight limitation bounds for standard NISQ proposals in both the noisy and noiseless regimes, with or without error-mitigation tools. The bounds limit the performance of both circuit model algorithms, such as QAOA, and also continuous-time algorithms, such as quantum annealing. In the noisy regime with local depolarizing noise $p$, we prove that at depths $L=\mathcal{O}(p^{-1})$ it is exponentially unlikely that the outcome of a noisy quantum circuit outperforms efficient classical algorithms for combinatorial optimization problems like Max-Cut. Although previous results already showed that classical algorithms outperform noisy quantum circuits at constant depth, these results only held for the expectation value of the output. Our results are based on newly developed quantum entropic and concentration inequalities, which constitute a homogeneous toolkit of theoretical methods from the quantum theory of optimal mass transport whose potential usefulness goes beyond the study of variational quantum algorithms.