arXiv 2408.170352024-08quant-pheess.SP

利用薛定谔动力学实现量子计算中幺正门的研究与实现

Study And Implementation of Unitary Gates in Quantum Computation Using Schrodinger Dynamics

Kumar Gautam

发表信息：: Quantum Information Processing; 2015 and 2017
DOI：: 10.1007/s11128-015-1059-0; 10.1007/s11128-015-1061-6; 10.1007/s11128-017-1564-4

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AI总结：

本论文研究利用受时变电场和磁场微扰的物理系统通过薛定谔动力学实现量子门，基于 Baker-Campbell-Hausdorff 公式构造非谐门和时变线性微扰门，并分析可控性条件，用 Dyson 级数截断获得部分解。

摘要：

本论文探讨了利用原子和振子等物理系统在电场和磁场微扰下实现量子门的概念。基本思想是：如果一个时间无关的哈密顿量 $H_0$ 受到形式为 $f(t)V$ 的时变哈密顿量的微扰，其中 $f(t)$ 是时间的标量函数，$V$ 是与 $H_0$ 不对易的厄米算符，那么可以通过对应于时变哈密顿量 $H_0+f(t)V$ 的薛定谔演化来实现一大类幺正算符。这是李群和李代数中 Baker-Campbell-Hausdorff 公式的结果。基于这一思想，本论文解决两个问题：首先，取一个谐振子并用时间无关的非谐项微扰它，然后计算 $U_g=e^{-ιT H_1}$。然后，用线性时间相关项微扰谐振子哈密顿量，并计算对应于 $H(t)$ 在时间 $T$ 的幺正演化。这个门可以表示为 $U(T)=U(T,ε,f)=T\{e^{-ι\int_0^TH(t)dt}\}$。非谐门 $U_g$ 被量子计算中许多常用门所替代，例如受控幺正门和量子傅里叶变换门。控制电场被适当选择。本论文还讨论了可控性问题，确定在什么条件下存在一个标量实值时间函数 $f(t), 0\leq t\leq T$，使得如果 $|ψ_ι\rangle$ 是任意初始波函数且 $|ψ_f\rangle$ 是任意末态波函数，则 $U(T,f)|ψ_i\rangle=|ψ_f\rangle$。通过用其 Dyson 级数截断版本替换幺正演化核，获得了部分解。在所有设计过程中，出现的门都是无限维的，原子与电磁场之间的相互作用由可控的时间函数调制。

英文摘要：

This thesis explores the concept of realizing quantum gates using physical systems like atoms and oscillators perturbed by electric and magnetic fields. The basic idea is that if a time-independent Hamiltonian $H_0$ is perturbed by a time-varying Hamiltonian of the form $f(t)V$, where $f(t)$ is a scalar function of time and $V$ is a Hermitian operator that does not commute with $H_0$, then a large class of unitary operators can be realized via the Schrodinger evolution corresponding to the time-varying Hamiltonian $H_0+f(t)V$. This is a consequence of the Baker-Campbell-Hausdorff formula in Lie groups and Lie algebras. The thesis addresses two problems based on this idea: first, taking a Harmonic oscillator and perturbing it with a time-independent anharmonic term, and then computing $U_g=e^{-ιT H_1}$. Then, perturbing the harmonic Hamiltonian with a linear time-dependent term, and calculating the unitary evolution corresponding to $H(t)$ at time $T$. This gate can be expressed as $U(T)=U(T,ε,f)=T\{e^{-ι\int_0^TH(t)dt}\}$.The anharmonic gate $U_g$ is replaced by a host of commonly used gates in quantum computation, such as controlled unitary gates and quantum Fourier transform gates. The control electric field is selected appropriately. The thesis also addresses the controllability issue, determining under what conditions there exists a scalar real valued function of time $f(t), 0\leq t\leq T$ such that if $|ψ_ι\rangle$ is any initial wave function and $|ψ_f\rangle$ is any final wave function, then $U(T,f)|ψ_i\rangle=|ψ_f\rangle$. A partial solution was obtained by replacing the unitary evolution kernel by its Dyson series truncated version. In all design procedures, the gates that appear are infinite-dimensional, with an interaction between the atom and the electromagnetic field modulated by a controllable function of time.

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