Geometric Quantum Computation Research Articles

Berry phase in Tavis-Cummings model

In this paper we investigate the Berry phase in Tavis-Cummings model in the rotating wave approximation. The dipole-dipole interaction between the atoms is considered. The eigenfunctions of the system are obtained and thus the Berry phase is evaluated explicitly in terms of the introduction of the phase shift. It is shown that the Berry phase can be easily controlled by the atom-cavity coupling strength, the cavity frequency detuning, which can be important in applications in geometric quantum computing.

Scheme for unconventional geometric quantum computation in cavity QED

We present a scheme for implementing the unconventional geometric two-qubit phase gate with nonzero dynamical phase by using the two-channel Raman interaction of two atoms in a cavity. We show that the dynamical phase acquired in a cyclic evolution is proportional to the geometric phase acquired in the same cyclic evolution, hence the the total phase possesses the same geometric features as the geometric phase. In our scheme the atomic excited state is adiabatically eliminated and the operation of the proposed logic gate involves only in the metastable states of the atom and hence is not affected by spontaneous emission.

The Berry phase in GaAs semiconductor with a quantized field

In this paper we investigate the Berry phase in GaAs semiconductor with a quantizedmagnetic field in the rotating wave approximation. The eigenfunctions of the nuclear spinin the quantized external field are obtained and thus the Berry phase is evaluated explicitlyin terms of the introduction of the phase shift. It is shown that the Berry phase can beeasily controlled by the coupling strength, the anisotropy constant and the frequency of theelectromagnetic wave, which can be important in applications in geometric quantumcomputing.

Influence from cavity decay on geometric quantum computation in the large-detuning cavity QED model

We introduce a general displacement operator to investigate the unconventional geometric quantum computation with dissipation under the model of many identical three-level atoms in a cavity, driven by a classical field. Our concrete calculation is made for the case of two atoms, based on a previous scheme [S.-B. Zheng, Phys. Rev. A 70, 052320 (2004)] for the large-detuning interaction of the atoms with the cavity mode. The analytical results we present will be helpful for experimental realization of geometric quantum computation in real cavities.

Experimental implementation of high-fidelity unconventional geometric quantum gates using an NMR interferometer

Following a key idea of unconventional geometric quantum computation developed earlier [Phys. Rev. Lett. 91, 197902 (2003)], here we propose a more general scheme in such an intriguing way: $\gamma_{d}=\alpha_{g}+\eta \gamma _{g}$, where $\gamma_{d}$ and $\gamma_{g}$ are respectively the dynamic and geometric phases accumulated in the quantum gate operation, with $\eta$ as a constant and $\alpha_{g}$ being dependent only on the geometric feature of the operation. More arrestingly, we demonstrate the first experiment to implement a universal set of such kind of generalized unconventional geometric quantum gates with high fidelity in an NMR system.

Berry phase of nuclear spins in GaAs semiconductor

In this Letter, we investigate the Berry phase of an anisotropy nuclear spin- 3 2 in GaAs semiconductor driven by an external magnetic field. The new observation is that the Berry phase can be controlled by both the magnitude of applied field and the anisotropy constant, which is useful in the geometric phase quantum computing.

Temperature effects on mixed-state geometric phase

Geometric phase of an open quantum system that is interacting with a thermal environment (bath) is studied through some simple examples. The system is considered to be a simple spin-half particle which is weakly coupled to the bath. It is seen that even in this regime the geometric phase can vary with temperature. In addition, we also consider the system under an adiabatically time-varying magnetic field which is weakly coupled to the bath. An important feature of this model is that it reveals existence of a temperature scale in which adiabaticity condition is preserved and beyond which the geometric phase is varying quite rapidly with temperature. This temperature is exactly the one in which the geometric phase vanishes. This analysis has some implications in realistic implementations of geometric quantum computation.

Nonadiabatic geometric quantum computation using a single-loop scenario

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.

Quantal interferometry with dissipative internal motion

In presence of dissipation, quantal states may acquire complex-valued phase effects. We suggest a notion of dissipative interferometry that accommodates this complex-valued structure and that may serve as a tool for analyzing the effect of certain kinds of external influences on quantal interference. The concept of mixed-state phase and concomitant gauge invariance is extended to dissipative internal motion. The resulting complex-valued mixed-state interference effects lead to well-known results in the unitary limit and in the case of dissipative motion of pure quantal states. Dissipative interferometry is applied to fault-tolerant geometric quantum computation.

Perturbative formulation and nonadiabatic corrections in adiabatic quantum-computing schemes

Adiabatic limit is the presumption of the adiabatic geometric quantum computation and of the adiabatic quantum algorithm. But in reality, the variation speed of the Hamiltonian is finite. Here we develop a general formulation of adiabatic quantum computing, which accurately describes the evolution of the quantum state in a perturbative way, in which the adiabatic limit is the zeroth-order approximation. As an application of this formulation, non-adiabatic correction or error is estimated for several physical implementations of the adiabatic geometric gates. A quantum computing process consisting of many adiabatic gate operations is considered, for which the total non-adiabatic error is found to be about the sum of those of all the gates. This is a useful constraint on the computational power. The formalism is also briefly applied to the adiabatic quantum algorithm.

Geometric quantum computation on solid-state qubits

Geometric quantum computation is a scheme to use non-Abelian holonomic operationsrather than the conventional dynamic operations to manipulate quantum states forquantum information processing. Here we propose a geometric quantum computationscheme which can be realized with current technology on nanoscale Josephson junctionnetworks, known as a promising candidate for solid-state quantum computers.

Unconventional geometric quantum computation.

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.

Josephson nanocircuit in the presence of linear quantum noise

We derive the effective hamiltonian for a charge-Josephson qubit in a circuit with no use of phenomenological arguments, showing how energy renormalizations induced by the environment appear with no need of phenomenological counterterms. This analysis may be important for multiqubit systems and geometric quantum computation.

Universal quantum gates based on a pair of orthogonal cyclic states: Application to NMR systems

We propose an experimentally feasible scheme to achieve quantum computation based on a pair of orthogonal cyclic states. In this scheme, quantum gates can be implemented based on the total phase accumulated in cyclic evolutions. In particular, geometric quantum computation may be achieved by eliminating the dynamic phase accumulated in the whole evolution. Therefore, both dynamic and geometric operations for quantum computation are workable in the present theory. Physical implementation of this set of gates is designed for NMR systems. Also interestingly, we show that a set of universal geometric quantum gates in NMR systems may be realized in one cycle by simply choosing specific parameters of the external rotating magnetic fields. In addition, we demonstrate explicitly a multiloop method to remove the dynamic phase in geometric quantum gates. Our results may provide useful information for the experimental implementation of quantum logical gates.

Noncyclic geometric quantum computation

Geometric quantum computation is extended to the noncyclic case. Adiabatic nuclear magnetic resonance implementations of geometric phase-shift gates at any precession angle and their concomitant fault tolerance with respect to the amplitude of the oscillating field are analyzed. Nonadiabatic one-qubit geometric quantum computation is considered from the noncyclic perspective, with particular emphasis on errors in precession angles.

Geometric phase shift in quantum computation using superconducting nanocircuits: Nonadiabatic effects

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.

Nonadiabatic geometric quantum computation with trapped ions

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.

Nonadiabatic Geometric Quantum Computation with Asymmetric Superconducting Quantum Interference Device**The project supported in part by National Natural Science Foundation of China under Grant No. 19975036, and the Foundation of the Science and Technology Committee of Hunan Province of China under Grant No. 21000205

We propose a method of controlling the dc-SQUID (superconducting quantum interference device) system by changing the gate voltages, which controls the amplitude of the fictitious magnetic fields , and the externally applied current that produces the piercing magnetic flux for the dc-SQUID system. We have also introduced a physical model for the dc-SQUID system. Using this physical model, one can obtain the non-adiabatic geometric phase gate for the single qubit and the non-adiabatic conditional geometric phase gate (controlled NOT gate) for the two qubits. It is shown that when the gate voltage and the externally applied current of the dc-SQUID system satisfies an appropriate constraint condition, the charge state evolution can be controlled exactly on a dynamic phase free path. The non-adiabatic evolution of the charge states is given as well.

The Hyperbolic Geometric Structure of the Density Matrix for Mixed State Qubits

Density matrices for mixed state qubits, parametrized by the Bloch vector in the open unit ball of the Euclidean 3-space, are well known in quantum computation theory. We bring the seemingly structureless set of all these density matrices under the umbrella of gyrovector spaces, where the Bloch vector is treated as a hyperbolic vector, called a gyrovector. As such, this article catalizes and supports interdisciplinary research spreading from mathematical physics to algebra and geometry. Gyrovector spaces are mathematical objects that form the setting for the hyperbolic geometry of Bolyai and Lobachevski just as vector spaces form the setting for Euclidean geometry. It is thus interesting, in geometric quantum computation, to realize that the set of all qubit density matrices has rich structure with strong link to hyperbolic geometry. Concrete examples for the use of the gyro-structure to derive old and new, interesting identities for qubit density matrices are presented.

Nonadiabatic conditional geometric phase shift with NMR.

A conditional geometric phase shift gate, which is fault tolerant to certain types of errors due to its geometric nature, was realized recently via nuclear magnetic resonance (NMR) under adiabatic conditions. However, in quantum computation, everything must be completed within the decoherence time. The adiabatic condition makes any fast conditional Berry phase (cyclic adiabatic geometric phase) shift gate impossible. Here we show that by using a newly designed sequence of simple operations with an additional vertical magnetic field, the conditional geometric phase shift gate can be run nonadiabatically. Therefore geometric quantum computation can be done at the same rate as usual quantum computation.