# Renormalized Circuit Complexity

## TL;DR

This paper introduces a modified approach to quantum circuit complexity for Hamiltonian simulation, optimizing gate count and establishing a holographic interpretation related to error tolerance and geometric structures.

## Contribution

It proposes a renormalized circuit complexity framework that improves gate efficiency and connects quantum circuit properties with holographic and geometric concepts.

## Key findings

- Gate count is linear in geodesic distance and volume.
- Optimized Suzuki-Trotter order depends on error tolerance.
- Holographic interpretation links circuit complexity to AdS geometry.

## Abstract

We propose a modification to Nielsen's circuit complexity for Hamiltonian simulation using the Suzuki-Trotter (ST) method, which provides a network like structure for the quantum circuit. This leads to an optimized gate counting linear in the geodesic distance and spatial volume, unlike in the original proposal. The optimized ST iteration order is correlated with the error tolerance and plays the role of an anti-de Sitter (AdS) radial coordinate. The density of gates is shown to be monotonic with the tolerance and a holographic interpretation using path-integral optimization is given.