Circuit Complexity From Supersymmetric Quantum Field Theory With Morse Function

TL;DR

This paper explores the connection between circuit complexity and Morse theory in supersymmetric quantum field theories, providing a geometric framework to analyze quantum chaos and complexity measures.

Contribution

It introduces a novel approach linking Morse theory with circuit complexity in supersymmetric quantum field theories, extending the geometric understanding of quantum chaos.

Findings

Circuit complexity expressed via the Hessian of Morse functions.

Established the relation between quantum chaos and circuit complexity.

Applied Morse theory to supersymmetric quantum field models.

Abstract

Computation of circuit complexity has gained much attention in the Theoretical Physics community in recent times to gain insights into the chaotic features and random fluctuations of fields in the quantum regime. Recent studies of circuit complexity take inspiration from Nielsen's geometric approach, which is based on the idea of optimal quantum control in which a cost function is introduced for the various possible path to determine the optimum circuit. In this paper, we study the relationship between the circuit complexity and Morse theory within the framework of algebraic topology, which will then help us study circuit complexity in supersymmetric quantum field theory describing both simple and inverted harmonic oscillators up to higher orders of quantum corrections. We will restrict ourselves to $\mathcal{N} = 1$ supersymmetry with one fermionic generator $Q_{\alpha}$. The expression of circuit complexity in quantum regime would then be given by the Hessian of the Morse function in supersymmetric quantum field theory. We also provide technical proof of the well known universal connecting relation between quantum chaos and circuit complexity of the supersymmetric quantum field theories, using the general description of Morse theory.