Critical Multi-Cubic Lattices: A Novel Implication Algebra for Infinite Systems of Qudit Gates

• URL: http://arxiv.org/abs/2306.12236v2
• Date: Tue, 26 Sep 2023 21:35:10 GMT
• Title: Critical Multi-Cubic Lattices: A Novel Implication Algebra for Infinite Systems of Qudit Gates
• Authors: Morrison Turnansky
• Abstract summary: We introduce a new structure, the critical multi-cubic lattice. We compute its automorphism group, and construct a Hilbert space over which we represent the group. We briefly explore the critical multi-cubic lattice as a novel implication algebra serving as a logical framework for qudit gates.
• Score: 0.0
• License: http://creativecommons.org/licenses/by-nc-sa/4.0/
• Abstract: We introduce a new structure, the critical multi-cubic lattice. Notably the critical multi-cubic lattice is the first true generalization of the cubic lattice to higher dimensional spaces. We then introduce the notion of a homomorphism in the category of critical multi-cubic lattices, compute its automorphism group, and construct a Hilbert space over which we represent the group. With this unitary representation, we re-derive the generalized Pauli matrices common in quantum computation while also defining an algebraic framework for an infinite system of qudits. We also briefly explore the critical multi-cubic lattice as a novel implication algebra serving as a logical framework for qudit gates.

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