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.
Highlights
Background to resultsFor the gate complexity eq (2.1), let A denote the group generated by elements of A and their inverses, which by assumption is dense in G
We present a unified perspective on various notions of quantum complexity, viewed as functions on the space of unitary operators
As → 0, C (U ) tends to a nowhere continuous function on the group G, analogous to the nowhere continuous Dirichlet function of real analysis. (Notice that for a finite group with A = G, C0 recovers the “word metric” studied in geometric group theory; we refer to ref. [14] for a discussion of analogies between geometric group theory and quantum complexity.)
Summary
The classical notion of computational complexity, for example based on Boolean circuits, operates in discrete time with a discrete set of elementary operations. Both the discrete gate complexity C (U ) and the continuous complexity distance C(U, V ) exhibit similar worst-case exponential scaling [4] in N to the circuit depth σ(U ) It is usually assumed that these definitions are more-or-less equivalent from the viewpoint of quantum computing [1, 13], and differ only in their ease of calculation. We first prove the existence of efficiently universal gate sets for which C (U ) exhibits worst-case scaling in , C (U ) = Ω(log 1/ ), densely in G This clarifies the manner in which C (U ) tends to a nowhere continuous function as → 0. These results give precise meaning to various qualitative discussions of the “fractal” geometry of quantum complexity in the literature
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