复杂量子动力学系统的量子计算鲁棒性研究
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摘要
量子计算利用量子态的相干叠加性实现并行的量子计算,从而达到经典计算无法比拟的信息处理功能。尽管在过去的十几年内量子计算的理论研究和物理实现都取得了巨大的进步,但是到目前为止还没有大规模量子信息处理装置出现。其中的一个主要原因在于量子信息处理系统内部近距离的相互作用以及量子系统与外部环境之间不可避免的耦合等严重影响了量子计算的可控性和可操作性,成为制约量子计算技术发展的重要障碍。因此分析量子信息处理系统在各种干扰下的演化规律,研究干扰对量子计算鲁棒性的影响,将非常有助于设计高可靠性的容错量子计算系统。
首先简要总结了量子计算的发展历程及其鲁棒性研究的最新进展;然后介绍了一种具有复杂动力学行为的周期驱动的量子Harper模型(Quantum Kicked Harper,QKH)及其量子仿真算法;随后利用随机矩阵理论(Random Matrix Theory)和Husimi分布函数分析了量子Harper模型在规则运动和混沌运动时Floquet算子的能谱统计特性和本征态的统计遍历性,并且运用保真度摄动分析、量子轨迹、量子Monte Carlo仿真等方法分别研究了理想环境和开放环境中QKH量子仿真算法的鲁棒性,以及随机噪声干扰、静态干扰和耗散干扰对量子计算可信计算时间尺度、保真度和动态局域化因子等特征物理量的影响。在干扰强度大于某一阈值时,QKH量子计算将产生混沌行为,产生不可信的计算结果。采用数值仿真的方法得出静态干扰导致量子混沌的干扰强度阈值远小于随机噪声干扰的阈值,而且随机噪声干扰时的保真度随系统演化呈指数衰减,静态干扰时为高斯衰减。相位阻尼信道噪声模型的仿真结果表明耗散干扰将破坏QKH本征态的动态局域化以及相空间的随机网结构,但是QKH的动力学特性对耗散干扰下量子计算的鲁棒性没有影响。由于耗散干扰破坏了QKH模型的酉演化性质,耗散干扰下的量子计算不再具有可逆性。在分析动态解耦法抑制退相干原理的基础上,利用随机动态解耦法提高量子计算的抗静态干扰能力,最后将量子计算鲁棒性的研究结果应用于提高二能级量子构造控制系统的稳定性。
A quantum computer could perform certain computations much more efficient than its classical counterpart by exploiting the superposition principle and the interferences of quantum mechanics. Although the advances in the field of quantum computation were great in the last two decades, there is no scalable quantum information processors yet come into existence. One of the major reasons of this fact is the noise and imperfections, arising due to the interactions among the qubits or the couplings with the environment, destroy the operationality of quantum computation severely. Thus exploring the robustness features of quantum information processing system in presence of various imperfections will be very helpful to design fault-tolerant quantum information processor.
A historical overview of quantum computation and the recent development on the research of its robustness are summarized firstly in this paper. Then the quantum kicked Harper model, which behaves a complex quantum dynamics, as well as its simulation algorithm are presented. Based on the random matrix theory and the Husimi function approach, the statistical properties of the quasi-energy and the eigenstates of the quantum kicked Harper, both in regular and chaotic regime, were studied. After that, the effects of various imperfections on the quantum computation which are indicated by several important physical quantities, such as fidelity, the timescales for reliable quantum computation and dynamical localization length etc., were analyzed using the fidelity perturbation method and the quantum trajectory approach. Once the strength of imperfections above a certain threshold, quantum chaos sets in and leads to unreliable quantum computation. It was demonstrated that the threshold for static imperfections are much smaller than that of the random noise in quantum gate operations. The fidelity drops exponentially in the presence of random noise, while it is Gaussian decrease in the case of static imperfections. Taking the phase damping channel as the dissipative model, it was shown that the dynamical localization for the quantum kicked Harper and the stochastic web in phase space are destroyed by moderate levels of dissipation. And the robustness of the quantum computation under dissipative imperfections is independent of the integrable or chaotic nature of the underlying dynamics. Since the unitary evolution is destroyed by the dissipative imperfections, the quantum computation of the open QKH model is no longer reversible. Furthermore, the random dynamical decoupling methods which are originally proposed to suppress the dissipative imperfections are used to combat the static imperfections in the quantum computation. Applications of the theory of robust quantum computation to the quantum constructive control of quantum two-level systems are described in the end.
首先简要总结了量子计算的发展历程及其鲁棒性研究的最新进展;然后介绍了一种具有复杂动力学行为的周期驱动的量子Harper模型(Quantum Kicked Harper,QKH)及其量子仿真算法;随后利用随机矩阵理论(Random Matrix Theory)和Husimi分布函数分析了量子Harper模型在规则运动和混沌运动时Floquet算子的能谱统计特性和本征态的统计遍历性,并且运用保真度摄动分析、量子轨迹、量子Monte Carlo仿真等方法分别研究了理想环境和开放环境中QKH量子仿真算法的鲁棒性,以及随机噪声干扰、静态干扰和耗散干扰对量子计算可信计算时间尺度、保真度和动态局域化因子等特征物理量的影响。在干扰强度大于某一阈值时,QKH量子计算将产生混沌行为,产生不可信的计算结果。采用数值仿真的方法得出静态干扰导致量子混沌的干扰强度阈值远小于随机噪声干扰的阈值,而且随机噪声干扰时的保真度随系统演化呈指数衰减,静态干扰时为高斯衰减。相位阻尼信道噪声模型的仿真结果表明耗散干扰将破坏QKH本征态的动态局域化以及相空间的随机网结构,但是QKH的动力学特性对耗散干扰下量子计算的鲁棒性没有影响。由于耗散干扰破坏了QKH模型的酉演化性质,耗散干扰下的量子计算不再具有可逆性。在分析动态解耦法抑制退相干原理的基础上,利用随机动态解耦法提高量子计算的抗静态干扰能力,最后将量子计算鲁棒性的研究结果应用于提高二能级量子构造控制系统的稳定性。
A quantum computer could perform certain computations much more efficient than its classical counterpart by exploiting the superposition principle and the interferences of quantum mechanics. Although the advances in the field of quantum computation were great in the last two decades, there is no scalable quantum information processors yet come into existence. One of the major reasons of this fact is the noise and imperfections, arising due to the interactions among the qubits or the couplings with the environment, destroy the operationality of quantum computation severely. Thus exploring the robustness features of quantum information processing system in presence of various imperfections will be very helpful to design fault-tolerant quantum information processor.
A historical overview of quantum computation and the recent development on the research of its robustness are summarized firstly in this paper. Then the quantum kicked Harper model, which behaves a complex quantum dynamics, as well as its simulation algorithm are presented. Based on the random matrix theory and the Husimi function approach, the statistical properties of the quasi-energy and the eigenstates of the quantum kicked Harper, both in regular and chaotic regime, were studied. After that, the effects of various imperfections on the quantum computation which are indicated by several important physical quantities, such as fidelity, the timescales for reliable quantum computation and dynamical localization length etc., were analyzed using the fidelity perturbation method and the quantum trajectory approach. Once the strength of imperfections above a certain threshold, quantum chaos sets in and leads to unreliable quantum computation. It was demonstrated that the threshold for static imperfections are much smaller than that of the random noise in quantum gate operations. The fidelity drops exponentially in the presence of random noise, while it is Gaussian decrease in the case of static imperfections. Taking the phase damping channel as the dissipative model, it was shown that the dynamical localization for the quantum kicked Harper and the stochastic web in phase space are destroyed by moderate levels of dissipation. And the robustness of the quantum computation under dissipative imperfections is independent of the integrable or chaotic nature of the underlying dynamics. Since the unitary evolution is destroyed by the dissipative imperfections, the quantum computation of the open QKH model is no longer reversible. Furthermore, the random dynamical decoupling methods which are originally proposed to suppress the dissipative imperfections are used to combat the static imperfections in the quantum computation. Applications of the theory of robust quantum computation to the quantum constructive control of quantum two-level systems are described in the end.
引文
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