论文标题
使用单量门门的纠缠Quaternionic功率非线性添加量子状态
Nonlinear Addition of Qubit States Using Entangled Quaternionic Powers of Single-Qubit Gates
论文作者
论文摘要
本文提出了一种使用单位四季度代数来表达单量门门的任意根或分数力的新方法，并将这种分数幂用作结合这些分数输入信号的代数的发电机，表现为一种非线性添加。该方法通过连接几个众所周知的等效性来起作用。所有单量门门的组为$ u（2）$，$ c^2 $的单一转换。使用适当的阶段乘数，$ u（2）$的每个元素都可以映射到$ su（2）$的相应元素，其量子机械行为相同。组$ su（2）$对单元四元组组是同构。可以通过选择-1的首选平方根，将de Moivre的副本定理扩展到四元组的de Moivre定理来构建单位四元组的功率和根。使用这条等价链，对于任何单Qubit Gate $ a $和实际指数$ k $，可以预见的是$ b^k = a $。可以通过将单独旋转的Qubits连接到使用2 Qubit CNOT门的常见的“总和”量子组合来结合以这种方式生成的不同分数。探索了此类代数的示例，包括Quaternion $ k $的根（对应于Bloch Sphere的$ x $ rotation），Quaternion $ \ frac {\ sqrt {2}}} {2} {2} {2}（i + k）$（对应于hastamard Gate），以及这些混合物的混合物。这项研究的目标之一是开发经典组件的量子版本，例如分类器的合奏和机器学习和人工智能中使用的激活功能。提出了文本分类的示例应用程序，该应用程序使用分数旋转门表示分类器权重，并通过使用CNOT门来在主题分数量子位中收集适当的分类器权重来对新输入进行分类。
This paper presents a novel way to use the algebra of unit quaternions to express arbitrary roots or fractional powers of single-qubit gates, and to use such fractional powers as generators for algebras that combine these fractional input signals, behaving as a kind of nonlinear addition. The method works by connecting several well-known equivalences. The group of all single-qubit gates is $U(2)$, the unitary transformations of $C^2$. Using an appropriate phase multiplier, every element of $U(2)$ can be mapped to a corresponding element of $SU(2)$ with unit determinant, whose quantum mechanical behavior is identical. The group $SU(2)$ is isomorphic to the group of unit quaternions. Powers and roots of unit quaternions can be constructed by extending de Moivre's theorem for roots of complex numbers to the quaternions by selecting a preferred square root of -1. Using this chain of equivalences, for any single-qubit gate $A$ and real exponent $k$, a gate $B$ can be predictably constructed so that $B^k = A$. Different fractions generated in this way can be combined by connecting the individually rotated qubits to a common 'sum' qubit using 2-qubit CNOT gates. Examples of such algebras are explored including those generated by roots of the quaternion $k$ (which corresponds to and $X$-rotation of the Bloch sphere), the quaternion $\frac{\sqrt{2}}{2}(i + k)$ (which corresponds to the Hadamard gate), and a mixture of these. One of the goals of this research is to develop quantum versions of classical components such as the classifier ensembles and activation functions used in machine learning and artificial intelligence. An example application for text classification is presented, which uses fractional rotation gates to represent classifier weights, and classifies new input by using CNOT gates to collect the appropriate classifier weights in a topic-scoring qubit.
