Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Renormalized Circuit Complexity

Arpan Bhattacharyya, Pratik Nandy, Aninda Sinha

TL;DR

The paper refines Nielsen’s circuit complexity for Hamiltonian simulation by introducing Suzuki-Trotter-based optimizations that yield a gate-count scaling linear in the geodesic length and system volume after optimizing the ST order. It demonstrates a holographic-like interpretation, where the optimal ST order $k^{\text{opt}}$ plays the role of an AdS radial coordinate and explores a renormalized circuit complexity via flows of penalty factors. The framework is illustrated with concrete examples across SYK-like, free and interacting scalar field theories, and time-evolution operators, showing how penalty choices control gate density and scaling behavior. These results connect quantum circuit complexity to holography and path-integral optimization, offering a scalable and geometry-inspired perspective on quantum simulation resources.

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.

Renormalized Circuit Complexity

TL;DR

The paper refines Nielsen’s circuit complexity for Hamiltonian simulation by introducing Suzuki-Trotter-based optimizations that yield a gate-count scaling linear in the geodesic length and system volume after optimizing the ST order. It demonstrates a holographic-like interpretation, where the optimal ST order plays the role of an AdS radial coordinate and explores a renormalized circuit complexity via flows of penalty factors. The framework is illustrated with concrete examples across SYK-like, free and interacting scalar field theories, and time-evolution operators, showing how penalty choices control gate density and scaling behavior. These results connect quantum circuit complexity to holography and path-integral optimization, offering a scalable and geometry-inspired perspective on quantum simulation resources.

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.

Paper Structure

This paper contains 6 sections, 48 equations, 2 figures.

Table of Contents

  1. Derivation of $N_{\mathrm{gates}}$ and connection with geometry
  2. Examples
  3. 2-Majorana ($N=2$) SYK like model:
  4. Complexity for ground state of free scalar field theory:
  5. Interacting scalar field theory:
  6. Complexity for time evolution operator: a perturbative computation

Figures (2)

  • Figure 1: The ST "holographic" network. The circuit above is a "compactified" version of the circuit below and is a pictorial representation of the ST recursion relation.
  • Figure 2: Schematic flow of the coupling $g=1/\ln p$ with scale $k^{\mathrm{opt}}=\ln(\Lambda/\Lambda_0)$. Here $g\,(\Lambda=\Lambda_0) = 1$.