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

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) Dr Balint Koczor of this department, and Dr Jonathan Foldager of DTU investigated to what degree practical shallow quantum circuits scramble local noise into global white noise.  

In their paper 'Can shallow quantum circuits scramble local noise into global white noise?' published in Journal of Physics A: Mathematical and Theoretical, they define two key metrics as (a) density matrix eigenvalue uniformity, and (b) commutator norm that quantifies stability of the dominant eigenvector.  While the former determines the distance from white noise, the latter determines the performance of purification based error mitigation.  

The authors derived analytical approximate bounds on their scaling and found that in most cases they matched the numerical results.  Conversely, they also simulated a broad class of practical quantum circuits and found that white noise was (in certain cases) a bad approximation posing significant limitations on the performance of some of the simpler error mitigation schemes.

More positively, they found that in all cases the commutator norm was sufficiently small and guaranteed a very good performance of purification-based error mitigation.  The paper also describes how they identified techniques that may decrease both metrics, such as increasing the dimensionality of the dynamical Lie algebra by gate insertions or randomised compiling.