How smooth is quantum complexity?

• URL: http://arxiv.org/abs/2106.08324v2
• Date: Wed, 11 Aug 2021 15:44:04 GMT
• Title: How smooth is quantum complexity?
• Authors: Vir B. Bulchandani and S. L. Sondhi
• Abstract summary: The "quantum complexity" of a unitary operator measures the difficulty of its construction from a set of elementary quantum gates. In this paper, we present a unified perspective on various notions of quantum complexity, viewed as functions on the space of unitary operators.
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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