List of codes which can perform non-Clifford gates fault-tolerantly

Question

I am looking to make a comprehensive list of all known quantum error correction codes which can perform one non-Clifford gate fault-tolerantly.

Currently, I have found two codes.

1. 3D Color code
2. Tri-orthogonal code

I am looking for more such codes.

Answer 1

First, I guess that you mean: which can perform one non-Clifford gate fault-tolerantly without state injection and distillation.

There are several such codes. For example, see https://arxiv.org/abs/1403.2734: "The 15-qubit Reed-Muller code also does not admit a universal fault-tolerant gate set but possesses fault-tolerant T and control-control-Z gates". The Reed-Muller code is behind many ideas of subsystem codes enabling fault-tolerant T gates without injection.

See here for CSS codes with transversal T gates: https://arxiv.org/abs/1910.09333: "We prove three corollaries: (a) For any non-degenerate [[n,k,d]] stabilizer code supporting a physical transversal T, there exists an [[n,k,d]] CSS code with the same property; (b) Triorthogonal codes are the most general CSS codes that realize logical transversal T via physical transversal T; (c) Triorthogonality is necessary for physical transversal T on a CSS code to realize the logical identity".

And see here for fault-tolerant non-Clifford gate with the surface codes (plus some tricks): https://arxiv.org/pdf/1903.11634.pdf for another direction.

Answer 2

The codes constructed in our paper https://arxiv.org/abs/2310.17652 all have distance at least $ 3 $ and essentially all of them implement non-Clifford gates fault tolerantly. There is an 11 qubit code with distance 3 and transversal $ T $ gate (recall that $ T $ gate is $ Z^{1/4} $). In this paper we actually construct, for any $ Z^{1/n} $, a distance 3 codes on a reasonably small number of qubits that can transversally implement $ Z^{1/n} $ (recall transversal gates are fault tolerant). Recall that for $ n \neq 1,2 $ the gate $ Z^{1/n} $ is always non-Clifford.

Also in our paper https://arxiv.org/abs/2305.07023 we construct a family of $ d=3 $ codes implementing lots of exotic non-Clifford gates transversally, including the strange gate from Is this single qubit gate in the Clifford hierarchy?