The implementation of holonomic quantum computation is meaningful. We can effectively resist local and collective noise in the process of physical implementation by using the advantage of non-Abelian geometric phase. In this paper, we set out from the simplest and most fundamental Jaynes-Cummings model of ion trap system to implement single-qubit logical operations, and taking advantage of the interaction of two ions with a pair of laser pulses to implement two-qubit logical operations, respectively. Compared with the previous proposal, the whole process of our proposal are in tunable way and the universal holonomic quantum qubit gates can be made robust to systematic error and decay which pushes the gate fidelities in the presence of decoherence and systematic error to well high level. Thus our scheme affords an experimentally feasible and simple way to make realizing the robust nonadiabatic holonomic quantum computation.
The implementation of holonomic quantum computation is meaningful. We can effectively resist local and collective noise in the process of physical implementation by using the advantage of non-Abelian geometric phase. In this paper, we set out from the simplest and most fundamental Jaynes-Cummings model of ion trap system to implement single-qubit logical operations, and taking advantage of the interaction of two ions with a pair of laser pulses to implement two-qubit logical operations, respectively. Compared with the previous proposal, the whole process of our proposal are in tunable way and the universal holonomic quantum qubit gates can be made robust to systematic error and decay which pushes the gate fidelities in the presence of decoherence and systematic error to well high level. Thus our scheme affords an experimentally feasible and simple way to make realizing the robust nonadiabatic holonomic quantum computation.
For circuit-based quantum computation, experimental implementation of a universal set of quantum logic gates with high-fidelity and strong robustness is essential and central. Quantum gates induced by geometric phases, which depend only on global properties of the evolution paths, have built-in noise-resilience features. Here, we propose and experimentally demonstrate nonadiabatic holonomic single-qubit quantum gates on two dark paths in a trapped 171Yb+ ion based on four-level systems with resonant drives. We confirm the implementation with measured gate fidelity through both quantum process tomography and randomized benchmarking methods. Meanwhile, we find that nontrivial holonomic two-qubit quantum gates can also be realized within current experimental technologies. Compared with previous implementations, our experiments share both the advantages of fast nonadiabatic evolution and robustness against systematic errors. Therefore, our experiments confirm a promising method for fast and robust holonomic quantum computation.
Quantum computation with quantum gates induced by geometric phases is regarded as a promising strategy in fault-tolerant quantum computation, owing to its robustness against operational noise. However, because of the parametric restrictions in previous schemes, the main robust advantage of holonomic quantum gates is reduced. Here, we experimentally demonstrate a solution scheme, obtaining nonadiabatic holonomic single-qubit quantum gates with optimal control in a trapped 171Yb+ ion based on a three-level system with resonant driving, which also has the advantages of rapid evolution and convenient implementation. Compared with previous geometric gates and conventional dynamical gates, the superiority of our scheme is that it is more robust against control amplitude errors, which is confirmed by the gate infidelity as measured by both quantum-process tomography and random benchmarking methods. In addition, we outline how nontrivial two-qubit holonomic gates can also be realized using currently available experimental technology. Thus, our experiment confirms the feasibility of this robust and fast holonomic quantum-computation strategy.
Quantum computation is much more effective than classical computation in solving many problems such as factoring large integers and searching unsorted databases. However, such effectiveness relies on the ability to perform universal high-fidelity quantum gates. One main challenge in achieving such high-fidelity gates is to reduce control errors of a quantum system. To overcome this problem, various proposals of fault-tolerant quantum computation are proposed. A promising one of such proposals is nonadiabatic holonomic quantum computation. Nonadiabatic holonomic quantum computation is realized by using a quantum system with a subspace satisfying both the cyclic evolution and parallel transport conditions. For an N dimensional quantum system with Hamiltonian H(t) and evolution operator U(t)=Texp[−i∫0tH(t')dt'] , if there exists a time-dependent L dimensional subspace S(t) spanned by the orthonormal vectors {|ϕk(t)〉=U(t)|ϕk(0)〉}k=1L that satisfy the two conditions: (1) ∑k=1L|ϕk(τ)〉〈ϕk(τ)|= ∑k=1L|ϕk(0)〉〈ϕk(0)| , with τ being the evolution period, and (2) 〈ϕk(t)|H(t)|ϕl(t)〉=0, k,l=1,2,⋯,L, then the unitary transformation U(τ) is a holonomic gate on the L dimensional subspace S(0) spanned by {|ϕk(0)〉}k=1L . The work of realizing nonadiabatic holonomic gate is to build a physical system with a Hamiltonian satisfying both the cyclic evolution condition and the parallel transport condition. Nonadiabatic holonomic quantum computation is based on nonadiabatic non-Abelian geometric phases. Since nonadiabatic non-Abelian geometric phases are only dependent on evolution paths but independent of evolution details, nonadiabatic holonomic gates are robust against control errors. Besides, nonadiabatic holonomic gates do not require the long run-time evolution that is necessary for adiabatic holonomic gates. Due to the merits of both robustness against control errors and high-speed realization, nonadiabatic holonomic quantum computation has received increasing attention in theory and experiment. Various schemes of nonadiabatic holonomic quantum computation have been proposed based on different physical systems, and a number of these schemes have been experimentally demonstrated with nuclear magnetic resonance systems, nitrogen-vacancy center systems, and superconducting systems. The progress of nonadiabatic holonomic quantum computation is also reflected in the improvement of the methods of designing quantum gates. The single-shot scheme, single-loop multiple-pulse scheme and composite gate scheme of nonadiabatic quantum gate are based on the improved methods. In particular, the single-shot scheme simplifies the operation of the one-qubit gate, so that the operation of any one-qubit gate, which needs to be realized by combining two non-commuting π-rotation gate in the original scheme, can be simply realized by a single-shot operation. In this paper, we propose a scheme of nonadiabatic holonomic quantum computation with atom-cavity system. The universal set of quantum gates in our scheme consists of an arbitrary one-qubit nonadiabatic holonomic gate and a two-qubit nonadiabatic holonomic controlled-PHASE gate. To realize the arbitrary one-qubit gate, we use two resonant laser pulses and an off-resonant laser pulse to drive a single four-level atom. With the aid of Stark shifts, the one-qubit gate can be performed in one-step only by changing the Rabi frequencies and the phases of the laser pulses. To realize the two-qubit controlled-PHASE gate, we use two off-resonant laser pulses to drive the two four-level atoms in the cavity. The transitions between the two-atom ground state and the two-atom auxiliary state is facilitated by exchanging virtual photons through a common cavity mode. In this process, the effective two-atom transition is disentangled from the cavity mode and therefore the two-qubit gate is insensitive to the cavity decay.
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.
Implementing nonadiabatic holonomic quantum computation in decoherence-free subspaces may consolidate the advantages of both strategies and thus leads to suppression of both local and collective noises. In a recent paper, Xu et al. [Phys. Rev. Lett. 109, 170501 (2012)] proposed an interesting implementation of such a scheme; however, four-body interaction, which is hard to achieve in a realistic system, is required there. In this paper we propose a simplified scheme to implement such single-qubit and two-qubit logical operations by utilizing only two-body interactions, which is much easier to realize in a trapped ion system.
Nonadiabatic holonomic quantum computation has received increasing attention due to its robustness against control errors. However, all the previous schemes have to use at least two sequentially implemented gates to realize a general one-qubit gate. In this paper we put forward a scheme by which one can directly realize an arbitrary holonomic one-qubit gate with a single-shot implementation, avoiding the extra work of combining two gates into one. Based on a three-level model driven by laser pulses, we show that any single-qubit holonomic gate can be realized by varying the detuning, amplitude, and phase of lasers. Our scheme is compatible with previously proposed nonadiabatic holonomic two-qubit gates, combining with which the arbitrary holonomic one-qubit gates can play universal nonadiabatic holonomic quantum computation. We also investigate the effects of some unavoidable realistic errors on our scheme.