A geometric view of quantum cellular automata

TL;DR

This paper introduces a geometric framework for quantum cellular automata, linking causal structure and information distances to topology, and demonstrates its application in tracking entanglement and optimizing quantum algorithms.

Contribution

It proposes a novel geometric perspective on quantum cellular automata, connecting information geometry with causal structure and static topologies for improved analysis.

Findings

Constructed a geometric model for quantum cellular automata

Linked causal structure to topology and information distances

Showed utility in entanglement tracking and algorithm optimization

Abstract

Nielsen, et al. [1, 2] proposed a view of quantum computation where determining optimal algorithms is equivalent to extremizing a geodesic length or cost functional. This view of optimization is highly suggestive of an action principle of the space of N-qubits interacting via local operations. The cost or action functional is given by the cost of evolution operators on local qubit operations leading to causal dynamics, as in Blute et. al. [3] Here we propose a view of information geometry for quantum algorithms where the inherent causal structure determines topology and information distances [4, 5] set the local geometry. This naturally leads to geometric characterization of hypersurfaces in a quantum cellular automaton. While in standard quantum circuit representations the connections between individual qubits, i.e. the topology, for hypersurfaces will be dynamic, quantum cellular automata have readily identifiable static hypersurface topologies determined via the quantum update rules. We demonstrate construction of quantum cellular automata geometry and discuss the utility of this approach for tracking entanglement and algorithm optimization.