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.
Original language | English (US) |
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Article number | 230 |
Journal | Journal of High Energy Physics |
Volume | 2021 |
Issue number | 10 |
DOIs | |
State | Published - Oct 2021 |
Externally published | Yes |
All Science Journal Classification (ASJC) codes
- Nuclear and High Energy Physics
Keywords
- AdS-CFT Correspondence
- Differential and Algebraic Geometry