从Wolfram模型多方式重写系统构建量子门
Quantum Gates from Wolfram Model Multiway Rewriting Systems
AI总结：
基于Leibnizian字符串的多方式系统可构建量子门及量子电路，实现非确定性计算的建模。
摘要：
我们展示如何利用非确定性重写系统来构造有限维量子算子的表示。特别是，我们研究基于字符串替换的Wolfram模型多方式重写系统。多方式系统由S.Wolfram提出，作为多计算过程的通用系统模型，强调其作为建模复杂性、非确定性和测量结果分支结构的基础意义。这里，我们研究一种特定的多方式系统，基于具有邻域约束的环状字符字符串——后者称为Leibnizian字符串。我们证明这些字符串表现出费米-狄拉克分布对于字符邻域的占据数的期望值。一个Leibnizian字符串是N个费米子系统的抽象。这些字符串的多方式系统以非确定性方式编码重写事件之间的因果关系。字符字符串的集合实现了一个具有对称$\mathbb{Z}$-双线性形式的$\mathbb{Z}$-模。对于离散空间，这推广了向量场上内积的概念。这允许离散路径积分的离散类比和Leibnizian字符串多方式系统的$S$-矩阵。该$S$-矩阵的元素给出基于在序列Leibnizian字符串上定义的动作的多方式系统状态之间的转换振幅。我们随后证明这些$S$-矩阵给出qubit和qudit的量子门以及由这些门组成的电路的显式表示。我们发现，作为非确定性计算形式模型的Leibnizian字符串重写系统，具有因果结构，编码了CNOT、π/8和Hadamard门的表示。因此，利用多方式系统可以表示qubit的量子电路。
英文摘要：
We show how representations of finite-dimensional quantum operators can be constructed using nondeterministic rewriting systems. In particular, we investigate Wolfram model multiway rewriting systems based on string substitutions. Multiway systems were proposed by S. Wolfram as generic model systems for multicomputational processes, emphasizing their significance as a foundation for modeling complexity, nondeterminism, and branching structures of measurement outcomes. Here, we investigate a specific class of multiway systems based on cyclic character strings with a neighborhood constraint - the latter called Leibnizian strings. We show that such strings exhibit a Fermi-Dirac distribution for expectation values of occupation numbers of character neighborhoods. A Leibnizian string serves as an abstraction of a $N$-fermion system. A multiway system of these strings encodes causal relations between rewriting events in a nondeterministic manner. The collection of character strings realizes a $\mathbb{Z}$-module with a symmetric $\mathbb{Z}$-bilinear form. For discrete spaces, this generalizes the notion of an inner product over a vector field. This admits a discrete analogue of the path integral and a $S$-matrix for multiway systems of Leibnizian strings. The elements of this $S$-matrix yield transition amplitudes between states of the multiway system based on an action defined over a sequence of Leibnizian strings. We then show that these $S$-matrices give explicit representations of quantum gates for qubits and qudits, and also circuits composed of such gates. We find that, as formal models of nondeterministic computation, rewriting systems of Leibnizian strings with causal structure encode representations of the CNOT, $π/8$, and Hadamard gates. Hence, using multiway systems one can represent quantum circuits for qubits.
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