aiqbench analysis
This paper advances fundamental quantum complexity theory by deriving Nielsen's complexity for unitaries within the oscillator group representation using group-theoretical methods. By explicitly solving geodesic equations on this specific group structure, the authors provide a framework to compute the complexity of any unitary operator in these representations, offering theoretical insight rather than immediate hardware application.
Group-theoretical analysis of quantum complexity: the oscillator group case
arXiv:2512.17552v2 Announce Type: replace 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.
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.
36 pages,3 figures. V2 considerably extened version
arXiv:2512.17552 [quant-ph]
(or arXiv:2512.17552v2 [quant-ph] for this version)
https://doi.org/10.48550/arXiv.2512.17552 Focus to learn more arXiv-issued DOI via DataCite
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Full text under the publisher's open licence · 252 words
