Geometrically Enhanced Topological Quantum Codes

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.