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

TL;DR

This paper derives the complexity of unitaries in oscillator group representations using group structure and geodesic equations, providing explicit solutions and a method to compute quantum complexity for these operators.

Contribution

It presents a complete derivation of Nielsen's complexity for oscillator group unitaries, linking group theory with quantum complexity analysis.

Findings

Explicit solutions to geodesic equations on oscillator group.

Method to compute complexity of any unitary in the representation.

Transcendental equations determine geodesic lengths.

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.