Nonadiabatic holonomic gates provide a promising way to realize high-fidelity quantum gates because of their robustness and high-speed implementation. Compared with quantum computation that consists of many nonadiabatic holonomic one-qubit and two-qubit gates, the nonadiabatic holonomic multiqubit gate, which can reduce the number of quantum gates, is more efficient to construct simpler and faster complicated quantum computation. We propose the simplest scheme to realize nonadiabatic holonomic three-qubit controlled gates. On the one hand, the parameters we set to realize the three-qubit controlled gates are all time independent, which makes it easy to control and operate the quantum systems. On the other hand, our scheme avoids the combination of the elementary gates and realizes the three-qubit controlled gates only by one-shot implementation.
Nonadiabatic holonomic gates provide a promising way to realize high-fidelity quantum gates because of their robustness and high-speed implementation. Compared with quantum computation that consists of many nonadiabatic holonomic one-qubit and two-qubit gates, the nonadiabatic holonomic multiqubit gate, which can reduce the number of quantum gates, is more efficient to construct simpler and faster complicated quantum computation. We propose the simplest scheme to realize nonadiabatic holonomic three-qubit controlled gates. On the one hand, the parameters we set to realize the three-qubit controlled gates are all time independent, which makes it easy to control and operate the quantum systems. On the other hand, our scheme avoids the combination of the elementary gates and realizes the three-qubit controlled gates only by one-shot implementation.
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.
Nonadiabatic holonomic quantum computation in decoherence-free subspaces has attracted increasing attention recently, as it allows for high-speed implementation and combines both the robustness of holonomic gates and the coherence stabilization of decoherence-free subspaces. Since the first protocol of nonadiabatic holonomic quantum computation in decoherence-free subspaces, a number of schemes for its physical implementation have been put forward. However, all previous schemes require two noncommuting gates to realize an arbitrary one-qubit gate, which doubles the exposure time of gates to error sources as well as the resource expenditure. In this paper, we propose an alternative protocol for nonadiabatic holonomic quantum computation in decoherence-free subspaces, in which an arbitrary one-qubit gate in decoherence-free subspaces is realized by a single-shot implementation. The present protocol not only maintains the merits of the original protocol but also avoids the extra work of combining two gates to implement an arbitrary one-qubit gate and thereby reduces the exposure time to various error sources.
Previous schemes of nonadiabatic holonomic quantum computation were focused mainly on realizing a universal set of elementary gates. Multiqubit controlled gates could be built by decomposing them into a series of the universal gates. In this article, we propose an approach for realizing nonadiabatic holonomic multiqubit controlled gates in which a (n+1)-qubit controlled-$(\mathrm{n}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbf{\ensuremath{\sigma}})$ gate is realized by $(2n\ensuremath{-}1)$-basic operations instead of decomposing it into the universal gates, whereas a (n+1)-qubit controlled arbitrary rotation gate can be obtained by combining only two such controlled-$(\mathrm{n}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbf{\ensuremath{\sigma}})$ gates. Our scheme greatly reduces the operations of nonadiabatic holonomic quantum computation.
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 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.
Abstract Multi-qubit controlled gates are frequently used in quantum information processing. Although they can be built with gates from the universal set of one- and two-qubit gates, this procedure typically becomes very demanding as the number of such gates rapidly grows with the size of the computational problem. Thus, finding a way to realize them with fewer steps and therefore with reduced decoherence errors is important. Particularly, quantum information is entering noisy intermediate-scale quantum era and in this era, quantum computers do not have enough resources for full fault tolerance and therefor can only support the computation with a short duration. This makes the reduction of the steps even more important. Here, we propose the realization of multi-qubit controlled nonadiabatic holonomic gates with connecting systems. Our proposal can efficiently reduce the operation steps of the realization, thereby reducing the affection from decoherence and increasing the precision of the computation. So, it is useful to realize efficient and robust quantum information processors.