Operator relaxation and the optimal depth of classical shadows

TL;DR

This paper analyzes how the depth of local unitary circuits in classical shadows affects the sample complexity for learning quantum state properties, revealing a balance between operator spreading and relaxation.

Contribution

It introduces a theoretical framework linking operator dynamics under random circuits to the sample complexity of classical shadows, including bounds and optimal depths.

Findings

Exponential sample complexity gain at depth ~ log(k)

Universal subleading correction to optimal depth in 1D

Agreement with matrix product state simulations

Abstract

Classical shadows are a powerful method for learning many properties of quantum states in a sample-efficient manner, by making use of randomized measurements. Here we study the sample complexity of learning the expectation value of Pauli operators via ``shallow shadows'', a recently-proposed version of classical shadows in which the randomization step is effected by a local unitary circuit of variable depth $t$. We show that the shadow norm (the quantity controlling the sample complexity) is expressed in terms of properties of the Heisenberg time evolution of operators under the randomizing (``twirling'') circuit -- namely the evolution of the weight distribution characterizing the number of sites on which an operator acts nontrivially. For spatially-contiguous Pauli operators of weight $k$, this entails a 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). From this simple picture we derive (i) an upper bound on the shadow norm which, for depth $t\sim \log(k)$, guarantees an exponential gain in sample complexity over the $t=0$ protocol in any spatial dimension, and (ii) quantitative results in one dimension within a mean-field approximation, including a universal subleading correction to the optimal depth, found to be in excellent agreement with infinite matrix product state numerical simulations. Our 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.