



01949nas a2200193   4500008004100000245005300041210005200094260001500146520133800161653004301499653002701542653003101569653002901600653003101629100001801660700002001678700002001698856003701718       2022                        eng d00aSample-optimal classical shadows for pure states0 aSampleoptimal classical shadows for pure states  c11/21/20223 a<p>We consider the classical shadows task for pure states in the setting of both joint and independent measurements. The task is to measure few copies of an unknown pure state ρ in order to learn a classical description which suffices to later estimate expectation values of observables. Specifically, the goal is to approximate Tr(Oρ) for any Hermitian observable O to within additive error ϵ provided Tr(O2)&le;B and ∥O∥=1. Our main result applies to the joint measurement setting, where we show Θ~(B&minus;&minus;&radic;ϵ&minus;1+ϵ&minus;2) samples of ρ are necessary and sufficient to succeed with high probability. The upper bound is a quadratic improvement on the previous best sample complexity known for this problem. For the lower bound, we see that the bottleneck is not how fast we can learn the state but rather how much any classical description of ρ can be compressed for observable estimation. In the independent measurement setting, we show that O(Bd&minus;&minus;&minus;&radic;ϵ&minus;1+ϵ&minus;2) samples suffice. Notably, this implies that the random Clifford measurements algorithm of Huang, Kueng, and Preskill, which is sample-optimal for mixed states, is not optimal for pure states. Interestingly, our result also uses the same random Clifford measurements but employs a different estimator.</p>10aFOS: Computer and information sciences10aFOS: Physical sciences10aInformation Theory (cs.IT)10aMachine Learning (cs.LG)10aQuantum Physics (quant-ph)1 aGrier, Daniel1 aPashayan, Hakop1 aSchaeffer, Luke  uhttps://arxiv.org/abs/2211.11810