2009/03/30 by Richard A. Low · 48 citations
Computer Science · Mathematics · Physics and Astronomy · #Entropy (arrow of time) #Haar #Haar measure #Markov Chains and Monte Carlo Methods #Mathematical Approximation and Integration #Measure (data warehouse) #Quantum Computing Algorithms and Architecture #Standard deviation #Unitary state #Von Neumann architecture #Von Neumann entropy #quant-ph
paper · pdf · doi:10.1098/rspa.2009.0232
published in Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences 465(2111), 3289-3308 (Royal Society) · 20 pages
arxiv created 2009/03/30 · openalex publication_date 2009/08/05 · arxiv updated 2015/05/13 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
We present a technique for de-randomizing large deviation bounds of functions on the unitary group. We replace the Haar measure with a pseudo-random distribution, a k -design. k -Designs have the first k moments equal to those of the Haar measure. The advantage of this is that (approximate) k -designs can be implemented efficiently, whereas Haar random unitaries cannot. We find large deviation bounds for unitaries chosen from a k -design and then illustrate this general technique with three applications. We first show that the von Neumann entropy of a pseudo-random state is almost maximal. Then we show that, if the dynamics of the universe produces a k -design, then suitably sized subsystems will be in the canonical state, as predicted by statistical mechanics. Finally we show that pseudo-random states are useless for measurement-based quantum computation.