arXiv subjects

Morrison Turnansky

Publications and source records attributed to Morrison Turnansky.

3 recordsLinked to original sources

arXiv2025

An Error Correctable Implication Algebra for a System of Qubits

We present the Lukasiewicz logic as a viable system for an implication algebra on a system of qubits. Our results show that the three valued Lukasiewicz logic can be embedded in the stabilized space of an arbitrary quantum error correcting stabilizer code. We then fully characterize the non trivial errors that may occur up to group isomorphism. Lastly, we demonstrate by explicit algorithmic example, how any algorithm consistent with the Lukasiewicz logic can immediately run on a quantum system and utilize the indeterminate state.

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arXiv2023

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

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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arXiv2023

Generalizing Pauli Spin Matrices Using Cubic Lattices

In quantum mechanics, the connection between the operator algebraic realization and the logical models of measurement of state observables has long been an open question. In the approach that is presented here, we introduce a new application of the cubic lattice. We claim that the cubic lattice may be faithfully realized as a subset of the self-adjoint space of a von Neumann algebra. Furthermore, we obtain a unitary representation of the symmetry group of the cubic lattice. In so doing, we re-derive the classic quantum gates and gain a description of how they govern a system of qubits of arbitrary cardinality.

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