Group-theoretical analysis of quantum complexity: the oscillator group case

TL;DR

The paper develops a group-theoretical approach to quantum complexity by analyzing Nielsen's geometric framework on the oscillator group. By classifying unitary irreducible representations via Casimir operators and solving the geodesic equations for a natural right-invariant metric, it yields explicit expressions for geodesic paths and their lengths, which serve as lower bounds for gate complexity. Through concrete examples, including harmonic oscillators and linear perturbations, the work shows how complexity depends on the chosen representation and the specific group element, and demonstrates the potential for multiple geodesics with different minimal lengths. The study illuminates how symmetry structures can render quantum complexity tractable in both finite- and certain infinite-dimensional settings, and provides a blueprint for extending the analysis to other groups and representations.

Abstract

Motivated by the recent rapid development of complexity theory applied to quantum mechanical processes we present the complete derivation of Nielsen's complexity of unitaries belonging to the representations of oscillator group. Our approach is based on the observation that the whole problem refers to the structure of the underlying group. The questions concerning the complexity of particular unitaries are solved by lifting the abstract structure to the operator level by considering the relevant unitary representation. For the class of right-invariant metrics obeying natural invariance condition we solve the geodesic equations on oscillator group. The solution is given explicitly in terms of elementary functions. Imposing the boundary conditions yield a transcendental equation and the length of the geodesic is given in terms of the solutions to the latter. Since the unitary irreducible representations of oscillator group are classified this allows us to compute, at least in principle, the complexity of any unitary operator belonging to the representation.

Figures (3)

• Figure 1: The shape of $f(\tilde{\nu};\Delta)$ for different value of $\Delta$.
• Figure 2: The shape of $f(\tilde{\nu},\Delta)$ for $\Delta<\frac{1}{2}$.
• Figure 3: The shape of $f(\tilde{\nu},\Delta)$ for $\Delta>\frac{1}{2}$.