Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: How smooth is quantum complexity?   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.

17
The categories of this paper are: q-fin
29
The categories of this paper are: funct-an
19
The categories of this paper are: eess
32
The categories of this paper are: atom-ph
21
The categories of this paper are: adap-org
1
The categories of this paper are: cs
26
The categories of this paper are: bayes-an
37
The categories of this paper are: supr-con
18
The categories of this paper are: econ
0
The categories of this paper are: hep-ph
33
The categories of this paper are: plasm-ph
34
The categories of this paper are: chem-ph
30
The categories of this paper are: q-alg
11
The categories of this paper are: hep-lat
28
The categories of this paper are: alg-geom
3
The categories of this paper are: physics