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How smooth is quantum complexity?
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DOI：10.1007/JHEP10(2021)230.png)
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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.
Keyword:
AdS-CFT Correspondence
Differential and Algebraic Geometry
期刊
IF:
5.5
论文数:
4.0W
被引数:
13.7W
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