Computing exact moments of local random quantum circuits via tensor networks

Abstract

A basic primitive in quantum information is the computation of the moments EU[Tr[UρU†O]t]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {E}_U[\textrm{Tr}[U\rho U^{\dagger }O]^t]$$\end{document}. These describe the distribution of expectation values obtained by sending a state ρ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho $$\end{document} through a random unitary U, sampled from some distribution and measuring the observable O. While the exact calculation of these moments is generally hard, if U is composed of local random gates, one can estimate EU[Tr[UρU†O]t]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {E}_U[\textrm{Tr}[U\rho U^{\dagger }O]^t]$$\end{document} by performing Monte Carlo simulations of a Markov chain-like process. However, this approach can require a prohibitively large number of samples, or suffer from the sign problem. In this work, we instead propose to estimate the moments via tensor networks, where the local gates moment operators are mapped to small dimensional tensors acting on their local commutant bases. By leveraging representation theoretical tools, we study the local tensor dimension and we provide bounds for the bond dimension of the matrix product states arising from deep circuits. We compare our techniques against Monte Carlo simulations, showing that we can significantly outperform them. Then, we showcase how tensor networks can exactly compute the second moment when U is a quantum neural network acting on thousands of qubits and having thousands of gates. To finish, we numerically study the anti-concentration phenomena of circuits with orthogonal random gates, a task that cannot be studied via Monte Carlo due to sign problems.