CSS code surgery as a universal construction

Abstract

We define code maps between Calderbank-Shor-Steane (CSS) codes using maps between chain complexes, and describe code surgery between such codes using a specific colimit in the category of chain complexes. As well as describing a surgery operation, this gives a general recipe for new codes. As an application we describe how to `merge' and `split' along a shared $\overline{X}$ or $\overline{Z}$ operator between arbitrary CSS codes in a fault-tolerant manner, so long as certain technical conditions concerning gauge fixing and code distance are satisfied. We prove that such merges and splits on LDPC codes yield codes which are themselves LDPC.

Corrected some errors based on reviewer feedback

References in corpus (7)

• Supplementary information for "Quantum supremacy using a programmable superconducting processor"
• Surface codes: Towards practical large-scale quantum computation
• Homological Error Correction: Classical and Quantum Codes
• Implementing Fault-tolerant Entangling Gates on the Five-qubit Code and the Color Code
• Quantum computing by color-code lattice surgery
• 3-d topological quantum memory with a power-law energy barrier
• Phase-free ZX diagrams are CSS codes (...or how to graphically grok the surface code)

Cited by in corpus (8)

• LDPC-cat codes for low-overhead quantum computing in 2D
• Fault-tolerant logical measurements via homological measurement
• Efficient fault-tolerant code switching via one-way transversal CNOT gates
• Low-overhead fault-tolerant quantum computation by gauging logical operators
• Feasibility of Logical Bell State Generation in Memory Assisted Quantum Networks
• ZX-calculus is Complete for Finite-Dimensional Hilbert Spaces
• Parallel Logical Measurements via Quantum Code Surgery
• Accelerating Fault-Tolerant Quantum Computation with Good qLDPC Codes