A single-loop scenario is proposed to realize nonadiabatic geometric quantum computation. Conventionally, a so-called multi-loop approach is used to remove the dynamical phase accumulated in the operation process for geometric quantum gates. More intriguingly, we here illustrate in detail how to use a special single-loop method to remove the dynamical phase and thus to construct a set of universal quantum gates based on the nonadiabatic geometric phase shift. The present scheme is applicable to NMR systems and may be feasible in other physical systems.
We clarify that the nonadiabatic scheme based on a parallel extension of the adiabatic scenario cannot realize the desired goal of quantum computation.
Nonadiabatic geometric quantum computation provides a means to perform fast and robust quantum gates. It has been implemented in various physical systems, such as trapped ions, nuclear magnetic resonance and superconducting circuits. Another system being adequate for implementation of nonadiabatic geometric quantum computation may be Rydberg atoms, since their internal states have very long coherence time and the Rydberg-mediated interaction facilitates the implementation of a two-qubit gate. Here, we propose a scheme of nonadiabatic geometric quantum computation based on Rydberg atoms, which …
Because of using geometric phases, nonadiabatic geometric gates have the robustness against control errors. On the other hand, decoherence still affects nonadiabatic geometric gates, which is a key factor in reducing their fidelities. In this paper, we show that based on the system Hamiltonian that realizes a nonadiabatic geometric gate, one may construct a new system Hamiltonian, by using which not only the geometric feature of the nonadiabatic geometric gate is preserved, but also the system's coherence is protected. As a result, a coherence-protected nonadiabatic geometric gate is realized …
The geometrical phase of a time-dependent singular oscillator is obtained in the framework of Gaussian wave packet. It is shown by a simple geometrical approach that the geometrical phase is connected to the classical nonadiabatic Hannay angle of the generalized harmonic oscillator.
Nonadiabatic geometric quantum computation in decoherence-free subspaces has received increasing attention due to the merits of its high-speed implementation and robustness against both control errors and decoherence. However, all the previous schemes in this direction have been based on the conventional geometric phases, of which the dynamical phases need to be removed. In this paper, we put forward a scheme of nonadiabatic geometric quantum computation in decoherence-free subspaces based on unconventional geometric phases, of which the dynamical phases do not need to be removed. Specifically…
Nonadiabatic geometric phases are only dependent on the evolution path of a quantum system but independent of the evolution details, and therefore quantum computation based on nonadiabatic geometric phases is robust against control errors. To realize nonadiabatic geometric quantum computation, it is necessary to ensure that the quantum system undergoes a cyclic evolution and the dynamical phases are removed from the total phases. To satisfy these conditions, the evolution paths in previous schemes are usually restricted to some special forms, e.g, orange-slice-shaped loops, which make the path…
Motivated for the fault tolerant quantum computation, quantum gate by adiabatic geometric phase shift is extensively investigated. In this paper, we demonstrate the nonadiabatic scheme for the geometric phase shift and conditional geometric phase shift. Essentially, the new scheme is simply to add an appropriate additional field. With this additional field, the state evolution can be controlled exactly on a dynamical phase free path. Geometric quantum gates for single qubit and the controlled NOT gate for two qubits are given.
We propose an experimentally feasible scheme to achieve quantum computation based on nonadiabatic geometric phase shifts, in which a cyclic geometric phase is used to realize a set of universal quantum gates. Physical implementation of this set of gates is designed for Josephson junctions and for NMR systems. Interestingly, we find that the nonadiabatic phase shift may be independent of the operation time under appropriate controllable conditions. A remarkable feature of the present nonadiabatic geometric gates is that there is no intrinsic limitation on the operation time, unlike adiabatic ge…
The nonadiabatic geometric quantum computation may be achieved using coupled low-capacitance Josephson juctions. We show that the nonadiabtic effects as well as the adiabatic condition are very important for these systems. Moreover, we find that it may be hard to detect the adiabatic Berry's phase in this kind of superconducting nanocircuits; but the nonadiabatic phase may be measurable with current techniques. Our results may provide useful information for the implementation of geometric quantum computation.
A fundamental requirement of quantum information processing is the protection from the adverse effects of decoherence and noise. Decoherence-free subspaces and geometric processing are important steps of quantum information protection. Here, we provide a new experimentally feasible scheme to combine decoherence-free subspaces with nonadiabatic geometric manipulations to attain a universal quantum computation. The proposed scheme is different from previous proposals and is based on the typical XY interaction coupling, which can be set up in various nano-engineered systems and therefore open up …
In a scheme of nonadiabatic purely geometric quantum gates in nuclear magnetic resonance(NMR) systems we propose proper magnitudes of magnetic fields that are suitable for an experiment. We impose a natural condition and reduce the degree of freedom of the magnetic fields to the extent. By varying the magnetic fields with essentially one-dimensional degree of freedom, any spin state can acquire arbitrary purely geometric phase ϕ_{g}=-2π(1-cos{theta}), 0 < cosθ < 1. This is an essential ingredient for constructing universal geometric quantum gates.
Recently, geometric phases, which is fault tolerate to certain errors intrinsically due to its geometric property, are getting considerable attention in quantum computing theoretically. So far, only one experiment about adiabatic geometric gate with NMR through Berry phase has been reported. However, there are two drawbacks in it. First, the adiabatic condition of Berry phase makes such gate very slowly. Second, the extra operation to eliminate the dynamic phase. As we know, geometric phase can exist both adiabatic(Berry phase) and nonadiabatic(Aharonov-Anandan phase). In this letter, we repor…
Nonadiabatic geometric quantum computation (NGQC) and nonadiabatic holonomic quantum computation (NHQC) have been proposed to reduce the run time of geometric quantum gates. However, in terms of robustness against experimental control errors, the existing NGQC and NHQC scenarios have no advantage over standard dynamical gates in most cases. Here, we give the reasons why nonadiabatic geometric gates are sensitive to the control errors and, further, we propose a scheme of super-robust nonadiabatic geometric quantum control, in which the super-robust condition can guarantee both high speed and ro…
A new scheme of realizing the nonadiabatic conditional geometric phase shift via a noncoplanar (and coiled) fiber system is presented in this Letter. It is shown that the effective Hamiltonian that describes the interaction of polarized photons with the fiber medium is just the Wang-Keiji type of Hamiltonian. This, therefore, means that the coiled fiber system may be an ideal implementation of realizing the nonadiabatic geometric phase gates for the topological quantum computation. The remarkable feature of the present method is that it can automatically meet the conditions and requirements pr…
We propose a simple but versatile protocol to engineer time-dependent Hamiltonians inversely for geometric quantum computation. By utilizing SU(2) transformation, a speedup goal on gate operation is achieved with more freedom to design the control parameters. As an application, this protocol enables the conventional and unconventional nonadiabatic geometric quantum gates with desired evolution paths by controlling the microwave pulses in the diamond nitrogen-vacancy center system. We show that the inversely designed Hamiltonian can fulfill the geometric gate with more economical evolution time…
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…
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…
We propose a protocol to realize nonadiabatic geometric quantum computation of small-amplitude Schrödinger cat qubits via invariant-based reverse engineering. We consider a system with a two-photon driven Kerr nonlinearity, which provides a pair of dressed even and odd coherent states, i.e., Schrödinger cat states for fault-tolerant quantum computations. An additional coherent field is applied to linearly drive a cavity mode, to induce oscillations between dressed cat states. By designing this linear drive with invariant-based reverse engineering, nonadiabatic geometric quantum computation wit…
Producing and maintaining entanglement reside at the heart of the optimal construction of quan- tum operations and are fundamental issues in the realization of universal quantum computation. We here introduce a setup of spin qubits that allows for geometric implementation of entangling gates between the register qubits with any arbitrary entangling power. We show this by demon- strating a circuit through a spin chain, which performs universal nonadiabatic holonomic two-qubit entanglers. The proposed gates are all electric and geometric, which would help to realize fast and robust entangling ga…
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 $Λ$ system: paths belonging to the original two-loop scheme [New J. Phys. {\bf 14}, 103035 (2012)], the single-loop multiple-pulse scheme [Phys. Rev. A {\bf 94}…
Noise-resilient nonadiabatic geometric quantum computation for bosonic binomial codes
Fujian Key Laboratory of Quantum Information and Quantum Optics, College of Physics and Information Engineering, Fuzhou University（福州大学物理与信息工程学院）；School of Physics, Hangzhou Normal University（杭州师范大学物理学院）；Center for Joint Quantum Studies and Department of Physics, School of Science, Tianjin University（天津大学理学院）；Institute of Quantum Science and Technology, Yanbian University（延边大学量子科学与技术研究院）；Quantum Information Physics Theory Research Team, Center for Quantum Computing, RIKEN（理研量子计算中心）
Quantum gates induced by geometric phases are intrinsically robust against noise due to their global properties of the evolution paths. Compared to conventional nonadiabatic geometric quantum computation (NGQC), the recently proposed nonadiabatic noncyclic geometric quantum computation (NNGQC) works in a faster fashion, while still remaining the robust feature of the geometric operations. Here, we experimentally implement the NNGQC in a single trapped ultracold $^{40}$Ca$^{+}$ ion for verifying the noise-resilient and fast feature. By performing unitary operations under imperfect conditions, w…
Quantum computation based on nonadiabatic geometric phases has attracted a broad range of interests, due to its fast manipulation and inherent noise resistance. However, it is limited to some special evolution paths, and the gate-times are typically longer than conventional dynamical gates, resulting in weakening of robustness and more infidelities of the implemented geometric gates. Here, we propose a path-optimized scheme for geometric quantum computation on superconducting transmon qubits, where high-fidelity and robust universal nonadiabatic geometric gates can be implemented, based on con…
Nonadiabatic geometric quantum computation (NGQC) has been developed to realize fast and robust geometric gate. However, the conventional NGQC is that all of the gates are performed with exactly the sameamount of time, whether the geometric rotation angle is large or small, due to the limitation of cyclic condition. Here, we propose an unconventional scheme, called nonadiabatic noncyclic geometric quantum computation(NNGQC), that arbitrary single- and two-qubit geometric gate can be constructed via noncyclic non-Abeliangeometric phase. Consequently, this scheme makes it possible to accelerate …
Composite nonadiabatic geometric quantum gates with optimization on superconducting circuits
School of Mathematics and Physics, Anqing Normal University（安庆师范大学数学与物理学院）；School of Electronic Engineering and Intelligent Manufacturing, Anqing Normal University（安庆师范大学电子工程与智能制造学院）；School of Electrical and Photoelectronic Engineering, West Anhui University（皖西大学电气与光电工程学院）；Key Laboratory of Atomic and Subatomic Structure and Quantum Control (Ministry of Education), Guangdong Basic Research Center of Excellence for Structure and Fundamental Interactions of Matter, and School of Physics, South China Normal University（华南师范大学原子和亚原子结构及量子控制教育部重点实验室、广东省物质结构和基本相互作用卓越研究中心、物理学院）；Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials, Guangdong-Hong Kong Joint Laboratory of Quantum Matter, and Frontier Research Institute for Physics, South China Normal University（华南师范大学广东省量子工程与量子材料重点实验室、粤港澳量子物质联合实验室、前沿物理研究院）
Quantum manipulation based on geometric phases provides a promising way towards robust quantum gates. However, in the current implementation of nonadiabatic geometric phases, operational and/or random errors tend to destruct the conditions that induce geometric phases, thereby smearing their noise-resilient feature. In a recent experiment [Y. Xu et al., Phys. Rev. Lett. 124, 230503 (2020)], high-fidelity universal geometric quantum gates have been implemented in a superconducting circuit, which are robust to different types of errors under different configurations of the geometric evolution pa…
Using geometric phases to realize noise-resilient quantum computing is an important method to enhance the control fidelity. In this work, we experimentally realize a universal nonadiabatic geometric quantum gate set in a superconducting qubit chain. We characterize the realized single- and two-qubit geometric gates with both quantum process tomography and randomized benchmarking methods. The measured average fidelities for the single-qubit rotation gates and two-qubit controlled-Z gate are 0.9977(1) and 0.977(9), respectively. Besides, we also experimentally demonstrate the noise-resilient fea…
Quantum gates, which are the essential building blocks of quantum computers, are very fragile. Thus, to realize robust quantum gates with high fidelity is the ultimate goal of quantum manipulation. Here, we propose a nonadiabatic geometric quantum computation scheme on superconducting circuits to engineer arbitrary quantum gates, which share both the robust merit of geometric phases and the capacity to combine with optimal control technique to further enhance the gate robustness. Specifically, in our proposal, arbitrary geometric single-qubit gates can be realized on a transmon qubit, by a res…
Besides the intrinsic noise resilience property, nonadiabatic geometric phases are of the fast evolution nature, and thus can naturally be used in constructing quantum gates with excellent performance, i.e., the so-called nonadiabatic geometric quantum computation (NGQC). However, previous single-loop NGQC schemes are sensitive to the operational control error, i.e., the $X$ error, due to the limitations of the implementation. Here, we propose a robust scheme for NGQC combining with the dynamical correction technique, which still uses only simplified pulses, and thus being experimental friendl…
Quantum gates based on geometric phases possess intrinsic noise-resilience features and therefore attract much attention. However, the implementations of previous geometric quantum computation typically require a long pulse time of gates. As a result, their experimental control inevitably suffers from the cumulative disturbances of systematic errors due to excessive time consumption. Here, we experimentally implement a set of noncyclic and nonadiabatic geometric quantum gates in a superconducting circuit, which greatly shortens the gate time. And also, we experimentally verify that our univers…
Quantum computation has demonstrated advantages over classical computation for special hard problems, where a set of universal quantum gates is essential. Geometric phases, which have built-in resilience to local noise, have been used to construct quantum gates with excellent performance. However, this advantage has been smeared in previous schemes. Here, we propose a state-independent nonadiabatic geometric quantum-gate scheme that is able to realize a more fully geometric gate than previous approaches, allowing for the cancelation of dynamical phases accumulated by an arbitrary state. Numeri…
We analyse the geometric phase due to the Stark shift in a system composed of a bosonic field, driven by time-dependent linear amplification, interacting dispersively with a two-level (fermionic) system. We show that a geometric phase factor in the joint state of the system, which depends on the fermionic state (resulting form the Stark shift), is introduced by the amplification process. A clear geometrical interpretation of this phenomenon is provided. We also show how to measure this effect in an interferometric experiment and to generate geometric "Schrodinger cat"-like states. Finally, con…
Geometric phases are only dependent on evolution paths but independent of evolution details so that they own some intrinsic noise-resilience features. Based on different geometric phases, various quantum gates have been proposed, such as nonadiabatic geometric gates based on nonadiabatic Abelian geometric phases and nonadiabatic holonomic gates based on nonadiabatic non-Abelian geometric phases. Up to now, nonadiabatic holonomic one-qubit gates have been experimentally demonstrated with the supercondunting transmon, where three lowest levels with cascaded configuration are all applied in the o…
Fast quantum spin manipulation is needed to design spin-based quantum logic gates and other quantum applications. Here, we construct exact evolution operator of the nitrogen-vacancy-center (NV) spin in diamond under external magnetic fields and investigate the nonadiabatic geometric phases, both cyclic and non-cyclic, in these fast-manipulated NV spin systems. It is believed that the nonadiabatic geometric phases can be measured in future experiments and these fast quantum manipulations can be useful in designing spin-based quantum applications.
The spin in a rotating frame has attracted a lot of attentions recently, as it deeply relates to both fundamental physics such as pseudo-magnetic field and geometric phase, and applications such as gyroscopic sensors. However, previous studies only focused on adiabatic limit, where the rotating frequency is much smaller than the spin frequency. Here we propose to use a levitated nano-diamond with a built-in nitrogen-vacancy (NV) center to study the dynamics and the geometric phase of a rotating electron spin without adiabatic approximation. We find that the transition between the spin levels a…
It is well known that any cyclic solution of a spin 1/2 neutral particle moving in an arbitrary magnetic field has a nonadiabatic geometric phase proportional to the solid angle subtended by the trace of the spin. For neutral particles with higher spin, this is true for cyclic solutions with special initial conditions. For more general cyclic solutions, however, this does not hold. As an example, we consider the most general solutions of such particles moving in a rotating magnetic field. If the parameters of the system are appropriately chosen, all solutions are cyclic. The nonadiabatic geome…
Nonadiabatic holonomic quantum computation (NHQC) is implemented by fast evolution processes in a geometric way to withstand local noises. However, recent works of implementing NHQC are sensitive to the systematic noise and error. Here, we present a path-optimized NHQC (PONHQC) scheme based on the non-Abelian geometric phase, and find that a geometric gate can be constructed by different evolution paths, which have different responses to systematic noises. Due to the flexibility of the PONHQC scheme, we can choose an optimized path that can lead to excellent gate performance. Numerical simulat…
The geometric phases of the cyclic states of a generalized harmonic oscillator with nonadiabatic time-periodic parameters are discussed in the framework of squeezed state. A class of cyclic states are expressed as a superposition of an infinte number of squeezed states. Then, their geometric phases are obtained explicitly and found to be $-(n+1/2)$ times the classical nonadiabatic Hannay angle. It is shown that the analysis based on squeezed state approach provide a clear picture of the geometric meaning of the quantal phase.
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 …
We show that the adiabatic approximation for nonselfadjoint hamiltonians seems to induce two non-equal expressions for the geometric phase. The first one is related to the spectral projector involved in the adiabatic theorem, the other one is the adiabatic limit of the nonadiabatic geometric phase. This apparent inconsistency is resolved by observing that the difference between the two expressions is compensated by a small deviation in the dynamical phases.
Geometric quantum computation is the idea that geometric phases can be used to implement quantum gates, i.e., the basic elements of the Boolean network that forms a quantum computer. Although originally thought to be limited to adiabatic evolution, controlled by slowly changing parameters, this form of quantum computation can as well be realized at high speed by using nonadiabatic schemes. Recent advances in quantum gate technology have allowed for experimental demonstrations of different types of geometric gates in adiabatic and nonadiabatic evolution. Here, we address some conceptual issues …
Based only on the parallel transport condition, we present a general method to compute Abelian or non-Abelian geometric phases acquired by the basis states of pure or mixed density operators, which also holds for nonadiabatic and noncyclic evolution. Two interesting features of the non-Abelian geometric phase obtained by our method stand out: i) it is a generalization of Wilczek and Zee's non-Abelian holonomy, in that it describes nonadiabatic evolution where the basis states are parallelly transported between distinct degenerate subspaces, and ii) the non-Abelian character of our geometric ph…
We study the Landau-Zener Problem for a decaying two-level-system described by a non-hermitean Hamiltonian, depending analytically on time. Use of a super-adiabatic basis allows to calculate the non-adiabatic transition probability P in the slow-sweep limit, without specifying the Hamiltonian explicitly. It is found that P consists of a ``dynamical'' and a ``geometrical'' factors. The former is determined by the complex adiabatic eigenvalues E_(t), only, whereas the latter solely requires the knowledge of α_(+-)(t), the ratio of the components of each of the adiabatic eigenstates. Both factors…
The nonadiabatic geometric quantum computation is promising as it is robust against certain types of local noises. However, its experimental implementation is challenging due to the need of complex control on multi-level and/or multiple quantum systems. Here, we propose to implement it on a two-dimensional square superconducting qubit lattice. In the construction of our geometric quantum gates, we only use the simplest and experimentally accessible control over the qubit states of the involved quantum systems, without introducing any auxiliary state. Specifically, our scheme is achieved by par…
Geometric quantum gates are performed by using the geometric phase, making them particularly robust to the pulse amplitude error due to the intrinsic global property. However, in many systems, such as the silicon-based spin qubits, the off-resonance error is the dominant noise, which can cause dephasing and is always difficult to deal with for a geometric gate. Thus how to deal with the off-resonance error is very significant for the application of the geometric gates. A recent work in \emph{Phy. Rev. Appl. 16, 044005 (2021)} reveals that by inserting two $π$-pulse dynamically corrected sequen…
The geometric phases of the cyclic states of a generalized harmonic oscillator with nonadiabatic time-periodic parameters are discussed in the framework of squeezed state. It is shown that the cyclic and quasicyclic squeezed states correspond to the periodic and quasiperiodic solutions of an effective Hamiltonian defined on an extended phase space, respectively. The geometric phase of the cyclic squeezed state is found to be a phase-space area swept out by a periodic orbit. Furthermore, a class of cyclic states are expressed as a superposition of an infinte number of squeezed states. Their geo…
