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Chinese Physics B

Operating a geometric quantum gate by external controllable parameters

Yinghua Ji, Shihua Cai, Le Jian-Xin, Wang Zisheng

Published Jan 1, 2010• 2 citations
Publisher

Abstract

A scheme to perfectly preserve an initial qubit state in geometric quantum computation is proposed for a single-qubit geometric quantum gate in a nuclear magnetic resonance system. At first, by adjusting some magnetic field parameters, one can let the dynamic phase be proportional to the geometric phase. Then, by controlling the azimuthal angle in the initial state, we may realize a geometric quantum gate whose fidelity is equal to one under cyclic evolution. This means that the quantum information is no distortion in the process of geometric quantum computation.

DOI
10.1088/1674-1056/19/1/010311

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A scheme to perfectly preserve an initial qubit state in geometric quantum computation is proposed for a single-qubit geometric quantum gate in a nuclear magnetic resonance system. At first, by adjusting some magnetic field parameters, one can let the dynamic phase be proportional to the geometric phase. Then, by controlling the azimuthal angle in the initial state, we may realize a geometric quantum gate whose fidelity is equal to one under cyclic evolution. This means that the quantum information is no distortion in the process of geometric quantum computation.
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.
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.
A proposal for applying nonadiabatic geometric phases to quantum computing, called double-loop method [S.-L. Zhu and Z. D. Wang, Phys. Rev. A 67, 022319 (2003)], is demonstrated in a liquid-state nuclear magnetic-resonance quantum computer. Using a spin-echo technique, the original method is modified so that quantum gates are implemented in a standard high-precision nuclear magnetic-resonance system for chemical analysis. We show that a dynamical phase is successfully eliminated and a one-qubit quantum gate is realized although the gate fidelity is not high.
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.
Fast and robust quantum gates are the cornerstones of fault-tolerance quantum computation. In this paper, as opposed to the previous proposals which use cyclic geometric phases, we propose to realize quantum computation based on noncyclic geometric evolution. The superiority of the noncyclic scheme is that the operation time is proportional to the rotation angle of the quantum state. Therefore, the noncyclic scheme becomes fairly fast in the case of quantum gates with small rotation angle, which will be more insensitive to the decoherence and the leakage to the states outside the computational basis. Similar to the cyclic version of geometric quantum computation, dynamical phases here during the evolution are canceled by spin-echo process and the adiabatic control can be sped up through shortcut to adiabaticity. The proposed scheme is robust against random noise due to the geometric characteristic of projective Hilbert space. Since the proposed scheme is fast and robust, it is a particularly suitable way to manipulate the physical systems with weak nonlinearity, such as superconducting systems.