Non-Adiabatic Holonomic Quantum Gates

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 $\mathcal{H}$ 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 $\mathcal{H}_{\text{geo}}$ of the Hilbert-space that is spanned by states $\{\vert \Phi_{k}\rangle\}_{k}$ that fulfill $(\Phi_{k}\vert \hat{H}\vert \Phi_{j}\rangle=0$ , where $\hat{H}$ is the system's Hamiltonian. Restricting the propagation to $\mathcal{H}_{\text{geo}}$ 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 $\mathcal{H}_{\text{geo}}$ and enabling the synthesis of holonomies from higher symmetry groups $\mathrm{U}(N)$ as larger and more capable computational units [6], [7].