A Quantum Computational Perspective on Spread Complexity

TL;DR

This paper links spread complexity to quantum circuit complexity, showing that spread complexity arises as a limit of a framework involving time-evolution and superposition, with practical computational benefits.

Contribution

It introduces a new perspective connecting spread complexity with quantum circuit complexity through a minimal cost synthesis approach.

Findings

Spread complexity emerges as a limit of a circuit complexity framework.

The approach offers computational advantages over traditional methods.

Illustrated with an explicit SU(2) example and broader applications.

Abstract

We establish a direct connection between spread complexity and quantum circuit complexity by demonstrating that spread complexity emerges as a limiting case of a circuit complexity framework built from two fundamental operations: time-evolution and superposition. Our approach leverages a computational setup where unitary gates and beam-splitting operations generate target states, with the minimal cost of synthesis yielding a complexity measure that converges to spread complexity in the infinitesimal time-evolution limit. This perspective not only provides a physical interpretation of spread complexity but also offers computational advantages, particularly in scenarios where traditional methods like the Lanczos algorithm fail. We illustrate our framework with an explicit SU(2) example and discuss broader applications, including cases where return amplitudes are non-perturbative or divergent