A theorem from control theory relating the Lie algebra generated by vector fields on a manifold to the controllability of the dynamical system is shown to apply to Holonomic Quantum Computation. Conditions for deriving the holonomy algebra are presented by taking covariant derivatives of the curvature associated to a non-Abelian gauge connection. When applied to the Optical Holonomic Computer, these conditions determine that the holonomy group of the two-qubit interaction model contains $SU(2) \times SU(2)$. In particular, a universal two-qubit logic gate is attainable for this model.
We present a nonlocal construction of universal gates by means of holonomic (geometric) quantum teleportation. The effect of the errors from imperfect control of the classical parameters, the looping variation of which builds up holonomic gates, is investigated. Additionally, the influence of quantum decoherence on holonomic teleportation used as a computational primitive is studied. Advantages of the holonomic implementation with respect to control errors and dissipation are presented.
Nonadiabatic holonomic quantum computation as one of the key steps to achieve fault tolerant quantum information processing has so far been realized in a number of physical settings. However, in some physical systems particularly in spin qubit systems, which are actively considered for realization of quantum computers, experimental challenges are undeniable and the lack of a practically feasible and scalable scheme that supports universal holonomic quantum computation all in a single well defined setup is still an issue. Here, we propose and discuss a scalable star-shape architecture with prom…
We implement a non-adiabatic universal set of holonomic quantum gates based on abelian holonomies using dynamical invariants, by Lie-algebraic methods. Unlike previous implementations, presented scheme does not rely on secondary methods such as double-loop or spin-echo and avoids associated experimental difficulties. It turns out that such gates exist purely in the non-adiabatic regime for these systems.
Geometric manipulation of a quantum system offers a method for fast, universal, and robust quantum information processing. Here, we propose a scheme for universal all-geometric quantum computation using non-adiabatic quantum holonomies. We propose three different realizations of the scheme based on an unconventional use of quantum dot and single-molecule magnet devices, which offer promising scalability and robust efficiency.
The non-adiabatic holonomic quantum computation with the advantages of fast and robustness attracts widespread attention in recent years. Here, we propose the first scheme for realizing universal single-qubit gates based on an optomechanical system working with the non-adiabatic geometric phases. Our quantum gates are robust to the control errors and the parameter fluctuations, and have unique functions to achieve the quantum state transfer and entanglement generation between cavities. We discuss the corresponding experimental parameters and give some simulations. Our scheme may have the pract…
We show how one can implement any local quantum gate on specific qubits in an array of qubits by carrying adiabatically a Hamiltonian around a closed loop. We find the exact form of the loop and the Hamiltonian for implementing general one and two qubits gates. Our method is analytical and is not based on numerical search in the space of all loops.
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-$(\boldsymbol{\mathrm{n}\cdot \mathrmσ})$ gate is realized by $(2n-1)$ basic operations instead of decomposing it into the universal gates, whereas an $(n+1)$-qubit controlled arbitrary rotation gate can be obtained by combin…
Experimental realization of a universal set of quantum logic gates with high-fidelity is critical to quantum information processing, which is always challenging by inevitable interaction between the quantum system and environment. Geometric quantum computation is noise immune, and thus offers a robust way to enhance the control fidelity. Here, we experimentally implement the recently proposed extensible nonadiabatic holonomic quantum computation with solid spins in diamond at room-temperature, which maintains both flexibility and resilience against decoherence and system control errors. Compar…
We develop a non-adiabatic generalization of holonomic quantum computation in which high-speed universal quantum gates can be realized by using non-Abelian geometric phases. We show how a set of non-adiabatic holonomic one- and two-qubit gates can be implemented by utilizing optical transitions in a generic three-level $Λ$ configuration. Our scheme opens up for universal holonomic quantum computation on qubits characterized by short coherence times.
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 …
In this paper, we present a universal control technique, the non-holonomic control, which allows us to impose any arbitrarily prescribed unitary evolution to any quantum system through the alternate application of two well-chosen perturbations.
We analyze the possibility and efficiency of non-holonomic control over quantum devices with exponentially large number of Hilbert space dimensions. We show that completely controllable devices of this type can be assembled from elementary units of arbitrary physical nature, and can be employed efficiently for universal quantum computations and simulation of quantum field dynamics.
Holonomic quantum computation is the idea to use non-Abelian geometric phases to implement universal quantum gates that are robust to fluctuations in control parameters. Here, we propose a compact design for a holonomic quantum computer based on coupled three-level systems. The scheme does not require adiabatic evolution and can be implemented in arrays of atoms or ions trapped in tailored standing wave potentials.
Geometric phases are well known to be noise-resilient in quantum evolutions/operations. Holonomic quantum gates provide us with a robust way towards universal quantum computation, as these quantum gates are actually induced by nonabelian geometric phases. Here we propose and elaborate how to efficiently implement universal nonadiabatic holonomic quantum gates on simpler superconducting circuits, with a single transmon serving as a qubit. In our proposal, an arbitrary single-qubit holonomic gate can be realized in a single-loop scenario, by varying the amplitudes and phase difference of two mic…
It is proposed that high-speed universal quantum gates can be realized by using non-Abelian holonomic transformation. A cyclic evolution path which brings the system periodically back to a degenerate qubit subspace is crucial to holonomic quantum computing. The cyclic nature and the resulting gate operations are fully dependent on the precise control of driving parameters, such as the modulated envelop function of Rabi frequency and the control phases. We investigate the effects of fluctuations in these driving parameters on the transformation fidelity of a universal set of single-qubit quantu…
Universal single-qubit operations based on purely geometric phase factors in adiabatic processes are demonstrated by utilizing a four-level system in a trapped single $^{40}$Ca$^+$ ion connected by three oscillating fields. Robustness against parameter variations is studied. The scheme demonstrated here can be employed as a building block for large-scale holonomic quantum computations, which may be useful for large qubit systems with statistical variations in system 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, non-adiabatic holonomic quantum computation in decoherence-free subspaces, which avoids long run-time requirement but with all the robust advantages, remains an open problem. Here, we demonstrate how to realize non-adiabatic holonomic quantum computation in decoherence-free…
We show how to realize, by means of non-abelian quantum holonomies, a set of universal quantum gates acting on decoherence-free subspaces and subsystems. In this manner we bring together the quantum coherence stabilization virtues of decoherence-free subspaces and the fault-tolerance of all-geometric holonomic control. We discuss the implementation of this scheme in the context of quantum information processing using trapped ions and quantum dots.
Due to its geometric nature, holonomic quantum computation is fault-tolerant against certain types of control errors. Although proposed more than a decade ago, the experimental realization of holonomic quantum computation is still an open challenge. In this Letter, we report the first experimental demonstration of nonadiabatic holonomic quantum computation in liquid NMR quantum information processors. Two non-commuting holonomic single-qubit gates, rotations about x-axis and about z-axis, and the two-qubit holonomic control-NOT gate are realized with high fidelity by evolving the work qubits a…
We investigate the influence of random errors in external control parameters on the stability of holonomic quantum computation in the case of arbitrary loops and adiabatic connections. A simple expression is obtained for the case of small random uncorrelated errors. Due to universality of mathematical description our results are valid for any physical system which can be described in terms of holonomies. Theoretical results are confirmed by numerical simulations.
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 scalabi…
Non-adiabatic 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 novelty 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,…
Our goal is to provide a new approach to the construction of geometry-induced entanglement between a pair of $Λ$ type atoms in a system consists of $N$ identical atoms by means of nonadiabatic quantum holonomies. By employing the quantum Zeno effect, we introduce a tripod type interaction Hamiltonian between two selected atoms trapped in an optical cavity, which allows arbitrary geometric entangling power. This would be a substantial step toward resolving the feasibility of realizing universal nonadiabatic holonomic entangling two-qubit gates.
Non-adiabatic holonomic quantum computation in decoherence-free subspaces protects quantum information from control imprecisions and decoherence. For the non-collective decoherence that each qubit has its own bath, we show the implementations of two non-commutable holonomic single-qubit gates and one holonomic nontrivial two-qubit gate that compose a universal set of non-adiabatic holonomic quantum gates in decoherence-free-subspaces of the decoupling group, with an encoding rate of $\frac{N-2}{N}$. The proposed scheme is robust against control imprecisions and the non-collective decoherence, …
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 [New J. Phys. {\bf 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 on…
A three-level system can be used in a $Λ$-type configuration in order to construct a universal set of quantum gates through the use of non-Abelian non-adiabatic geometrical phases. Such construction allows for high-speed operation times which diminish the effects of decoherence. This might be, however, accompanied by a breakdown of the validity of the rotating wave approximation (RWA) due to the comparable time scale between counter-rotating terms and the pulse length, which greatly affects the dynamics. Here, we investigate the trade-off between dissipative effects and the RWA validity, obtai…
We show an equivalence relation between fault-tolerant circuits for a stabilizer code and fault-tolerant adiabatic processes for holonomic quantum computation (HQC), in the case where quantum information is encoded in the degenerated ground space of the system Hamiltonian. By this equivalence, we can systematically construct a fault-tolerant HQC scheme, which can geometrically implement a universal set of encoded quantum gates by adiabatically deforming the system Hamiltonian. During this process, quantum information is protected from low weight thermal excitations by an energy gap that does n…
Non-adiabatic holonomic quantum computation is a method used to implement high-speed quantum gates with non-Abelian geometric phases associated with paths in state space. Due to their noise tolerance, these phases can be used to construct error resilient quantum gates. We extend the holonomic dark path qubit scheme in [M.-Z. Ai {\t et al.}, Fundam. Res. {\bf 2}, 661 (2022)] to qudits. Specifically, we demonstrate one- and two-qudit universality by using the dark path technique. Explicit qutrit ($d=3$) gates are demonstrated and the scaling of the number of loops with the dimension $d$ is addre…
To implement a set of universal quantum logic gates based on non-Abelian geometric phases, it is a conventional wisdom that quantum systems beyond two levels are required, which is extremely difficult to fulfil for superconducting qubits, appearing to be a main reason why only single qubit gates was implemented in a recent experiment [A. A. Abdumalikov Jr \emph{et al}., Nature 496, 482 (2013)]. Here we propose to realize non-adiabatic holonomic quantum computation in decoherence-free subspace on circuit QED, where one can use only the two levels in transmon qubits, a usual interaction, and a m…
The key for realizing fault-tolerant quantum computation lies in maintaining the coherence of all qubits so that high-fidelity and robust quantum manipulations on them can be achieved. One of the promising approaches is to use geometric phases in the construction of universal quantum gates, due to their intrinsic robustness against certain types of local noises. However, due to limitations in previous implementations, the noise-resilience feature of nonadiabatic holonomic quantum computation (NHQC) still needs to be improved. Here, combining with the dynamical correction technique, we propose …
Realization of fast fault-tolerant quantum gates on a single spin is the core requirement for solid-state quantum-information processing. As polarized light shows geometric interference, spin coherence is also geometrically controlled with light via the spin-orbit interaction. Here, we show that a geometric spin in a degenerate subspace of a spin-1 electronic system under a zero field in a nitrogen vacancy center in diamond allows implementation of optical non-adiabatic holonomic quantum gates. The geometric spin under quasi-resonant light exposure undergoes a cyclic evolution in the spin-orbi…
Based on the geometrical nature of quantum phases, non-adiabatic holonomic quantum control (NHQC) has become a standard technique for enhancing robustness in constructing quantum gates. However, the conventional approach of NHQC is sensitive to control instability, as it requires the driving pulses to cover a fixed pulse area. Furthermore, even for small-angle rotations, all operations need to be completed with the same duration of time. Here we experimentally demonstrate a time-optimal and unconventional approach of NHQC (called TOUNHQC), which can optimize the operation time of any holonomic…
Holonomic quantum computation exploits a quantum state's non-trivial, matrix-valued geometric phase (holonomy) to perform fault-tolerant computation. Holonomies arising from systems where the Hamiltonian traces a continuous path through parameter space have been well-researched. Discrete holonomies, on the other hand, where the state jumps from point to point in state space, have had little prior investigation. Using a sequence of incomplete projective measurements of the spin operator, we build an explicit approach to universal quantum computation. We show that quantum error correction codes …
Geometric phase is an indispensable element for achieving robust and high-fidelity quantum gates due to its built-in noise-resilience feature. However, due to the complexity of manipulation and the intrinsic leakage of the encoded quantum information to non-logical-qubit basis, the experimental realization of universal nonadiabatic holonomic quantum computation is very difficult. Here, we propose to implement scalable nonadiabatic holonomic quantum computation with decoherence-free subspace encoding on a two-dimensional square superconducting transmon-qubit lattice, where only the two-body int…
A cavity QED implementation of the non-adiabatic holonomic quantum computation in decoherence-free subspaces is proposed with nitrogen-vacancy centers coupled commonly to the whispering-gallery mode of a microsphere cavity, where a universal set of quantum gates can be realized on the qubits. In our implementation, with the assistant of the appropriate driving fields, the quantum evolution is insensitive to the cavity field state, which is only virtually excited. The implemented non-adiabatic holonomies, utilizing optical transitions in the Λ type of three-level configuration of the nitrogen-v…
In this paper we examine the use of an adiabatic quantum data transfer protocol to build a universal quantum computer. Single qubit gates are realized by using a bus protocol to transfer qubits of information down a spin chain with a unitary twist. This twist arises from altered couplings on the chain corresponding to unitary rotations performed on one region of the chain. We show how a controlled NOT gate can be realized by using a control qubit with Ising type coupling. The method discussed here can be extended to non-adiabatic quantum bus protocols. We also examine the potential of realizin…
High-fidelity quantum gates are essential for large-scale quantum computation. However, any quantum manipulation will inevitably affected by noises, systematic errors and decoherence effects, which lead to infidelity of a target quantum task. Therefore, implementing high-fidelity, robust and fast quantum gates is highly desired. Here, we propose a fast and robust scheme to construct high-fidelity holonomic quantum gates for universal quantum computation based on resonant interaction of three-level quantum systems via shortcuts to adiabaticity. In our proposal, the target Hamiltonian to induce …
High-fidelity quantum gates are essential for large-scale quantum computation, which can naturally be realized in a noise-resilient way. Geometric manipulation and decoherence-free subspace encoding are promising ways toward robust quantum computation. Here, by combining the advantages of both strategies, we propose and experimentally realize universal holonomic quantum gates in both a singleloop scheme and a composite scheme, based on nonadiabatic and non-Abelian geometric phases, in a decoherence-free subspace with nuclear magnetic resonance. Our experiment uses only two-body resonant spin-s…
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 $Δ$-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…
We investigate an all-out optical setup allowing for generation of non-Abelian geometric phases on its large degenerate eigenspaces. The proposal has the form of an M -pod system and can be implemented in terms of integrated photonic waveguide structures. We show that by injecting a larger number of photons into the optical setup, the degeneracy of eigenspaces scales rapidly. After studying the spectral properties of our system for the general case, we show how arbitrary U(4) transformations can be generated on the dark subspace of an optical tripod filled with two photons. Moreover, a degener…
We propose an implementation of holonomic (geometrical) quantum gates by means of semiconductor nanostructures. Our quantum hardware consists of semiconductor macroatoms driven by sequences of ultrafast laser pulses ({\it all optical control}). Our logical bits are Coulomb-correlated electron-hole pairs (excitons) in a four-level scheme selectively addressed by laser pulses with different polarization. A universal set of single and two-qubit gates is generated by adiabatic change of the Rabi frequencies of the lasers and by exploiting the dipole coupling between excitons.
Nonadiabatic holonomic quantum computation has received increasing attention due to its robustness against control errors as well as high-speed realization. Several schemes of its implementation have been put forward based on various physical systems, each of which has some particular merits. In this paper, we put forward an alternative scheme of nonadiabatic holonomic quantum computation, in which a universal set of quantum gates is realized based on Rydberg superatoms. A Rydberg superatom is a mesoscopic atomic ensemble that allows for only a single Rydberg excitation shared by many atoms wi…
In recent years qubit designs such as transmons approached the fidelities of up to 0.999. However, even these devices are still insufficient for realizing quantum error correction requiring better than 0.9999 fidelity. Topologically protected superconducting qubits are arguably most prospective for building a realistic quantum computer as they are intrinsically protected from noise and leakage errors that occur in transmons. We propose a topologically protected qubit design based on a $π$-periodic Josephson element and a universal set of gates: protected Clifford group and highly robust (with …
High-fidelity and robust quantum manipulation is the key for scalable quantum computation. Therefore, due to the intrinsic operational robustness, quantum manipulation induced by geometric phases is one of the promising candidates. However, the longer gate time for geometric operations and more physical-implementation difficulties hinder its practical and wide applications. Here, we propose a simplified implementation of universal holonomic quantum gates on superconducting circuits with experimentally demonstrated techniques, which can remove the two main challenges by introducing the time-opt…
Reliable quantum information processing requires high-fidelity universal manipulation of quantum systems within the characteristic coherence times. Non-adiabatic holonomic quantum computation offers a promising approach to implement fast, universal, and robust quantum logic gates particularly useful in nano-fabricated solid-state architectures, which typically have short coherence times. Here, we propose an experimentally feasible scheme to realize high-speed universal geometric quantum gates in nano-engineered pseudo-spin charge qubits. We use a system of three coupled quantum dots containing…
