Oracle Separation between Noisy Quantum Polynomial Time and the Polynomial Hierarchy

TL;DR

This paper demonstrates that noisy quantum circuits with constant depth and error rates can separate classical complexity classes like NP and the polynomial hierarchy, even without error correction, under various noise models.

Contribution

It establishes oracle separations between noisy quantum circuits and classical complexity classes, extending previous results to constant depth circuits without error correction.

Findings

Constant error rate achieves NP separation.

Omega(log n/n) error rate extends to PH separation.

No error correction needed for the separations.

Abstract

This work investigates the oracle separation between the physically motivated complexity class of noisy quantum circuits, inspired by definitions such as those presented by Chen, Cotler, Huang, and Li (2022). We establish that with a constant error rate, separation can be achieved in terms of NP. When the error rate is $\Omega(\log n/n)$, we can extend this result to the separation of PH. Notably, our oracles, in all separations, do not necessitate error correction schemes or fault tolerance, as all quantum circuits are of constant depth. This indicates that even quantum computers with minor errors, without error correction, may surpass classical complexity classes under various scenarios and assumptions. We also explore various common noise settings and present new classical hardness results, generalizing those found in studies by Raz and Tal (2022) and Bassirian, Bouland, Fefferman, Gunn, and Tal (2021), which are of independent interest.