从几何代数角度看量子计算中的量子计算门和普适性
A Geometric Algebra Perspective On Quantum Computational Gates And Universality In Quantum Computing
摘要：
我们研究了几何（Clifford）代数（GA）方法在量子信息科学中的两个具体应用。首先，利用多粒子时空代数（MSTA，相对论位形空间的几何代数），给出了单量子位和双量子位量子态的显式代数描述，以及单量子位和双量子位量子计算门的MSTA刻画。其次，利用上述刻画和基于转子群Spin+（3，0）形式的李代数SO（3）和SU（2）的GA描述，我们重新检验了Boykin关于量子门普适性的证明。我们的结论是，MSTA方法确实导致了一个有用的概念统一，其中复量子比特空间和作用于它们的幺正算子的复空间变得统一，两者都是由实空间中的多向量构成的。最后，基于转子组的GA旋转方法与标准矢量和矩阵方法相比，确实带来了概念和计算上的优势。
英文摘要：
We investigate the utility of geometric (Clifford) algebras (GA) methods in two specific applications to quantum information science. First, using the multiparticle spacetime algebra (MSTA, the geometric algebra of a relativistic configuration space), we present an explicit algebraic description of one and two-qubit quantum states together with a MSTA characterization of one and two-qubit quantum computational gates. Second, using the above mentioned characterization and the GA description of the Lie algebras SO(3) and SU(2) based on the rotor group Spin+(3, 0) formalism, we reexamine Boykin's proof of universality of quantum gates. We conclude that the MSTA approach does lead to a useful conceptual unification where the complex qubit space and the complex space of unitary operators acting on them become united, with both being made just by multivectors in real space. Finally, the GA approach to rotations based on the rotor group does bring conceptual and computational advantages compared to standard vectorial and matricial approaches.
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