量子线的基尔霍夫法则II：可能应用于量子计算机的逆问题
Kirchhoff's Rule for Quantum Wires. II: The Inverse Problem with Possible Applications to Quantum Computers
摘要：
在这篇文章中，我们继续我们的调查单粒子量子散射理论的薛定谔运营商的一组连接（理想化的一维）线形成一个图形与任意数量的开放端。哈密顿量被给出为减去拉普拉斯算子，在顶点处具有适当的线性边界条件（局部基尔霍夫定律）。在“量子线的基尔霍夫规则”[J. Phys. A：Math. Gen. 32，595 - 630（1999）]中，我们提供了一个关于边界条件和内部线长度的（壳层上）S-矩阵的显式代数表达式，并证明了它的幺正性。在这里，我们在最简单的上下文中解决逆问题，只有一个顶点，但有任意数量的开放端。我们提供了一个明确的公式的边界条件的S-矩阵在一个固定的，规定的能量。我们表明，任何酉$n\times n$矩阵可以实现为S-矩阵在一个给定的能量，通过选择适当的（唯一的）边界条件。这可能用于量子计算中基本门的设计。作为一个例子，我们计算了量子计算机的一些基本门的幺正算符的边界条件，并提出了一个问题，即一般情况下，与量子门相关的幺正算符是否应该被视为散射算符，而不是与量子力学哈密顿量相关的给定时间的时间演化算符。
英文摘要：
In this article we continue our investigations of one particle quantum scattering theory for Schroedinger operators on a set of connected (idealized one-dimensional) wires forming a graph with an arbitrary number of open ends. The Hamiltonian is given as minus the Laplace operator with suitable linear boundary conditions at the vertices (the local Kirchhoff law). In ``Kirchhoff's rule for quantum wires'' [J. Phys. A: Math. Gen. 32, 595 - 630 (1999)] we provided an explicit algebraic expression for the resulting (on-shell) S-matrix in terms of the boundary conditions and the lengths of the internal lines and we also proved its unitarity. Here we address the inverse problem in the simplest context with one vertex only but with an arbitrary number of open ends. We provide an explicit formula for the boundary conditions in terms of the S-matrix at a fixed, prescribed energy. We show that any unitary $n\times n$ matrix may be realized as the S-matrix at a given energy by choosing appropriate (unique) boundary conditions. This might possibly be used for the design of elementary gates in quantum computing. As an illustration we calculate the boundary conditions associated to the unitary operators of some elementary gates for quantum computers and raise the issue whether in general the unitary operators associated to quantum gates should rather be viewed as scattering operators instead of time evolution operators for a given time associated to a quantum mechanical Hamiltonian.
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