Geometrically Enhanced Topological Quantum Codes

TL;DR

This paper introduces geometric rotation techniques for higher-dimensional toric codes to reduce qubit requirements, enabling efficient state preparation and logical operations, with potential applications in measurement-based quantum computing.

Contribution

It proposes a novel geometric rotation approach for toric codes and general stabilizer codes, enhancing state preparation and logical operation methods in quantum error correction.

Findings

Optimal rotations identified in low dimensions through computer analysis

Methods for logical Clifford operations using crystalline symmetries and surgery

New state injection techniques for stabilizer codes at low noise levels

Abstract

We consider geometric methods of ``rotating" the toric code in higher dimensions to reduce the qubit count. These geometric methods can be used to prepare higher dimensional toric code states using single shot techniques, and in turn these may be used to prepare entangled logical states such as Bell pairs or GHZ states. This bears some relation to measurement-based quantum computing in a twisted spacetime. We also propose a generalization to more general stabilizer codes, and we present computer analysis of optimal rotations in low dimensions. We present methods to do logical Clifford operations on these codes using crystalline symmetries and surgery, and we present a method for state injection at low noise into stabilizer quantum codes generalizing previous ideas for the two-dimensional toric code.