A topological theory for qLDPC: non-Clifford gates and magic state fountain on homological product codes with constant rate and beyond the $N^{1/3}$ distance barrier

2025-01-31

AI Summary

This work addresses the dual challenge of implementing non-Clifford gates and efficiently preparing high-fidelity magic states in fault-tolerant quantum computation, while simultaneously achieving high code distance, constant-weight stabilizers, and high encoding rate. We introduce a unifying framework for qLDPC codes and topological codes grounded in simplicial complex geometry, mapping CSS product codes to triangulated codings on thickened product manifolds. Our contributions include: (i) the first constant-depth CCZ gate implementation on constant-weight, constant-rate qLDPC codes; (ii) breaking the Bravyi–König distance barrier by achieving minimum distance Ω(√N); and (iii) a “magic fountain” mechanism enabling single-shot, parallel injection of Θ(√N) high-fidelity magic states without distillation. The technical advances encompass a manifold-based generalization of Freedman–Hastings codes, homological product code optimization, and geometric reconstruction of Tanner graphs.

Technology Category

• Machine Learning: Quantum Machine Learning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work
• Security and Privacy: Large-scale security measurements

Abstract

We develop a unified theory for fault-tolerant quantum computation in quantum low-density parity-check (qLDPC) and topological codes. We show that there exist hidden simplicial complex structures encoding the topological data for all qLDPC and CSS 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 topological quantum field theory. When applying this mapping to a 3D hypergraph product code obtained from the product of 3 copies of good classical expander codes, we obtain the first non-Clifford logical CCZ gates via constant depth circuits on a code with constant stabilizer weight $w=O(1)$, constant rate $K=Theta(N)$, and polynomial distance $D=Omega(N^{1/3})$. When applied to 3D homological product codes consisting of the product of a pair of good quantum and classical LDPC codes, we can further improve the distance to $D=Omega(sqrt{N})$ exceeding the $N^{1/3}$ distance barrier implied by the Bravyi-K""onig bound for conventional topological codes. Our 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 $Theta(sqrt{N})$ magic states in parallel in a single round.

Problem

• Quantum computing
• Non-Clifford operations
• Magic state distillation

Innovation

• qLDPC Coding
• Non-Clifford Operations
• Topological Codes