Can shallow quantum circuits scramble local noise into global white noise?

TL;DR

This paper investigates how well shallow quantum circuits scramble local noise into global white noise, analyzing metrics like eigenvalue uniformity and commutator norm to assess implications for error mitigation.

Contribution

It provides analytical bounds and numerical simulations to evaluate the effectiveness of noise scrambling in practical shallow quantum circuits, highlighting limitations and potential improvements.

Findings

White noise approximation can be inaccurate for practical circuits.

Commutator norm remains small, supporting purification error mitigation.

Techniques like increasing Lie algebra dimensionality can reduce noise metrics.

Abstract

Shallow quantum circuits are believed to be the most promising candidates for achieving early practical quantum advantage - this has motivated the development of a broad range of error mitigation techniques whose performance generally improves when the quantum state is well approximated by a global depolarising (white) noise model. While it has been crucial for demonstrating quantum supremacy that random circuits scramble local noise into global white noise - a property that has been proved rigorously - we investigate to what degree practical shallow quantum circuits scramble local noise into global white noise. We define two key metrics as (a) density matrix eigenvalue uniformity and (b) commutator norm. While the former determines the distance from white noise, the latter determines the performance of purification based error mitigation. We derive analytical approximate bounds on their scaling and find in most cases they nicely match numerical results. On the other hand, we simulate a broad class of practical quantum circuits and find that white noise is in certain cases a bad approximation posing significant limitations on the performance of some of the simpler error mitigation schemes. On a positive note, we find in all cases that the commutator norm is sufficiently small guaranteeing a very good performance of purification-based error mitigation. Lastly, we identify techniques that may decrease both metrics, such as increasing the dimensionality of the dynamical Lie algebra by gate insertions or randomised compiling.