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Multi-Segment Generated Subspace Evolution for Holonomic Quantum Gates

Jonathan Larsson

Published Jan 1, 2026• 0 citations
OpenAlex

Abstract

We investigate whether a single-loop construction in non-adiabatic holonomic quantum computation can be generalized to multi-segment evolutions while preserving separation between geometric and dynamical contributions. Starting from a three-level Λ-system, we first study phase-only basis transformations between segments and find an exact non-trivial three-segment solution under correlated pulse-area and phase constraints. We then use a numerical loss-function approach to search for zero-leakage solutions with more than three segments, and find evidence of irreducible multi-segment evolutions in the tested low-segment cases. We further investigate more general unitary basis transformations and analyze the separation condition using projected dynamical generators in the computational subspace. The analysis suggests that admissible transformations must preserve a common dark direction, effectively reducing the problem to a two-dimensional bright-excited subspace. While an enlargement of the set of reachable single-qubit gates is not possible, the results may provide additional implementation freedom for realizing holonomic gates.

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We investigate whether a single-loop construction in non-adiabatic holonomic quantum computation can be generalized to multi-segment evolutions while preserving separation between geometric and dynamical contributions. Starting from a three-level Λ-system, we first study phase-only basis transformations between segments and find an exact non-trivial three-segment solution under correlated pulse-area and phase constraints. We then use a numerical loss-function approach to search for zero-leakage solutions with more than three segments, and find evidence of irreducible multi-segment evolutions in the tested low-segment cases. We further investigate more general unitary basis transformations and analyze the separation condition using projected dynamical generators in the computational subspace. The analysis suggests that admissible transformations must preserve a common dark direction, effectively reducing the problem to a two-dimensional bright-excited subspace. While an enlargement of the set of reachable single-qubit gates is not possible, the results may provide additional implementation freedom for realizing holonomic gates.
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 offers a promising route to robust quantum gates, but decoherence remains a central obstacle in realistic implementations. Here we develop a nonadiabatic holonomic scheme for a driven three-level system in the no-jump regime described by an effective non-Hermitian Hamiltonian. Within a biorthogonal framework, tailored complex pulses enforce exact closure of the computational-subspace evolution at the final time despite the underlying nonunitary dynamics, enabling arbitrary single-qubit holonomic gates without requiring cyclic evolution in its orthogonal complement. In contrast to existing non-Hermitian treatments, which either neglect the overall exponential prefactor or, in adiabatic settings, include dissipation only on the auxiliary excited level, our scheme incorporates decay and dephasing of all bare eigenstates directly into the pulse design, so that dissipation does not reduce the no-jump gate fidelity.
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.
In the Λ-system, non-adiabatic holonomic quantum phases are used to construct holonomic quantum gates. An interesting approach would be to implement open path holonomies in the Λ-system. By dividing the loop into two curve segments with a unitary transformation between them, universality can be reached. In doing so the exibility of the scheme has been increased by the fact that one single full pulse is now enough for universality, and we have achieved a clearer proof of the geometric property of the Λ system.
Abstract Adiabatic holonomic gates possess the geometric robustness of adiabatic geometric phases, i.e., dependence only on the evolution path of the parameter space but not on the evolution details of the quantum system, which, when coordinated with decoherence-free subspaces, permits additional resilience to the collective dephasing environment. However, the previous scheme [Phys. Rev. Lett. 95 130501 (2005)] of adiabatic holonomic quantum computation in decoherence-free subspaces requires four-body interaction that is challenging in practical implementation. In this work, we put forward a scheme to realize universal adiabatic holonomic quantum computation in decoherence-free subspaces using only realistically available two-body interaction, thereby avoiding the difficulty of implementing four-body interaction. Furthermore, an arbitrary one-qubit gate in our scheme can be realized by a single-shot implementation, which eliminates the need to combine multiple gates for realizing such a gate.