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Physical Review A

Noncyclic geometric quantum computation in a nuclear-magnetic-resonance system

Z. S. Wang, G.Q. Liu, Y. H. Ji

Published May 1, 2009• 88 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.79.054301

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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 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.
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.
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 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.