Holonomic quantum gates offer a promising approach for achieving high fidelity and robustness in quantum computation by leveraging geometric phases to suppress the influence of noise. However, existing schemes often suffer from limitations such as complex segmented evolution paths and an inability to implement certain gates, including those with a geometric phase of 𝜋𝜋. In this work, we propose a circular-path holonomic quantum gate scheme with multiple degrees of freedom, which enables the realization of any single-qubit gate, including gates with a geometric phase of 𝜋𝜋. The scheme allows for the optimization of pulse performance, effectively mitigating control errors in both the addressing frequency and Rabi frequency. Applied to a Rydberg-atoms system, numerical simulations demonstrate high fidelity and robustness against control errors, with fidelities exceeding 99.97% under various conditions. This work addresses the limitations of previous schemes and provides a flexible framework for implementing robust quantum gates in three-level quantum systems.
Holonomic quantum gates offer a promising approach for achieving high fidelity and robustness in quantum computation by leveraging geometric phases to suppress the influence of noise. However, existing schemes often suffer from limitations such as complex segmented evolution paths and an inability to implement certain gates, including those with a geometric phase of 𝜋𝜋. In this work, we propose a circular-path holonomic quantum gate scheme with multiple degrees of freedom, which enables the realization of any single-qubit gate, including gates with a geometric phase of 𝜋𝜋. The scheme allows for the optimization of pulse performance, effectively mitigating control errors in both the addressing frequency and Rabi frequency. Applied to a Rydberg-atoms system, numerical simulations demonstrate high fidelity and robustness against control errors, with fidelities exceeding 99.97% under various conditions. This work addresses the limitations of previous schemes and provides a flexible framework for implementing robust quantum gates in three-level quantum systems.
Nonadiabatic holonomic quantum computation (NHQC) based on non-Abelian geometric phases can possess potential robustness advantages due to its intrinsic error-resilient features. However, rigorous geometric conditions, including cyclic evolution and parallel transport conditions, typically demand complex and error-prone control and also involve additional auxiliary resources. Moreover, the limitations of optional evolution paths in previous NHQC schemes make it difficult to avoid path segments that are heavily affected by systematic errors, thereby weakening the error resilience of holonomic gates. To address this problem, we propose a multipath NHQC scheme, which can sufficiently improve the robustness of quantum gates by flexibly correcting geometric evolution paths. Numerical simulation results show that our scheme surpasses previous NHQC schemes in suppressing systematic errors for arbitrary gate types. In addition, we provide a physical implementation based on superconducting quantum circuits, with the decoherence-free-subspace encoding and an experiment-friendly two-body exchange interaction, demonstrating the excellent compatibility of our scheme. Therefore, our scheme offers a promising alternative for future fault-tolerant quantum computation.
We propose a one-step scheme to implement the time-optimal nonadiabatic holonomic three-qubit controlled gates in Rydberg atoms by modulating Rabi frequencies of driving fields on the target atom. We perform the time-optimal operations following the brachistochrone curve, after which the scheme manifests completely different characteristics in contrast to the conventional nonadiabatic holonomic quantum computation (NHQC) that all operations require exactly the same evolution time, even for small angular rotations. Since the present three-qubit controlled-gate scheme significantly reduces the evolution time, the influence of decoherence can be greatly reduced. On the other hand, the robustness on the systematic error can also be guaranteed by the feature of geometric phase. The simulation through master equation verifies the high fidelity and robustness of our scheme. Although we use the Rydberg-atom system to verify the feasibility of our scheme, any quantum platforms capable of achieving the same form of effective Hamiltonian is feasible, which means the proposed three-qubit time-optimal-control nonadiabatic holonomic gates is independent of the particular physical platform and may have the potential for wider application.
Holonomic quantum computation (HQC) offers an inherently robust approach to quantum gate implementation by exploiting quantum holonomies. While adiabatic HQC benefits from robustness against certain control errors, its long runtime limits practical utility due to increased exposure to environmental noise. Nonadiabatic HQC addresses this issue by enabling faster gate operations but compromises robustness. In this work, we propose a scheme for fast holonomic quantum gates based on the π -pulse method, which accelerates adiabatic evolution while preserving its robustness. By guiding the system Hamiltonian along geodesic paths in the parameter space and applying phase-modulating π pulses at discrete points, we realize a universal set of holonomic gates beyond the conventional adiabatic limit. Our scheme allows for arbitrary single-qubit and two-qubit controlled gates within a single-loop evolution and provides additional tunable parameters for flexible gate design. These results demonstrate a promising path toward high-fidelity, fast, and robust quantum computation .
Nonadiabatic holonomic quantum computation provides a promising approach toward fault-tolerant quantum control, due to its simple requirements for energy level structure and intrinsic robustness stemming from non-Abelian geometric phases. However, conventional nonadiabatic holonomic quantum computation relies on segmented evolution along a specific trajectory, which not only complicates experimental control but also exacerbates decoherence effects. Meanwhile, minor deviations in systematic parameters can directly disrupt the cyclic evolution process necessary to construct holonomic gates, leading to degraded gate robustness. To address these disadvantages, we here propose a general strategy to incorporate cyclic evolution protection into the holonomic gate construction. The aim is to design on-demand trajectories by modulating pulse shapes, thereby circumventing the detrimental impact of systematic errors on cyclic evolution. Consequently, universal holonomic gates implemented through a stable cyclic evolution process can maintain lower error sensitivity. Meanwhile, in our scheme, compressing the state population in the ancillary state ensures less energy consumption, resulting in higher gate fidelity. Therefore, our work serves as a practical solution for achieving high-fidelity and robust universal quantum gates, paving the way for large-scale quantum computation.
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.
To realize one desired nonadiabatic holonomic gate, various equivalent evolution paths can be chosen. However, in the presence of errors, these paths become inequivalent. In this paper we investigate the difference of these evolution paths in the presence of systematic Rabi-frequency errors and aim to find paths with optimal robustness to realize one-qubit nonadiabatic holonomic gates. We focus on three types of evolution paths in the $\mathrm{\ensuremath{\Lambda}}$ system: paths belonging to the original two-loop scheme [E. Sj\"oqvist et al., New J. Phys. 14, 103035 (2012)], the single-loop multiple-pulse scheme [E. Herterich and E. Sj\"oqvist, Phys. Rev. A 94, 052310 (2016)], and the off-resonance single-shot scheme [G. F. Xu et al., Phys. Rev. A 92, 052302 (2015); E. Sj\"oqvist, Phys. Lett. A 380, 65 (2016)]. Whereas both the single-loop multiple-pulse and single-shot schemes aim to improve the robustness of the original two-loop scheme by shortening the exposure to decoherence, we find here that the two-loop scheme is more robust to systematic errors in the Rabi frequencies. More importantly, we derive conditions under which the resilience to this kind of error can be optimized, thereby strengthening the robustness of nonadiabatic holonomic gates.