How smooth is quantum complexity?

Journal article

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.

Authors: Bulchandani, VB; Sondhi, S

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Springer Nature
• Journal: Journal of High Energy Phyics
• Volume: 2021
• Issue: 10
• Article number: 230
• Publication date: 2021-10-28
• Acceptance date: 2021-09-13
• DOI: 10.1007/JHEP10%282021%29230
• EISSN: 1029-8479
• ISSN: 1126-6708
• Language: English
• Keywords: differential and algebraic geometry; AdS-CFT correspondence; FFR