Unitary circuits of finite depth and infinite width from quantum channels

Understand

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.