Cross-cap defects and fault-tolerant logical gates in the surface code and the honeycomb Floquet code

Abstract

We consider the ${\mathbb{Z}}_2$ toric code, surface code and Floquet code defined on a non-orientable surface, which can be considered as families of codes extending Shor's 9-qubit code. We investigate the fault-tolerant logical gates of the ${\mathbb{Z}}_2$ toric code in this setup, which corresponds to $e\leftrightarrow m$ exchanging symmetry of the underlying ${\mathbb{Z}}_2$ gauge theory. We find that non-orientable geometry provides a new way the emergent symmetry acts on the code space, and discover the new realization of the fault-tolerant Hadamard gate of 2d ${\mathbb{Z}}_2$ toric code on a surface with a single cross-cap, dubbed a non-orientable toric code. This Hadamard gate can be realized by a constant-depth local unitary circuit modulo non-locality caused by a cross-cap. Via folding, the non-orientable surface code can be turned into a bilayer local quantum code, where the folded cross-cap is equivalent to a bi-layer twist terminated on a gapped boundary and the logical Hadamard only contains local gates with intra-layer couplings. We further obtain the complete logical Clifford gate set for a stack of non-orientable surface codes. We then construct the honeycomb Floquet code in the presence of a single cross-cap, and find that the period of the sequential Pauli measurements acts as a $HZ$ logical gate on the single logical qubit, where the cross-cap enriches the dynamics compared with the orientable case. We find that the dynamics of the honeycomb Floquet code is precisely described by a condensation operator of the ${\mathbb{Z}}_2$ gauge theory, and illustrate the exotic dynamics of our code in terms of a condensation operator supported at a non-orientable surface.

26 pages, 19 figures. Added Sec. II D, major revision in Sec. III F, and numerous improvements. To appear in PRX Quantum

References in corpus (10)

• Surface codes: Towards practical large-scale quantum computation
• Logical quantum processor based on reconfigurable atom arrays
• Topological Quantum Distillation
• Restrictions on Transversal Encoded Quantum Gate Sets
• High-threshold and low-overhead fault-tolerant quantum memory
• Genons, twist defects, and projective non-Abelian braiding statistics
• Dynamically Generated Logical Qubits
• Floquet codes without parent subsystem codes
• Codimension-2 defects and higher symmetries in (3+1)D topological phases
• Locality-Preserving Logical Operators in Topological Stabiliser Codes

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• Tessellation codes: encoded quantum gates by geometric rotation
• Decoder Dependence in Surface-Code Threshold Estimation under Digitized Hybrid Continuous-Variable and Discrete Noise