# Unitary circuits of finite depth and infinite width from quantum channels

## TL;DR

This paper presents a quantum channel-based method to efficiently compute reduced density matrices and entanglement properties in finite-depth, infinite-width quantum circuits, enabling analysis of complex quantum dynamics.

## Contribution

It introduces a novel quantum channel approach to analyze reduced density matrices in infinite quantum circuits, combining numerical efficiency with analytical insights.

## Key findings

- Efficient numerical evaluation of entanglement spectra and Rènyi entropies.
- Analytical recovery of the kicked Ising model behavior at the self-dual point.
- Benchmarking results on random unitary circuits with available analytic solutions.

## Abstract

We introduce an approach to compute reduced density matrices for local quantum unitary circuits of finite depth and infinite width. Suppose the time-evolved state under the circuit is a matrix-product state with bond dimension $D$; then the reduced density matrix of a half-infinite system has the same spectrum as an appropriate $D\times D$ matrix acting on an ancilla space. We show that reduced density matrices at different spatial cuts are related by quantum channels acting on the ancilla space. This quantum channel approach allows for efficient numerical evaluation of the entanglement spectrum and R\'enyi entropies and their spatial fluctuations at finite times in an infinite system. We benchmark our numerical method on random unitary circuits, where many analytic results are available, and also show how our approach analytically recovers the behaviour of the kicked Ising model at the self-dual point. We study various properties of the spectra of the reduced density matrices and their spatial fluctuations in both the random and translation-invariant cases.