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arXiv:quant-ph/0607041 · Modern Physics Letters B

NON-ADIABATIC ARBITRARY GEOMETRIC PHASE GATE IN 2-QUBIT SPIN MODEL

YU TONG

Published Jun 20, 2007• 0 citations
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Abstract

We study a 2-qubit spin model for the possibility of realizing an arbitrary geometric quantum phase gate in terms of a single coherent magnetic pulse with multi-harmonic frequency. Using resonant transition approximation, the time-dependent Hamiltonian of two coupled spins can be solved analytically. The time evolution of the wave function is obtained without adiabatic approximation. The parameters of magnetic pulse, such as the frequency, amplitude, phase of each harmonic part as well as the time duration of the pulse are determined for achieving an arbitrary non-adiabatic geometric phase gate. The requirement of materials for realizing such a gate is analyzed. As a result, the non-adiabatic geometric controlled phase gates and A–A phase are also addressed.

DOI
10.1142/s0217984907013353

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We study a 2-qubit spin model for the possibility of realizing an arbitrary geometric quantum phase gate in terms of a single coherent magnetic pulse with multi-harmonic frequency. Using resonant transition approximation, the time-dependent Hamiltonian of two coupled spins can be solved analytically. The time evolution of the wave function is obtained without adiabatic approximation. The parameters of magnetic pulse, such as the frequency, amplitude, phase of each harmonic part as well as the time duration of the pulse are determined for achieving an arbitrary non-adiabatic geometric phase gate. The requirement of materials for realizing such a gate is analyzed. As a result, the non-adiabatic geometric controlled phase gates and A–A phase are also addressed.
Arrays of neutral atoms have emerged as promising platforms for quantum computing. The realization of high-fidelity two-qubit gates with robustness is currently a significantly important task for large-scale operations. In this paper, we present a convenient approach for implementing a two-qubit controlled-phase gate with Rydberg blockade. We achieve noncyclic and geometric control within a single modulated pulse. Compared with the control scheme by cyclic evolution determined by dynamical parameters, the robustness of our proposal against systematic errors will be remarkably improved due to its geometric characteristics. Importantly, the noncyclic geometric control reduces the gate time for small rotation angles and is more insensitive to decoherence effects. Furthermore, we accelerate the adiabatic control with the aid of shortcuts to adiabaticity to further shorten the operation time. We apply our protocol to the quantum Fourier transformation algorithm to demonstrate the actual acceleration. Therefore, the proposed scheme provides analytical wave forms for arbitrary two-qubit gates and may play an important role in experiments involving atomic arrays.
Arrays of neutral atoms have emerged as promising platforms for quantum computing. Realization of high-fidelity two-qubit gates with robustness is currently a significant important task for large-scale operations. In this paper, we present a convenient approach for implementing a two-qubit controlled-phase gate using Rydberg blockade. We achieve the noncyclic geometric control with a single modulated pulse. As compared with the control scheme by cyclic evolution that determined by dynamical parameters, the robustness of the proposal against systematic errors will be remarkably improved due to the geometric characteristic. Importantly, the noncyclic geometric control reduces the gate time for small rotation angles and will be more insensitive to the decoherence effect. We accelerate the adiabatic control with the aid of shortcuts to adiabaticity to further shorten the operation time. We apply our protocol to the algorithm of quantum Fourier transformation to show the actual acceleration. Therefore, the proposed scheme will provide an analytical waveforms for arbitrary two-qubit gates and may have important use in the experiments of atomic arrays.
ABSTRACT Geometric quantum gates, leveraging the geometric properties of evolution paths, offer inherent robustness against certain types of operational errors. In this work, we extend the concept of adiabatic geometric quantum gates to non‐Hermitian two‐level systems. Starting from the biorthogonal formalism, we derive the general expression for the complex Berry phase in non‐Hermitian quantum systems. By constructing a non‐Hermitian Hamiltonian driven by complex pulses and employing a single‐loop multiple‐pulse strategy, we demonstrate that the adiabatic cyclic evolution accumulates a purely geometric Abelian phase, enabling the realization of a universal set of single‐qubit geometric gates. We further analyze the impact of systematic errors on geometric gates, demonstrating their robustness against small pulse parameter variations. Finally, we propose a feasible scheme to implement a controlled‐phase () gate in a non‐Hermitian system of two Rydberg‐interacting atoms. Our results establish non‐Hermitian adiabatic geometric quantum gates as a viable and error‐resilient approach for quantum information processing. This work provides comprehensive theoretical support for non‐Hermitian geometric quantum computation and lays the foundation for the design and experimental realization of high‐fidelity, noise‐resilient quantum gates in open quantum systems.
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 investigate how to obtain a non-trivial geometric phase gate for a two-qubit spin chain, with Ising interaction in different magnetic fields. Indeed, one of the spins is driven by a time-varying rotating magnetic field, and the other is coupled with a static magnetic field in the direction of the rotation axis. This is an interesting problem both for the purpose of measuring the geometric phases and in quantum computing applications. It is shown that the static magnetic field does not change the adiabatic states of the system, and it does not affect the geometric phases, whereas it may be used to control the dynamic phases. In addition, by considering the exact two-spin adiabatic geometric phases, we find that a non-trivial two-spin unitary transformation, purely based on Berry phases, can be obtained by using two consecutive cycles with opposite directions of the magnetic fields, opposite signs of the interaction constant and the phase shift of the rotating magnetic field. In addition, in the non-adiabatic case, starting with a certain initial state, a cycle can be achieved and thus the Aharonov–Anandan phase is calculated.