Holonomic quantum computation (HQC) may not show its full potential in quantum speedup due to the prerequisite of a long coherent runtime imposed by the adiabatic condition. Here we show that the conventional HQC can be dramatically accelerated by using external control fields, of which the effectiveness is exclusively determined by the integral of the control fields in the time domain. This control scheme can be realized with net zero energy cost and it is fault-tolerant against fluctuation and noise, significantly relaxing the experimental constraints. We demonstrate how to realize the scheme via decoherence-free subspaces. In this way we unify quantum robustness merits of this fault-tolerant control scheme, the conventional HQC and decoherence-free subspace, and propose an expedited holonomic quantum computation protocol.
Holonomic quantum computation (HQC) may not show its full potential in quantum speedup due to the prerequisite of a long coherent runtime imposed by the adiabatic condition. Here we show that the conventional HQC can be dramatically accelerated by using external control fields, of which the effectiveness is exclusively determined by the integral of the control fields in the time domain. This control scheme can be realized with net zero energy cost and it is fault-tolerant against fluctuation and noise, significantly relaxing the experimental constraints. We demonstrate how to realize the scheme via decoherence-free subspaces. In this way we unify quantum robustness merits of this fault-tolerant control scheme, the conventional HQC and decoherence-free subspace, and propose an expedited holonomic quantum computation protocol.
Holonomic quantum computation (HQC) may not show its full potential in quantum speedup due to the prerequisite of a long coherent runtime imposed by the adiabatic condition. Here we show that the conventional HQC can be dramatically accelerated by using external control fields, of which the effectiveness is exclusively determined by the integral of the control fields in the time domain. Remarkably this control scheme can be realized with net zero energy cost and it is fault-tolerant against fluctuation and noise, significantly relaxing the experimental constraints. We demonstrate how to realize the scheme via decoherence-free subspaces. In this way we unify quantum robustness merits of this fault-tolerant control scheme, the conventional HQC and decoherence-free subspace, and propose an {\em expedited} holonomic quantum computation protocol.
By using transitionless quantum driving algorithm (TQDA), we present an efficient scheme for the shortcuts to the holonomic quantum computation (HQC). It works in decoherence-free subspace (DFS) and the adiabatic process can be speeded up in the shortest possible time. More interestingly, we give a physical implementation for our shortcuts to HQC with nitrogen-vacancy centers in diamonds dispersively coupled to a whispering-gallery mode microsphere cavity. It can be efficiently realized by controlling appropriately the frequencies of the external laser pulses. Also, our scheme has good scalability with more qubits. Different from previous works, we first use TQDA to realize a universal HQC in DFS, including not only two noncommuting accelerated single-qubit holonomic gates but also a accelerated two-qubit holonomic controlled-phase gate, which provides the necessary shortcuts for the complete set of gates required for universal quantum computation. Moreover, our experimentally realizable shortcuts require only two-body interactions, not four-body ones, and they work in the dispersive regime, which relax greatly the difficulty of their physical implementation in experiment. Our numerical calculations show that the present scheme is robust against decoherence with current experimental parameters.
Quantum computation that combines the coherence stabilization virtues of decoherence-free subspaces and the fault tolerance of geometric holonomic control is of great practical importance. Some schemes of adiabatic holonomic quantum computation in decoherence-free subspaces have been proposed in the past few years. However, nonadiabatic holonomic quantum computation in decoherence-free subspaces, which avoids a long run-time requirement but with all the robust advantages, remains an open problem. Here, we demonstrate how to realize nonadiabatic holonomic quantum computation in decoherence-free subspaces. By using only three neighboring physical qubits undergoing collective dephasing to encode one logical qubit, we realize a universal set of quantum gates.
Quantum computation based on geometric phase is generally believed to be more robust against certain errors or noises than the conventional dynamical strategy. However, the gate error caused by the decoherence effect is inevitable, and thus faster gate operations are highly desired. Here, we propose a nonadiabatic holonomic quantum-computation (NHQC) scheme with detuned interactions on a $\mathrm{\ensuremath{\Delta}}$-type three-level system, which combines the time-optimal control technique with the time-independent detuning adjustment to further accelerate universal gate operations, so that the gate time can be greatly shortened within the hardware limitation, and thus high-fidelity gates can be obtained. Meanwhile, our numerical simulations show that the gate robustness is also stronger than previous schemes. Finally, we present an implementation of our proposal on superconducting quantum circuits, with a decoherence-free subspace encoding, based on the experimentally demonstrated parametrically tunable coupling technique, which simplifies previous investigations. Therefore, our protocol provides a more promising alternative for future fault-tolerant quantum computation.
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 .
In Chapter 16 it was shown how holonomic quantum computation (HQC) can be combined with the method of decoherence-free subspaces (DFSs), leading to passive protection against certain types of correlated errors. However, this is not enough for fault tolerance since other types of errors can accumulate detrimentally unless corrected. Scalability of HQC therefore requires going beyond that scheme, e.g., by combining the holonomic approach with active error correction. One way of combining HQC with active quantum error-correcting codes, which is similar to the way HQC is combined with DFSs, was also mentioned in Chapter 16. This approach, however, is not scalable since it requires Hamiltonians that commute with the stabilizer elements, and when the code increases in size, this necessitates couplings that become increasingly nonlocal. In this chapter, we will show how HQC can be made fault tolerant by combining it with the techniques for fault-tolerant quantum error correction (FTQEC) on stabilizer codes using Hamiltonians of finite locality. The fact that the holonomic method can be mapped directly to the circuit model allows us to construct procedures that resort almost entirely to these techniques. We will discuss an approach that makes use of the encoding already present in a stabilizer code and does not require additional qubits [O08, OBL08]. An alternative approach, which requires ancillary qubits for implementing transversal operations between qubits in the code, can be found in [O09].