Morphing quantum codes

TL;DR

This paper presents a morphing technique to generate new quantum codes with enhanced fault-tolerance, including a minimal stabilizer code with a fault-tolerant T gate and hybrid color-toric codes with efficient decoding.

Contribution

It introduces a novel morphing procedure to create new quantum codes, enabling fault-tolerant gates and efficient decoding, expanding the toolbox for quantum error correction.

Findings

Successfully morphed the 15-qubit Reed-Muller code to a smaller $[ exttt{10,1,2}]$ code.

Constructed hybrid color-toric codes with inherited fault-tolerant gates.

Provided an efficient decoding algorithm and benchmarked its performance.

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.