Quantum Gates from Wolfram Model Multiway Rewriting Systems

TL;DR

This work demonstrates that finite-dimensional quantum operators and circuits can be represented within a background-free framework of nondeterministic Wolfram multiway rewriting on Leibnizian strings. By formulating a discrete path-sum (S-matrix) over sequences of Leibnizian strings and enforcing unitarity through edge-weight choices, the authors obtain explicit representations of quantum gates (e.g., Hadamard, π/8, CNOT, SWAP) and constructions of quantum circuits. They connect a fermionic occupation statistic for local string views to a discrete path integral formalism, enabling a statistical mechanics viewpoint on computation. The approach provides a modular, symbolic foundation for quantum information processing that may offer insights for quantum gravity and non-spatiotemporal quantum process theories, with potential for universal quantum computation via a one-to-many mapping from multiway systems to gate sets.

Abstract

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.

Figures (15)

• Figure 1: The plot of BSD variety vs conditional Shannon entropy for a set of Leibnizian strings for different alphabets. It is seen that BSD variety is positively correlated with the information content of words. Two-letter alphabet: 1281 words of length between 8--50. Three-letter alphabet: 1289 words of length between 8--50. Four-letter alphabet: 1704 words of length between 8--65. Five-letter alphabet: 2490 words of length between 8--90.
• Figure 2: Initial section of the multiway system graph of string "BBBAAACC" with the rules: $\text{BA}\to\text{AB},\text{CB}\to\text{BC}$. The physical multiway system that includes only the Leibnizian strings is highlighted in red.
• Figure 3: Representation of initial part (depth 4) of the multiway system specified by the following: Initial state: "AABAABBABAB". Rewriting rule: BA$\to$AB. The paths highlighted in red are Leibnizian (physical) paths where as the path in green is the maximal variety path.
• Figure 4: An illustration of a three-layer system. An in-word is denoted with $i$ and an out-word with $j$. The elements of the second layer are denoted with labels $c$, which one sums over.
• Figure 5: Anatomy of 2D systems. (a) non-interacting case, (b) interacting case. All the nodes are supposed to be Leibnizian strings. Note that there is a symmetry of permutation of words. So the case when in-1 and out-2, and in-2 and out-1 are connected is also a non-interacting set-up.

Theorems & Definitions (31)

• Definition 2.1: Neighborhood of a character barbour1992extremal
• Definition 2.2: String isomorphism barbour1992extremal
• Definition 2.3: Relative indifference barbour1992extremal
• Definition 2.4: Leibnizian strings barbour1992extremal
• Example 2.1
• Example 2.2
• Definition 2.5: Absolute indifference barbour1992extremal
• Definition 2.6: BSD Variety barbour1992extremal
• Example 2.3
• Definition 3.1: Shannon entropy of a string of characters shannon_entropy