Morphing quantum codes

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, producing a [[10,1,2]] code with a fault-tolerant logical T gate (the smallest code to our knowledge with this property). In addition, we construct a family of hybrid toric-color (HTC) 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. We use this property to construct toric codes with fault-tolerant multi-qubit control-Z gates. We also provide an effcient decoding algorithm for HTC codes in 2D, 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.