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arXiv:1902.02559 · Physical Review Research

Optimal design strategy for non-Abelian geometric phases using Abelian gauge fields based on quantum metric

Mark Kremer, Lucas Teuber, Alexander Szameit, Stefan Scheel

Published Nov 20, 2019• 31 citations
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Abstract

A photonic implementation of a SU(2) holonomic quantum gate, i.e. a single-qubit gate, is realized via a tripod-like structure of four coupled waveguides and characterized by its invariant Wilson loop. The gate operates on the degenerate dark subspace of the structure. The properties of the dark subspace are determined by the quantum metric, which provides an optimization scheme for the structure's design, thereby ensuring the required adiabaticity and providing a useful error measure.

DOI
10.1103/physrevresearch.1.033117

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A photonic implementation of a SU(2) holonomic quantum gate, i.e. a single-qubit gate, is realized via a tripod-like structure of four coupled waveguides and characterized by its invariant Wilson loop. The gate operates on the degenerate dark subspace of the structure. The properties of the dark subspace are determined by the quantum metric, which provides an optimization scheme for the structure's design, thereby ensuring the required adiabaticity and providing a useful error measure.
We construct holonomic quantum gates for qubits that are encoded in the near-degenerate vibrational EEE-doublet of a deformable three-body system. Using Kendall's shape theory, we derive the Wilczek--Zee connection governing adiabatic transport within the EEE-manifold. We show that its restricted holonomy group is SU(2)\mathrm{SU}(2)SU(2), implying universal single-qubit control by closed loops in shape space. We provide explicit loops implementing a π/2π/2π/2 phase gate and a Hadamard-type gate. For two-qubit operations, we outline how linked holonomic cycles in arrays generate a controlled Chern--Simons phase, enabling an entangling controlled-XXX (CNOT) gate. We present a Ramsey/echo interferometric protocol that measures the Wilson loop trace of the Wilczek--Zee connection for a control cycle, providing a gauge-invariant signature of the non-Abelian holonomy. As a physically realizable demonstrator, we propose bond-length modulations of a Cs(6s6s6s)--Cs(6s6s6s)--Cs(nd3/2nd_{3/2}nd3/2​) Rydberg trimer in optical tweezers and specify operating conditions that suppress leakage out of the EEE-manifold.
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Holonomic quantum computing functions by transporting an adiabatically degenerate manifold of computational states around a closed loop in a control-parameter space; this cyclic evolution results in a non-Abelian geometric phase which may couple states within the manifold. Realizing the required degeneracy is challenging and typically requires auxiliary levels or intermediate-level couplings. One potential way to circumvent this is through Floquet engineering, where the periodic driving of a nondegenerate Hamiltonian leads to degenerate Floquet bands, and subsequently non-Abelian gauge structures may emerge. Here we present an experiment in ultracold Rb87 atoms where atomic spin states are dressed by modulated RF fields to induce periodic driving of a family of Hamiltonians linked through a fully tuneable parameter space. The adiabatic motion through this parameter space leads to the holonomic evolution of the degenerate spin states in SU(2) , characterized by a non-Abelian connection. We study the holonomic transformations of spin eigenstates in the presence of a background magnetic field, characterizing the fidelity of these single-qubit gate operations. Results indicate that while the Floquet engineering technique removes the need for explicit degeneracies, it inherits many of the same limitations present in degenerate systems. Published by the American Physical Society 2024
Nonadiabatic holonomic quantum computation provides the means to perform fast and robust quantum gates by utilizing the resilience of non-Abelian geometric phases to fluctuations of the path in state space. While the original scheme [E. Sj\"oqvist et al., New J. Phys. 14, 103035 (2012)] needs two loops in the Grassmann manifold (i.e., the space of computational subspaces of the full state space) to generate an arbitrary holonomic one-qubit gate, we propose single-loop one-qubit gates that constitute an efficient universal set of holonomic gates when combined with an entangling holonomic two-qubit gate. Our one-qubit gate is realized by dividing the loop into path segments, each of which is generated by a $\mathrm{\ensuremath{\Lambda}}$-type Hamiltonian. We demonstrate that two path segments are sufficient to realize arbitrary single-loop holonomic one-qubit gates. We describe how our scheme can be implemented experimentally in a generic atomic system exhibiting a three-level $\mathrm{\ensuremath{\Lambda}}$-coupling structure by utilizing carefully chosen laser pulses.
Holonomic quantum computation (HQC) realizes quantum gates through non-Abelian geometric phases, providing an experimentally accessible approach to quantum control. While the nonadiabatic HQC framework has been extensively developed for three-level ΛΛΛ systems encoding qubits, its systematic extension to higher-dimensional qudits remains largely unexplored. In this work, we generalize nonadiabatic HQC to a ddd-pod configuration, where a single excited state is coupled to ddd ground states, the latter forming the computational subspace. This scheme enables universal holonomic single- and two-qudit gates using only optical or microwave pulses on trapped atoms or ions, offering an efficient route to implement a discrete universal gate set with minimal pulse coordination. As an explicit example, we analyze in detail the qutrit (d=3d=3d=3) case, demonstrating compact realizations of single- and two-qutrit holonomic gates, each gate requiring at most two loops in the Grassmannian generated by at most three pulses.