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.
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 \ensuremathΛ-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 \ensuremathΛ-coupling structure by utilizing carefully chosen laser pulses.
Control errors and decoherence are two main obstacles for realization of quantum computation. Nonadiabatic holonomic quantum computation in decoherence-free subspaces can protect quantum gates from both control errors and decoherence, and, therefore, it has received increasing attention. The original protocol of nonadiabatic holonomic quantum computation in decoherence-free subspaces needs two loops in the Grassmann manifold to generate an arbitrary nonadiabatic holonomic one-qubit gate. In this paper, we proposed a protocol of nonadiabatic holonomic quantum computation in decoherence-free subspaces, where an arbitrary one-qubit gate is generated by only a single loop in the Grassmann manifold and an entangling two-qubit gate assisted by two single-qubit gates can be used to construct a two-qubit controlled unitary gate. In addition, the encoding of our protocol for nonadiabatic holonomic gates requires a lower number of physical qubits compared with previous schemes. Our scheme not only maintains the merits of the original protocol but it also avoids the extra work of combining two gates to realize an arbitrary one-qubit gate.
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.
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.
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.
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.
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.