On the role of coherence for quantum computational advantage

TL;DR

This paper introduces path coherence as a new measure of quantum coherence related to interference paths, providing insights into quantum advantage and enabling classical simulation of certain quantum computations.

Contribution

It defines path coherence, links it to the complexity of classical estimation of quantum amplitudes, and offers a classical algorithm leveraging this measure.

Findings

Path coherence quantifies interference in quantum computations.

Classical algorithms can estimate quantum amplitudes with complexity depending on path coherence.

Results enable simulation of large quantum computations classically.

Abstract

Quantifying the resources available to a quantum computer appears to be necessary to separate quantum from classical computation. Among them, entanglement, nonstabilizerness and coherence are arguably of great significance. We introduce path coherence as a measure of the coherent paths interferences arising in a quantum computation. Leveraging the sum-over-paths formalism, we obtain a classical algorithm for estimating quantum transition amplitudes, the complexity of which scales with path coherence. As path coherence relates to the hardness of classical estimation of quantum transition amplitudes, it provides a new perspective on the role of coherence in quantum computational advantage. Beyond their fundamental significance, our results have practical applications for simulating large classes of quantum computations with classical computers.