Although geometric phases in quantum evolution are historically overlooked, their active control now stimulates strategies for constructing robust quantum technologies. Here, we demonstrate arbitrary single-qubit holonomic gates from a single cycle of nonadiabatic evolution, eliminating the need to concatenate two separate cycles. Our method varies the amplitude, phase, and detuning of a two-tone optical field to control the non-Abelian geometric phase acquired by a nitrogen-vacancy center in diamond over a coherent excitation cycle. We demonstrate the enhanced robustness of detuned gates to excited-state decoherence and provide insights for optimizing fast holonomic control in dissipative quantum systems.
Although geometric phases in quantum evolution are historically overlooked, their active control now stimulates strategies for constructing robust quantum technologies. Here, we demonstrate arbitrary single-qubit holonomic gates from a single cycle of nonadiabatic evolution, eliminating the need to concatenate two separate cycles. Our method varies the amplitude, phase, and detuning of a two-tone optical field to control the non-Abelian geometric phase acquired by a nitrogen-vacancy center in diamond over a coherent excitation cycle. We demonstrate the enhanced robustness of detuned gates to excited-state decoherence and provide insights for optimizing fast holonomic control in dissipative quantum systems.
The holonomic approach to controlling (nitrogen-vacancy) NV-center qubits provides an elegant way of theoretically devising universal quantum gates that operate on qubits via calculable microwave pulses. There is, however, a lack of simulated results from the theory of holonomic control of quantum registers with more than two qubits describing the transition between the dark states. Considering this, we have been experimenting with the IBM Quantum Experience technology to determine the capabilities of simulating holonomic control of NV-centers for three qubits describing an eight-level system that produces a non-Abelian geometric phase. The tunability of the geometric phase via the detuning frequency is demonstrated through the high fidelity (~85%) of three-qubit off-resonant holonomic gates over the on-resonant ones. The transition between the dark states shows the alignment of the gate's dark state with the qubit's initial state hence decoherence of the multi-qubit system is well-controlled through a π/3 rotation.
The holonomic approach to controlling (nitrogen-vacancy) NV-center qubits provides an elegant way of theoretically devising universal quantum gates that operate on qubits via calculable microwave pulses. There is, however, a lack of simulated results from the theory of holonomic control of quantum registers with more than two qubits describing the transition between the dark states. In light of this, we have been experimenting with the IBM Quantum Experience technology to determine the capabilities of simulating holonomic control of NV-centers for three qubits describing an eight-level system that produces a non-Abelian geometric phase. The tunability of the geometric phase via the detuning frequency is demonstrated through the high fidelity (about 80%) of 3-qubit off-resonant holonomic gates over the on-resonant ones. The transition between the dark states shows the alignment of the gate dark state with the qubits initial state hence decoherence of the multi-qubit system is well-controlled through a 0.33pi rotation. The electron return probability can exhibit spin-orbit coupling-like behavior as observed in topological materials based on the extra geometric phase.
Quantum computation has been shown to be superior to classical computation in solving some problems, and therefore can substantially change our life. However, the realization of quantum computation is still challenging, even if quantum technologies have been improved significantly. Two main obstacles to the realization of practical quantum computation are control errors and decoherence. Control errors are caused by inaccurate manipulations of quantum systems and decoherence is caused by the inevitable interaction between the system and its environment. Geometric phases are only dependent on evolution paths of quantum systems but independent of the evolution details and therefore quantum computation based on geometric phases, i.e., geometric quantum computation, is robust against control errors, benefiting the realization of practical quantum computation. The early proposals of geometric quantum computation are based on adiabatic geometric phases. These proposals require quantum systems to undergo adiabatic evolution, which makes quantum systems evolve for a long time. To circumvent this, nonadiabatic geometric quantum computation based on nonadiabatic Abelian geometric phases was proposed soon after. In 2012, nonadiabatic holonomic quantum computation based on nonadiabatic non-Abelian geometric phases was proposed, which also circumvents long-time evolutions. Moreover, compared with nonadiabatic geometric quantum computation that uses the geometric phase as one parameter of a quantum gate, nonadiabatic holonomic quantum computation uses the holonomic matrix itself as a quantum gate. This makes nonadiabatic holonomic quantum computation possess whole-geometric property. Due to the merits of both geometric robustness and high-speed implementation without the limit of adiabatic evolution, nonadiabatic holonomic quantum computation has been attracting much attention. Until now, much progress has been achieved in the field of nonadiabatic holonomic quantum computation. On one hand, various methods have been proposed to design more efficient nonadiabatic holonomic gates. Nonadiabatic holonomic gates were first realized by using resonant laser fields to drive a three-level system. After this, the single-shot proposal and the single-loop proposal were proposed, allowing us to realize one-qubit gates by a shorter path and thereby reducing the exposure time of nonadiabatic holonomic gates to the environment. To further shorten the exposure time, the path-shortening protocol was put forward, where nonadiabatic holonomic gates can be realized based on a class of extended evolution paths that are shorter than the former ones. Recently, a general approach of constructing Hamiltonians for nonadiabatic holonomic quantum computation was put forward, by using which one can easily find a Hamiltonian making the quantum system evolve along a desired path so that nonadiabatic holonomic gates can be realized with an economical evolution time. On the other hand, various methods have been proposed to combine nonadiabatic holonomic gates and various decoherence-resilient methods, making the resulting schemes robust against both control errors and decoherence. The first proposal in this aspect is combining nonadiabatic holonomic gates and decoherence-free subspaces. Gradually, proposals combining nonadiabatic holonomic gates with noiseless subsystems, dynamical decoupling and surface codes were put forward. Last but not least, various proposals suitable for specific physical systems have been proposed and particularly various experimental platforms have been used to demonstrate nonadiabatic holonomic gates. This also significantly improves the development of nonadiabatic holonomic quantum computation. In this paper, we review the above research advances on nonadiabatic holonomic quantum computation, aiming to help readers understand the main developments of nonadiabatic holonomic quantum computation.
We demonstrate universal non-adiabatic non-abelian holonomic single quantum gates over a geometric electron spin with phase-modulated polarized light and 93% average fidelity. This allows purely geometric rotation around an arbitrary axis by any angle defined by light polarization and phase using a degenerate three-level Λ-type system in a negatively charged nitrogen-vacancy center in diamond. Since the control light is completely resonant to the ancillary excited state, the demonstrated holonomic gate not only is fast with low power, but also is precise without the dynamical phase being subject to control error and environmental noise. It thus allows pulse shaping for further fidelity.
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) provides an essential way to construct robust and high-fidelity quantum gates due to its geometric features. However, NHQC is more sensitive to decay and dephasing errors than a conventional dynamical gate since it requires an ancillary intermediate state. Here, we utilize the Hamiltonian reverse engineering technique to study the influence of intermediate-state decoherence on NHQC gate fidelity, and propose schemes to construct an arbitrary single-qubit holonomic gate and nontrivial two-qubit holonomic gate with high fidelity and robustness to the decoherence. Although the proposed method is generic and can be applied to many experimental platforms, such as superconducting qubits, trapped ions, and quantum dots, here we take a nitrogen-vacancy center as an example to show that the gate fidelity can be significantly enhanced from 89% to 99.6% in contrast to recent experimental NHQC schemes [Phys. Rev. Lett. 119, 140503 (2017); Nat. Photonics 11, 309 (2017); Opt. Lett. 43, 2380 (2018)], and the robustness against decoherence can also be significantly improved. All in all, our scheme provides a promising way for fault-tolerant geometric quantum computation.