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Yunnan Daxue xuebao. Shehui kexue ban

Quantum gates based on the geometric and topologic phases

Yu-Mei Gao, Hu Lian

Published Jan 1, 2005• 0 citations
OpenAlex

Abstract

This article summarizes the recent results about geometric and topologic quantum computation.Based on a rigorous theorem concerned on orthogonal superposition quantum state proved in this paper,an orthogonal state method is proposed to carry out the non-adiabatic geometric quantum computation.And these geometric quantum gates are demonstrated in detail by two two-level systems: one is the system of spin-1/2 particle under NMR uniform rotating magnetic field;the other is asymmetric nano-SQUID circuit.The paper presents a perfect explanation for the early controversial helical fiber experiment done by Chiao R.Y.,also proposes a scheme of constructing topologic gates and illustrates these gates still with above two two-level systems.Finally,a brief comment is given on the controllability of quantum gates with the asymmetric nano-SQUID circuit.

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理论物理通讯：英文版•Jan 1, 2002•0 citations
This article summarizes the recent results about geometric and topologic quantum computation.Based on a rigorous theorem concerned on orthogonal superposition quantum state proved in this paper,an orthogonal state method is proposed to carry out the non-adiabatic geometric quantum computation.And these geometric quantum gates are demonstrated in detail by two two-level systems: one is the system of spin-1/2 particle under NMR uniform rotating magnetic field;the other is asymmetric nano-SQUID circuit.The paper presents a perfect explanation for the early controversial helical fiber experiment done by Chiao R.Y.,also proposes a scheme of constructing topologic gates and illustrates these gates still with above two two-level systems.Finally,a brief comment is given on the controllability of quantum gates with the asymmetric nano-SQUID circuit.
Geometric phase that manifests itself in number of optic and nuclear experiments is shown to be a useful tool for realization of quantum computations in so called holonomic quantum computer model (HQCM). This model is considered as an externally driven quantum system with adiabatic evolution law and finite number of the energy levels. The corresponding evolution operators represent quantum gates of HQCM. The explicit expression for the gates is derived both for one-qubit and for multi-qubit quantum gates as Abelian and non-Abelian geometric phases provided the energy levels to be time-independent or in other words for rotational adiabatic evolution of the system. Application of non-adiabatic geometric-like phases in quantum computations is also discussed for a Caldeira-Legett-type model (one-qubit gates) and for the spin 3/2 quadrupole NMR model (two-qubit gates). Generic quantum gates for these two models are derived. The possibility of construction of the universal quantum gates in both cases is shown.
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.
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.
We propose a method of controlling the dc-SQUID (superconducting quantum interference device) systemby changing the gate voltages, which controls the amplitude of the fictitious magnetic fields Bz, and the externallyapplied current that produces the piercing magnetic fiux φx for the dc-SQUID system. We have also introduced aphysical model for the dc-SQUID system. Using this physical model, one can obtain the non-adiabatic geometric phasegate for the single qubit and the non-adiabatic conditional geometric phase gate (controlled NOT gate) for the twoqubits. It is shown that when the gate voltage and the externally applied current of the dc-SQUID system satisfies anappropriate constraint condition, the charge state evolution can be controlled exactly on a dynamic phase free path. Thenon-adiabatic evolution of the charge states is given as well.