A. E. Margolin, V. I. Strazhev, A. Ya. Tregubovich
Calculation aspects of holonomic quantum computer (HQC) are considered. Wilczek--Zee potential defining the set of quantum calculations for HQC is explicitly evaluated. Principal possibility of realization of the logical gates for this case is discussed.
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Geometric measure of entanglement and geometric phase have recently been used to analyze quantum phase transition in the XY spin chain. We unify these two approaches by showing that the geometric entanglement and the geometric phase are respectively the real and imaginary parts of a complex-valued geometric entanglement, which can be investigated in typical quantum interferometry experiments. We argue that the singular behavior of the complex-value geometric entanglement at a quantum critical point is a characteristic of any quantum phase transition, by showing that the underlying mechanism is…
Nodal free geometric phases are the eigenvalues of the final member of a parallel transporting family of unitary operators. These phases are gauge invariant, always well-defined, and can be measured interferometrically. Nodal free geometric phases can be used to construct various types of quantum phase gates.
We propose an experimentally feasible scheme to achieve quantum computation based solely on geometric manipulations of a quantum system. The desired geometric operations are obtained by driving the quantum system to undergo appropriate adiabatic cyclic evolutions. Our implementation of the all-geometric quantum computation is based on laser manipulation of a set of trapped ions. An all-geometric approach, apart from its fundamental interest, promises a possible way for robust quantum computation.
In this letter, the generalization of geometric phase in density matrix is presented, we show that the extended sub-geometric phase have unified expression whatever in adiabatic or nonadiabatic procedure, the relations between them and the usual Berry phase or Aharonov-Anandan phase are established. We also demonstrated the influence of sub-geometric phases on the physical observables. Finally, our treatment is naturally used to investigate the geometric phase in mixed state.
We propose a new class of unconventional geometric gates involving nonzero dynamic phases, and elucidate that geometric quantum computation can be implemented by using these gates. Comparing with the conventional geometric gate operation, in which the dynamic phase shift must be removed or avoided, the gates proposed here may be operated more simply. We illustrate in detail that unconventional nontrivial two-qubit geometric gates with built-in fault tolerant geometric features can be implemented in real physical systems.
The connection between the geometric phase and quantum phase transition has been discussed extensively in the two-band model. By introducing the twist operator, the geometric phase can be defined by calculating its ground-state expectation value. In contrast to the previous numerical examinations, our discussion presents an exact calculation for the determination of the geometric phase. Through two representative examples, our calculation shows the intimate connection between the geometric phase and phase transition: different behaviors of the geometric phase can be identified in this paper, w…
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…
Beyond the quantum Markov approximation, we calculate the geometric phase of a two-level system driven by a quantized magnetic field subject to phase dephasing. The phase reduces to the standard geometric phase in the weak coupling limit and it involves the phase information of the environment in general. In contrast with the geometric phase in dissipative systems, the geometric phase acquired by the system can be observed on a long time scale. We also show that with the system decohering to its pointer states, the geometric phase factor tends to a sum over the phase factors pertaining to the …
We explore geometric phases of coherent states and some of their properties. A better and elegant expression of geometric phase for coherent state is derived. It is used to obtain the explicit form of the geometric phase for entangled coherent states, several interesting results followed by considering different cases for the parameters. The effects of entanglement and harmonic potential on the geometric phase are discussed.
We show that the geometric phase of Levy-Leblond arises from a low of parallel transport for wave functions and point out that this phase belongs to a new class of geometric phases due to the presence of a quantum potential.
The relation between the geometric phase and quantum phase transition has been discussed in the Lipkin-Meshkov-Glick model. Our calculation shows the ability of geometric phase of the ground state to mark quantum phase transition in this model. The possibility of the geometric phase or its derivatives as the universal order parameter of characterizing quantum phase transitions has been also discussed.
The geometric phase of a bi-particle model is discussed. For different initial states, especially when the initial state is pure or mixed, the geometric phase will show different properties. The relationship between the geometric phase and the entanglement or mixedness of the initial state is considered.
We study the geometric phase of a two-level atom coupled to an environment with Lorentzian spectral density. The non-Markovian effect on the geometric phase is explored analytically and numerically. In the weak coupling limit the lowest-order correction to the geometric phase is derived analytically and the general case is calculated numerically. It is found that the correction to the geometric phase is significantly large if the spectral width is small and in this case the non-Markovian dynamics has a significant impact to the geometric phase. When the spectral width increases, the correction…
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 …
We examine the adiabatic dynamics of a quantum system coupled to a noisy classical control field. A stochastic phase shift is shown to arise in the off-diagonal elements of the system's density matrix which can cause decoherence. We derive the condition for onset of decoherence, and identify the noise properties that drive decoherence. We show how this decoherence mechansim causes: (1) a dephasing of the observable consequences of the adiabatic geometric phase; and (2) the loss of computational efficiency of the Shor algorithm when run on a sufficiently noisy geometric quantum computer.
The geometric phase associated with a many body ground state exhibits a signature of quantum phase transition. In this context, we have studied the behaviour of the geometric phase during a linear quench caused by a gradual turning off of the magnetic field interacting with a spin chain.
We interpret quantum computing as a geometric evolution process by reformulating finite quantum systems via Connes' noncommutative geometry. In this formulation, quantum states are represented as noncommutative connections, while gauge transformations on the connections play a role of unitary quantum operations. Thereby, a geometrical model for quantum computation is presented, which is equivalent to the quantum circuit model. This result shows a geometric way of realizing quantum computing and as such, provides an alternative proposal of building a quantum computer.
It is shown that geometric phase in non-relativistic quantum mechanics is not Galilean invariant.
In this paper, the off-diagonal geometric phase of thermal state in hydrogen atom under the effects of external magnetic field is considered. Increasing temperature tends to suppress the off-diagonal geometric phase, including 1-order and 2-order cases. On the other hand, the relationship between the geometric phase and external magnetic field is discussed.
A general scheme for an adiabatic geometric phase gate is proposed which is maximally robust against parameter fluctuations. While in systems with SU(2) symmetry geometric phases are usually accompanied by dynamical phases and are thus not robust, we show that in the more general case of a SU(2) x SU(2) symmetry it is possible to obtain a non-vanishing geometric phase without dynamical contributions. The scheme is illustrated for a phase gate using two systems with dipole-dipole interactions in external laser fields which form an effective four-level system.
All the geometric phases, adiabatic and non-adiabatic, are formulated in a unified manner in the second quantized path integral formulation. The exact hidden local symmetry inherent in the Schrödinger equation defines the holonomy. All the geometric phases are shown to be topologically trivial. The geometric phases are briefly compared to the chiral anomaly which is naturally formulated in the path integral.
We propose a non-adiabatic scheme for geometric quantum computation with trapped ions. By making use of the Aharonov-Anandan phase, the proposed scheme not only preserves the globally geometric nature in quantum computation, but also provides the advantage of non-adiabaticity that overcomes the problem of slow evolution in the existing adiabatic schemes. Moreover, the present scheme requires only two atomic levels in each ion, making it an appealing candidate for quantum computation.
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 …
We consider stimulated Raman adiabatic passage (STIRAP) processes in tripod systems and show how to generate purely geometric phase changes of the quantum states involved. The geometric phases are controlled by three laser fields where pulse shapes, relative field strength and phases can be controlled. We present a robust set of universal gates for quantum computing based on these geometric phases: a one-qubit phase gate, a Hadamard gate and a two-qubit phase gate.
The concept of relative state is used to introduce geometric phases that originate from correlations in states of composite quantum systems. In particular, we identify an entanglement-induced geometric phase in terms of a weighted average of geometric phase factors associated with a decomposition that define the entanglement of formation. An explicit procedure to calculate the entanglement-induced geometric phase for qubit pairs is put forward. We illustrate it for maximally entangled mixed states (MEMS) of two qubits.
The applications of geometric control theory methods on Lie groups and homogeneous spaces to the theory of quantum computations are investigated. These methods are shown to be very useful for the problem of constructing an universal set of gates for quantum computations: the well-known result that the set of all one-bit gates together with almost any one two-bit gate is universal is considered from the control theory viewpoint.
We show that the geometric phase between any two states, including orthogonal states, can be computed and measured using the notion of projective measurement, and we show that a topological number can be extracted in the geometric phase change in an infinitesimal loop near an orthogonal state. Also, the Pancharatnam phase change during the passage through an orthogonal state is shown to be either $π$ or zero (mod $2π$). All the off-diagonal geometric phases can be obtained from the projective geometric phase calculated with our generalized connection.
We obtain the ground-state energy level and associated geometric phase in the Dicke model analytically by means of the Holstein-Primakoff transformation and the boson expansion approach in the thermodynamic limit. The non-adiabatic geometric phase induced by the photon field is derived with the time-dependent unitary transformation. It is shown that the quantum phase transition characterized by the non-analyticity of the geometric phase is remarkably of the first-order. We also investigate the scaling behavior of the geometric phase at the critical point, which can be measured in a practical e…
We examine the effect of spin-orbit coupling on geometric phases in hydrogenlike atoms exposed to a slowly varying magnetic field. The marginal geometric phases associated with the orbital angular momentum and the intrinsic spin fulfill a sum rule that explicitly relates them to the corresponding geometric phase of the whole system. The marginal geometric phases in the Zeeman and Paschen-Back limit are analyzed. We point out the existence of nodal points in the marginal phases that may be detected by topological means.
Time-resolved Faraday rotation spectroscopy is currently exploited as a powerful technique to probe spin dynamics in semiconductors. We propose here an all-optical approach to geometrically manipulate electron spin and to detect the geometric phase by this type of extremely sensitive experiment. The global nature of the geometric phase can make the quantum manipulation more stable, which may find interesting application in quantum devices.
We investigate the geometric phase of a two-level atom (qubit) coupled to a bosonic reservoir with Lorentzian spectral density, and find that for the non-Markovian dynamics in which rotating-wave approximation (RWA) is performed, geometric phase has a $π$-phase jump at the nodal point. However, the exact result without RWA given by hierarchy equation of motion method shows that there is no such a phase jump or nodal structure in geometric phase. Thus our results demonstrate that the counter-rotating terms significantly contribute to the geometric phase in multi-mode Hamiltonian under certain c…
Geometric phase has been proposed as one of the promising methodologies to perform fault tolerant quantum computations. However, since decoherence plays a crucial role in such studies, understanding of mixed state geometric phase has become important. While mixed state geometric phase was first introduced mathematically by Uhlmann, recently Sjoqvist et al. [Phys. Rev. Lett. 85(14), 2845 (2000)] have described the mixed state geometric phase in the context of quantum interference and shown theoretically that the visibility as well as the shift of the interference pattern are functions of geomet…
In this paper, the geometric phase of thermal state in hydrogen atom under the effects of external magnetic field is considered. Especially the effects of the temperature upon the geometric phase is discussed. Also we discuss the time evolution of entanglement of the system. They show some similar behaviors.
The properties of the geometric phases between three quantum states are investigated in a high-dimensional Hilbert space using the Majorana representation of symmetric quantum states. We found that the geometric phases between the three quantum states in an N-state quantum system can be represented by N-1 spherical triangles on the Bloch sphere. The parameter dependence of the geometric phase was analyzed based on this picture. We found that the geometric phase exhibits rich nonlinear behavior in a high-dimensional Hilbert space.
Geometric phase for a nonstatic coherent light-wave: nonlinear evolution harmonized with the dynamical phase
本文分析了静态环境中非静态相干光波的几何相位，发现其非线性演化与动力学相位协调，总相位呈现依赖非静态程度的独特模式，且相干态几何相位更显著。
The geometric phases of cyclic evolutions for mixed states are discussed in the framework of unitary evolution. A canonical one-form is defined whose line integral gives the geometric phase which is gauge invariant. It reduces to the Aharonov and Anandan phase in the pure state case. Our definition is consistent with the phase shift in the proposed experiment [Phys. Rev. Lett. \textbf{85}, 2845 (2000)] for a cyclic evolution if the unitary transformation satisfies the parallel transport condition. A comprehensive geometric interpretation is also given. It shows that the geometric phases for mi…
Geometric phases have been shown to be feasible in implementing quantum gates to perform quantum information processing. For all the realistic applications, the environmental influence on the geometric phase and decoherence such as memory effects must be properly considered in order to achieve the required precision in geometric quantum computation. In this paper, we study the geometric phase for a generic open quantum system based on a microscopic model. A remarkable feature of the open system's geometric phase obtained from our theoretical formulation is that the geometric phase can be obtai…
The influence of intersubsystem coupling on the cyclic adiabatic geometric phase in bipartite systems is investigated. We examine the geometric phase effects for two uniaxially coupled spin$-{1/2}$ particles, both driven by a slowly rotating magnetic field. It is demonstrated that the relation between the geometric phase and the solid angle enclosed by the magnetic field is broken by the spin-spin coupling, in particular leading to a quenching effect on the geometric phase in the strong coupling limit.
Everett's concept of relative state is used to introduce a geometric phase that depends nontrivially on entanglement in a pure quantum state. We show that this phase can be measured in multiparticle interferometry. A correlation-dependent generalization of the relative state geometric phase to mixed quantum states is outlined.
Geometric phases of reduced states in the transverse-field Ising chain
本文通过计算横向场伊辛模型中两体位点约化密度矩阵的干涉几何相位和Uhlmann相位，发现干涉相位比Uhlmann相位更能可靠地指示该模型的量子相变。
We extend the off-diagonal geometric phase [Phys. Rev. Lett. {\bf 85}, 3067 (2000)] to mixed quantal states. The nodal structure of this phase in the qubit (two-level) case is compared with that of the diagonal mixed state geometric phase [Phys. Rev. Lett. {\bf 85}, 2845 (2000)]. Extension to higher dimensional Hilbert spaces is delineated. A physical scenario for the off-diagonal mixed state geometric phase in polarization-entangled two-photon interferometry is proposed.
In an open system, the geometric phase should be described by a distribution. We show that a geometric phase distribution for open system dynamics is in general ambiguous, but the imposition of reasonable physical constraints on the environment and its coupling with the system yields a unique geometric phase distribution that applies even for mixed states, non-unitary dynamics, and non-cyclic evolutions.
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.
We study the general-setting quantum geometric phase acquired by a particle in a vibrating cavity. Solving the two-level theory with the rotating-wave approximation and the SU(2) method, we obtain analytic formulae that give excellent descriptions of the geometric phase, energy, and wavefunction of the resonating system. In particular, we observe a sudden $π$-jump in the geometric phase when the system is in resonance. We found similar behaviors in the geometric phase of a spin-1/2 particle in a rotating magnetic field, for which we developed a geometrical model to help visualize its evolution…
The theory of geometric phase is generalized to a cyclic evolution of the eigenspace of an invariant operator with $N$-fold degeneracy. The corresponding geometric phase is interpreted as a holonomy inherited from the universal connection of a Stiefel U(N)-bundle over a Grassmann manifold. Most significantly, for an arbitrary initial state, this geometric phase captures the inherent geometric feature of the state evolution. Moreover, the geometric phase in the evolution of the eigenspace of an adiabatic action operator is also addressed, which is elaborated by a pullback U(N)-bundle. Several i…
Bargmann invariants and null phase curves are known to be important ingredients in understanding the essential nature of the geometric phase in quantum mechanics. Null phase manifolds in quantum-mechanical ray spaces are submanifolds made up entirely of null phase curves, and so are equally important for geometric phase considerations. It is shown that the complete characterization of null phase manifolds involves both the Riemannian metric structure and the symplectic structure of ray space in equal measure, which thus brings together these two aspects in a natural manner.
Garrison and Wright showed that upon undergoing cyclic quantum evolution a meta-stable state acquires both a geometric phase and a geometric decay probability. This is described by a complex geometric ``phase'' associated with the cyclic evolution of two states and is closely related to the two state formalism developed by Aharonov et al.. Applications of the complex geometric phase to the Born--Oppenheimer approximation and the Aharonov--Bohm effect are considered. A simple experiment based on the optical properties of absorbing birefringent crystals is proposed.
We develop the widest possible generalisation of the well-known connection between quantum mechanical Bargmann invariants and geometric phases. The key notion is that of null phase curves in quantum mechanical ray and Hilbert spaces. Examples of such curves are developed. Our generalisation is shown to be essential to properly understand geometric phase results in the cases of coherent states and of Gaussian states. Differential geometric aspects of null phase curves are also briefly explored.
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…
The behavior of the geometric phase gained by a single spin-1/2 nucleus immersed into a thermal or a squeezed environment is investigated. Both the time dependence of the phase and its value at infinity are examined against several physical parameters. It is observed that for some intermediate ranges of the temperature and the coupling strength, the presence of squeezing enhances the geometric phase.
The geometric phases of a two-level atom interacting with non-Markovian environments are calculated and the non-Markovian effects on the geometric phases are discussed in this paper. Three kinds of methods that describe the non-Markovian process, projection superoperator technique, memory kernel master equation and post-Markovian master equation, are used in the discussions. The results show that when the dissipation rate is large, the non-Markovian effects change the geometric phase more strikingly than the small one.
This paper presents an alternative approach to geometric phases from the observable point of view. Precisely, we introduce the notion of observable-geometric phases, which is defined as a sequence of phases associated with a complete set of eigenstates of the observable. The observable-geometric phases are shown to be connected with the quantum geometry of the observable space evolving according to the Heisenberg equation. They are indeed distinct from Berry's phase \cite{Berry1984, Simon1983} as the system evolves adiabatically. It is shown that the observable-geometric phases can be used to …
The geometric phases for standard coherent states which are widely used in quantum optics have attracted a large amount of attention. Nevertheless, few physicists consider about the counterparts of non-linear coherent states, which are useful in the description of the motion of a trapped ion. In this paper, the non-unitary and non-cyclic geometric phases for two nonlinear coherent and one squeezed states are formulated respectively. Moreover, some of their common properties are discussed respectively, such as gauge invariance, non-locality and non-linear effects. The non-linear functions have …
This paper has been withdrawn by the author
The geometric phase acquired by the vector states under an adiabatic evolution along a noncyclic path can be calculated correctly in any instantaneous basis of a Hamiltonian that varies in time due to a time-dependent classical field.
An experimentally feasible realization of testing quantum-vacuum geometric phases of photons by using a gyrotropic-medium optical fibre via Casimir's effect is proposed.
Geometric phases of simple harmonic oscillator system are studied. Complete sets of "eigenfunctions" are constructed, which depend on the way of choosing classical solutions. For an eigenfunction, two different motions of the probability distribution function (pulsation of the width and oscillation of the center) contribute to the geometric phase which can be given in terms of the parameters of classical solutions. The geometric phase for a general wave function is also given. If a wave function has a parity under the inversion of space coordinate, then the geometric phase can be defined under…
In this article, we provide theoretical support for the use of geometric measures of nonclassicality as a general tool to identify quantum phase transitions. We argue that divergences in the susceptibility of any geometric measure of nonclassicality are sufficient conditions to identify phase transitions at arbitrary temperature. This establishes that geometric measures of nonclassicality, in any quantum resource theory, are generic tools to investigate phase transitions in quantum systems. At zero temperature, we show that geometric measures of quantum coherence are especially useful for iden…
We examine the dynamic and geometric phases of the electron in quantum mechanics using Hestenes' spacetime algebra formalism. First the standard dynamic phase formula is translated into the spacetime algebra. We then define new formulas for the dynamic and geometric phases that can be used in Hestenes' formalism.
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.
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