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Physical Review A

Geometric quantum computation and dynamical invariant operators

Zhen Wang

Published Feb 6, 2009• 105 citations
Publisher

Abstract

An entangling quantum gate based entirely on purely geometric operations is proposed in quantum computation for the Jaynes-Cummings model by the invariant theory, where the qubits include information about the states of photons. By controlling some arbitrary parameters in the invariant operators, the phase accumulated in the quantum gate is a pure geometric phase. This way may be extended to other physical systems.

DOI
10.1103/physreva.79.024304

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An entangling quantum gate based entirely on purely geometric operations is proposed in quantum computation for the Jaynes-Cummings model by the invariant theory, where the qubits include information about the states of photons. By controlling some arbitrary parameters in the invariant operators, the phase accumulated in the quantum gate is a pure geometric phase. This way may be extended to other physical systems.
A different way to realize nonadiabatic geometric quantum computation is proposed by varying parameters in the Hamiltonian for nuclear-magnetic resonance, where the dynamical and geometric phases are implemented separately without the usual operational process. Therefore the phase accumulated in the geometric gate is a pure geometric phase for any input state. In comparison with the conventional geometric gates by rotating operations, our approach simplifies experimental implementations making them robust to certain experimental errors. In contrast to the unconventional geometric gates, our approach distinguishes the total and geometric phases and offers a wide choice of the relations between the dynamical and geometric phases.
A scheme is proposed to include both cyclic and noncyclic geometric quantum computations in nuclear-magnetic-resonance system by the invariant theory. By controlling magnetic field and arbitrary parameters in the invariant operator, the phases accumulated in the entangling quantum gates for single- and two-qubit systems are pure geometric phases. Thus, fault tolerance may occur in some critical magnetic field parameters for either cyclic or noncyclic evolution by differently choosing for gate time.
A fundamental challenge of quantum computation is being able to scale up a large number of high fidelity quantum gates while noise and error are affecting the gate’s physical control parameters. This dissertation focuses on the fidelity of single-qubit quantum gates constructed by a change in quantum geometric phase, while the control parameters are affected by random noise and systematic error. A unified model of geometric quantum computation is developed, in which a qubit state is controlled by a composite Hamiltonian, resulting in a multiple-segment rotation of the quantum state and allowing characterization of evolution paths depending on the associated geometric and dynamic phase. The fidelity of the quantum gates in the presence of different noise error is compared for pu rely geometric, hybrid (having both geometric and dynamic phase), and conventional dynamic quantum gates built on single Hamiltonians. Results showed hybrid quantum gates had the highest fidelities, followed by geometric gates, and conventional dynamic gates had the lowest fidelities. In addition, there was indication in some cases higher fidelities result from gates created from a larger number of segments in the quantum state rotation. These results can be understood by the relation of the control parameters with the evolution path geometry. By translating between control parameters, our model can be applied to different systems for quantum computation, including: the laser manipulation of a two-level atom, laser manipulation of trapped ions, nuclear magnetic resonance, polarization states of photons, superconducting qubits in cavity QED, and quantum dots.
We propose a strategy to physically implement a universal set of quantum gates based on geometric phases accumulated in the nondegenerate eigenstates of a designated invariant operator in a periodic physical system. The system is driven to evolve in such a way that the dynamical phase shifts of the invariant operator eigenstates are identical (or mod $2\ensuremath{\pi}$) while the corresponding geometric phases are nontrivial. We illustrate how this strategy works in a simple but typical NMR-type qubit system.
Geometric phases arise naturally in a variety of quantum systems with observable consequences. They also arise in quantum computations when dressed states are used in gating operations. Here we show how they arise in these gating operations and how one may take advantage of the dressed states producing them. Specifically, we show that for a given, but arbitrary Hamiltonian, and at an arbitrary time $\ensuremath{\tau}$, there always exists a set of dressed states such that a given gate operation can be performed by the Hamiltonian up to a phase $\ensuremath{\varphi}$. The phase is a sum of a dynamical phase and a geometric phase. We illustrate the dressed phase for several systems.