Operator relaxation and the optimal depth of classical shadows
报告时间：
2023年4月28日（周五）13:00
报告嘉宾：
Vedika Khemani (Stanford University)
主办单位：
北京大学理论物理研究所
北京大学高能物理研究中心
直播通道
蔻享学术直播间
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摘要
Classical shadows are a powerful method for learning many properties of quantum states in a sample-efficient manner, by making use of randomized measurements. I will discuss the sample complexity of learning the expectation value of local Pauli operators via “shallow shadows”, a recently-proposed version of classical shadows in which the randomization step is affected by a random unitary circuit of variable depth t. I will show that the sample complexity can be derived from properties of the dynamics induced by the circuit, which can be calculated by mapping to simple statistical mechanics models. Specifically, the sample complexity of learning the expectation of local operators of weight k (i.e., acting nontrivially on k contiguous sites) is determined by the time-evolution of the operator weight starting from a fully-packed product initial operator of size k. This entails the competition between two processes: operator spreading (whereby the support of an operator grows over time, increasing its weight) and operator relaxation (whereby the bulk of the operator develops an equilibrium density of identity operators, decreasing its weight). Our results give a bound which, for depth t ~ log(k) guarantees an exponential gain in sample complexity. This work connects fundamental ideas in quantum many-body dynamics to applications in quantum information science, and paves the way to highly-optimized protocols for learning different properties of quantum states.