Nicholas Laracuente

A major line of questions in quantum information and computing asks how quickly locally random circuits converge to resemble global randomness. In particular, approximate k-designs are random unitary ensembles that resemble random circuits up to their first k moments. It was recently shown that on n qudits, random circuits with slightly structured architectures converge to k-designs in depth O(log n), even on one-dimensional connectivity. It has however remained open whether the same shallow depth applies more generally among random circuit architecture and connectivity, or if the structure is truly necessary. We recall the study of exponential relative entropy decay, another topic with a long history in quantum information theory. We show that a constant number of layers of a parallel random circuit on a family of architectures including one-dimensional `brickwork' has O(1 / logn) per-layer multiplicative entropy decay. We further show that on general connectivity graphs of bounded degree, randomly placed gates achieve O(1 / nlogn)-decay (consistent with log n depth convergence). Both of these results imply that random circuit ensembles with O(polylog(n)) average depth achieve k-designs in diamond norm. Hence our results address the question of whether extra structure is truly necessary for log-depth convergence. Furthermore, the relative entropy recombination techniques might be of independent interest.

Abstract Trace inequalities are powerful techniques in studying quantum entropy. The physics of quantum field theory and holography nonetheless motivate entropy inequalities in von Neumann algebras that lack a useful notion of a trace. Haagerup and Kosaki L_p spaces enable re-expressing trace inequalities in scenarios that start with a non-tracial von Neumann algebra, and we show how the generalized Araki-Lieb-Thirring and Golden-Thompson inequalities from (Sutter, Berta & Tomamichel 2017) port to general von Neumann algebras. Using an approximation method of Haagerup, we prove the tightened recovery map correction to data processing for relative entropy in (potentially non-tracial) von Neumann algebras. We also prove a p-fidelity version. Furthermore, we prove that non-decrease of relative entropy is equivalent to the existence of an L_1-isometry implementing the channel on both input states.