Nonadiabatic holonomic quantum computation has been proposed as a method to implement quantum logic gates with robustness comparable to that of adiabatic holonomic gates but with shorter execution times. In this paper, we establish an isoholonomic inequality for quantum gates, which provides a lower bound on the lengths of cyclic transformations of the computational space that generate a specific gate. Then, as a corollary, we derive a nonadiabatic execution time estimate for holonomic gates. In addition, we demonstrate that under certain dimensional conditions, the isoholonomic inequality is tight in the sense that every gate on the computational space can be implemented holonomically and unitarily in a time-optimal way. We illustrate the results by showing that the procedures for implementing a universal set of holonomic gates proposed in a pioneering paper on nonadiabatic holonomic quantum computation saturate the isoholonomic inequality and are thus time optimal. Published by the American Physical Society 2024
Nonadiabatic holonomic quantum computation has been proposed as a method to implement quantum logic gates with robustness comparable to that of adiabatic holonomic gates but with shorter execution times. In this paper, we establish an isoholonomic inequality for quantum gates, which provides a lower bound on the lengths of cyclic transformations of the computational space that generate a specific gate. Then, as a corollary, we derive a nonadiabatic execution time estimate for holonomic gates. In addition, we demonstrate that under certain dimensional conditions, the isoholonomic inequality is tight in the sense that every gate on the computational space can be implemented holonomically and unitarily in a time-optimal way. We illustrate the results by showing that the procedures for implementing a universal set of holonomic gates proposed in a pioneering paper on nonadiabatic holonomic quantum computation saturate the isoholonomic inequality and are thus time optimal. Published by the American Physical Society 2024
In holonomic quantum computation, quantum logic gates are realized by cyclic parallel transport of the computational space. The resulting quantum gate corresponds to the holonomy associated with the closed path traced by the computational space. The isoholonomic inequality for gates establishes a fundamental lower bound on the path length of such cyclic transports, which depends only on the spectrum of the holonomy, that is, the eigenvalues of the implemented quantum gate. The isoholonomic inequality also gives rise to an estimate of the minimum time required to execute a holonomic quantum gate, underscoring the central role of the inequality in quantum computation. In this paper, we show that when the codimension of the computational space is sufficiently large, any quantum gate can be implemented using a parallel transporting Hamiltonian in a way that saturates the isoholonomic inequality and the corresponding time estimate. We call such implementations tight. The treatment presented here is constructive and lays the foundation for the development of efficient and optimal implementation strategies in holonomic quantum computation.
Nonadiabatic holonomic quantum computation is a promising approach for implementing quantum gates that offers both efficiency and robustness against certain types of errors. A key element of this approach is a geometric constraint known as the parallel transport condition. According to the principle of covariance, this condition must be appropriately modified when changing reference frames. In this paper, we detail how to adjust the parallel transport condition when transitioning from the laboratory frame to a rotating reference frame. Furthermore, building on gauge invariance considerations, we develop a framework for nonadiabatic holonomic quantum computation with projective gates. The parallel transport condition of this framework effectively addresses the problem of global dynamical phases inherent in conventional nonadiabatic holonomic quantum computation. We extend the isoholonomic inequality, which provides a fundamental bound on the efficiency of protocols used to implement holonomic quantum gates, to encompass projective quantum gates. We also determine a minimum execution time for projective holonomic quantum gates and show that this time can be attained when the codimension of the computational space is sufficiently large.
The nonadiabatic holonomic quantum computation based on three-level systems has wide applicability experimentally due to its simpler energy level structure requirement and inherent robustness from the geometric phase. However, in previous conventional schemes, the states of the calculation subspace have always leaked to the noncomputation subspace, resulting in less robustness than anticipated. Recent efforts to address this problem are at the cost of excessively long gate time, which will lead to more decoherence-induced errors. Here, we propose a solution to the problem without the severe limitation of the much longer gate time. Specifically, we implement arbitrary holonomic gates via a three-segment Hamiltonian, where the gate time depends on the rotation angle, and the smaller the rotation angle, the shorter the gate time will be. Compared with the previous solutions, our numerical simulations indicate that the decoherence-induced gate errors of our scheme are greatly decreased and the robustness of our scheme is also better, particularly for small-angle rotation gates. Moreover, we provide a detailed physical realization of our proposal on a two-dimensional superconducting quantum circuit. Therefore, our protocol provides a promising alternative for future fault-tolerant 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.
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.
Nonadiabatic holonomic quantum computation uses non-Abelian geometric phases to implement a universal set of quantum gates that are robust against control imperfections and decoherence. Until now, a number of three-level-based schemes of nonadiabatic holonomic computation have been put forward, and several of them have been experimentally realized. However, all these works are based on the same class of nonadiabatic paths, which originates from the first nonadiabatic holonomic proposal. Here, we propose a universal set of nonadiabatic holonomic gates based on an extended class of nonadiabatic paths. We find that nonadiabatic holonomic gates can be realized with paths shorter than the known ones, which provides the possibility of realizing nonadiabatic holonomic gates with less exposure to decoherence. Furthermore, inspired by the form of this type of paths, we find a way to eliminate decoherence from nonadiabatic holonomic gates without resorting to redundancies.