How smooth is quantum complexity?

TL;DR

This paper explores the mathematical properties of quantum complexity, revealing its potential non-smooth and fractal nature, and discusses implications for understanding its physical significance.

Contribution

It provides a unified framework for various quantum complexity notions and introduces rigorous methods to quantify their non-smooth behavior using advanced mathematical tools.

Findings

Quantum complexity functions can exhibit non-smooth and fractal behavior.

Mathematical tools from Diophantine approximation and sub-Riemannian geometry are used to quantify complexity irregularities.

Implications for the physical interpretation of quantum complexity are discussed.

Abstract

The "quantum complexity" of a unitary operator measures the difficulty of its construction from a set of elementary quantum gates. While the notion of quantum complexity was first introduced as a quantum generalization of the classical computational complexity, it has since been argued to hold a fundamental significance in its own right, as a physical quantity analogous to the thermodynamic entropy. In this paper, we present a unified perspective on various notions of quantum complexity, viewed as functions on the space of unitary operators. One striking feature of these functions is that they can exhibit non-smooth and even fractal behaviour. We use ideas from Diophantine approximation theory and sub-Riemannian geometry to rigorously quantify this lack of smoothness. Implications for the physical meaning of quantum complexity are discussed.