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.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.

arXiv:1608.07418Physical Review ANov 9, 2016100 citations

$d$ -pod realization of nonadiabatic holonomic quantum computation

Oskar Axelsson, Claes Fälth, Elias Henriksson Lindberg +1 more

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 $Λ$ systems encoding qubits, its systematic extension to higher-dimensional qudits remains largely unexplored. In this work, we generalize nonadiabatic HQC to a $d$ -pod configuration, where a single excited state is coupled to $d$ 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 ( $d=3$ ) 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.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 $Λ$ systems encoding qubits, its systematic extension to higher-dimensional qudits remains largely unexplored. In this work, we generalize nonadiabatic HQC to a $d$-pod configuration, where a single excited state is coupled to $d$ 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 ($d=3$) 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.

quant-pharXiv:2609.16216Sep 16, 20260 citations

Nonadiabatic holonomic quantum gates in a decoherence-insensitive space

Fangzhou Jin, Jianpei Geng, Hui Zhou

Nonadiabatic holonomic quantum computation (NHQC) has shown promise in mitigating control errors, but requires additional methods to further eliminate the effects of decoherence errors. Existing NHQC schemes capable of combating both types of errors often necessitate auxiliary qubits, thereby increasing qubit resource consumption. This paper proposes nonadiabatic holonomic quantum gates based on a three-level dressed state space, which have inherent insensitivity to decoherence and eliminate the need for auxiliary qubits. Through numerical simulations, the presented gates within the dressed-state framework display enhanced resistance to both control field errors and decoherence errors. Moreover, we discuss the implementation of our approach on the nitrogen-vacancy center in diamond, demonstrating its feasibility and potential for geometric quantum computation.Nonadiabatic holonomic quantum computation (NHQC) has shown promise in mitigating control errors, but requires additional methods to further eliminate the effects of decoherence errors. Existing NHQC schemes capable of combating both types of errors often necessitate auxiliary qubits, thereby increasing qubit resource consumption. This paper proposes nonadiabatic holonomic quantum gates based on a three-level dressed state space, which have inherent insensitivity to decoherence and eliminate the need for auxiliary qubits. Through numerical simulations, the presented gates within the dressed-state framework display enhanced resistance to both control field errors and decoherence errors. Moreover, we discuss the implementation of our approach on the nitrogen-vacancy center in diamond, demonstrating its feasibility and potential for geometric quantum computation.

Physical Review AFeb 18, 20254 citations

Nonadiabatic holonomic single-qubit gates in off-resonant Λ systems

Erik Sjöqvist

arXiv:1511.00911Physics Letters AOct 17, 201585 citations

Λ

systems encoding qubits, its systematic extension to higher-dimensional qudits remains largely unexplored. In this work, we generalize nonadiabatic HQC to a

d

-pod configuration, where a single excited state is coupled to

d

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=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.