Loubenets Elena R, Käding Christian
Applied Mathematics Department, National Research University Higher School of Economics, Moscow 101000, Russia.
Steklov Mathematical Institute of Russian Academy of Sciences, Moscow 119991, Russia.
Entropy (Basel). 2020 May 3;22(5):521. doi: 10.3390/e22050521.
Optimal realizations of quantum technology tasks lead to the necessity of a detailed analytical study of the behavior of a -level quantum system (qudit) under a time-dependent Hamiltonian. In the present article, we introduce a new general formalism describing the unitary evolution of a qudit ( d ≥ 2 ) in terms of the Bloch-like vector space and specify how, in a general case, this formalism is related to finding time-dependent parameters in the exponential representation of the evolution operator under an arbitrary time-dependent Hamiltonian. Applying this new general formalism to a qubit case ( d = 2 ) , we specify the unitary evolution of a qubit via the evolution of a unit vector in R 4 , and this allows us to derive the precise analytical expression of the qubit unitary evolution operator for a wide class of nonstationary Hamiltonians. This new analytical expression includes the qubit solutions known in the literature only as particular cases.
量子技术任务的最优实现导致有必要对处于含时哈密顿量下的(d)能级量子系统（量子位）的行为进行详细的分析研究。在本文中，我们引入一种新的通用形式体系，用类布洛赫向量空间来描述量子位（(d\geq2)）的幺正演化，并详细说明在一般情况下，这种形式体系如何与在任意含时哈密顿量下寻找演化算符指数表示中的含时参数相关。将这种新的通用形式体系应用于量子比特情形（(d = 2)）时，我们通过(\mathbb{R}^4)中单位向量的演化来确定量子比特的幺正演化，这使我们能够推导出一大类非定常哈密顿量下量子比特幺正演化算符的精确解析表达式。这个新的解析表达式涵盖了文献中仅作为特殊情况已知的量子比特解。
