Abstract

Implementing quantum gates as non-Abelian holonomies, a class of topologically protected unitary operators, is a particularly promising paradigm for the design of intrinsically stable quantum computers [1]. In contrast to dynamic phases, the geometric phase accumulated by a quantum system propagating through a Hilbert space ℋ depends exclusively on its path. In general, geometric phases can exhibit arbitrary dimensionality. Wilczek and Zee introduced the idea of multi-dimensional, non-Abelian geometric phases – so called holonomies [2]. Anandan later dropped the requirement of adiabaticity to create holonomies, that are truly time-independent [3]. Non-adiabatic holonomies rely on a subspace ℋ_geo of the Hilbert-space that is spanned by states {|Φ_{⟩} that fulfill $〈{\Phi }_k\left|\widehat{H}\right|{\Phi }_j〉=0$, where $\widehat{H}$ is the system’s Hamiltonian. Restricting the propagation to ℋgeo ensures parallel transport and, thus, a purely geometric phase (see Fig. 1a) [4,5]. Quantum optics constitutes a particularly versatile platform for quantum information processing, and in particular for the construction of non-adiabatic holonomic quantum computers: In addition to integration and miniaturization provided by the platform, the bosonic nature of photons also conveniently allows for multiple excitations of the same mode, readily expanding ℋgeo and enabling the synthesis of holonomies from higher symmetry groups U() as larger and more capable computational units [6, 7].}

© 2023 IEEE