Title: LDPC=TQFT: non-Clifford transversal gates and magic state fountain from generalized symmetries

Abstract: Quantum low-density parity-check (qLDPC) codes are promising candidates for fault-tolerant quantum computation which greatly reduce the space-time overhead.   In this talk, I will give an overview of the basic properties of qLDPC codes and introduce a new topological theory on fault-tolerant transversal logical gates on these codes via generalized symmetries in topological quantum field theories (TQFT). 

In particular, I will show that there exist hidden simplicial complex structures encoding the topological data for qLDPC codes obtained from product construction by generalizing the Freedman-Hastings code-to-manifold mapping. This is achieved by building manifolds corresponding to high-dimensional topological expanders from the Tanner graphs of the skeleton classical or quantum codes, which further form a product manifold and an associated thickened product code defined on its triangulation with only a constant qubit overhead. This suggests that qLDPC or more generally CSS codes obtained from product constructions are topological, and hence can admit cohomology operations such as cup products, physically corresponding to higher symmetries in the underlying TQFT. When applied to a 3D homological product codes consisting of the product of a pair of good quantum and classical LDPC codes, one obtains the first non-Clifford logical CCZ gates via constant depth circuits on a code with constant stabilizer weight w=O(1), constant rate K=Θ(N), and polynomial distance D=Ω(N^1/2), exceeding the N^1/3 distance barrier implied by the Bravyi-König bound for conventional topological codes. This work suggests that it is feasible to apply native logical non-Clifford gates on qLDPC codes or directly inject high-fidelity magic states as resources (`magic state fountain') without the distillation process. For the homological product construction, the fountain can inject Θ(N^1/2) magic states in parallel in a single round.

References:

[1] arXiv:2501.19375

[2] arXiv:2310.16982

A light lunch will be served in the Bowen Atrium at noon.