From black hole interior to quantum complexity through operator rank

TL;DR

This paper provides microscopic evidence linking black hole interior size to quantum circuit complexity by relating the area of a specific surface to circuit depth using operator Schmidt rank.

Contribution

It establishes a rigorous relation between the interior surface area and quantum circuit complexity at early times, extending it to later times via circuit cuts.

Findings

Relation between interior surface area and circuit complexity confirmed at early times.

Operator Schmidt rank used as a key tool for microscopic analysis.

Extrapolation to later times supports the conjectured connection.

Abstract

It has been conjectured that the size of the black hole interior captures the quantum gate complexity of the underlying boundary evolution. In this short note we aim to provide a further microscopic evidence for this by directly relating the area of a certain codimension-two surface traversing the interior to the depth of the quantum circuit. Our arguments are based on establishing such relation rigorously at early times using the notion of operator Schmidt rank and then extrapolating it to later times by mapping bulk surfaces to cuts in the circuit representation.