Morphing quantum codes

Abstract

We introduce a morphing procedure that can be used to generate new quantum codes from existing quantum codes. In particular, we morph the 15-qubit Reed-Muller code to obtain a $[[10,1,2]]$ code that is the smallest known stabilizer code with a fault-tolerant logical $T$ gate. In addition, we construct a family of hybrid color-toric codes by morphing the color code. Our code family inherits the fault-tolerant gates of the original color code, implemented via constant-depth local unitaries. As a special case of this construction, we obtain toric codes with fault-tolerant multi-qubit control-$Z$ gates. We also provide an efficient decoding algorithm for hybrid color-toric codes in two dimensions, and numerically benchmark its performance for phase-flip noise. We expect that morphing may also be a useful technique for modifying other code families such as triorthogonal codes.

arXiv comment10 + 16 pages, 19 figures, v2: published version

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Cited by in corpus (6)

• Efficient color code decoders in $d\ge 2$ dimensions from toric code decoders
• Tailoring three-dimensional topological codes for biased noise
• Designing the Quantum Channels Induced by Diagonal Gates
• Decoding Merged Color-Surface Codes and Finding Fault-Tolerant Clifford Circuits Using Solvers for Satisfiability Modulo Theories
• Non-Pauli Errors in the Three-Dimensional Surface Code
• Single-shot quantum error correction with the three-dimensional subsystem toric code