Transversal non-Clifford gates on almost-good quantum LDPC and quantum locally testable codes

TL;DR

This paper demonstrates fault-tolerant non-Clifford gates on quantum LDPC and locally testable codes, using algebraic-topological methods to establish their fundamental topological nature.

Contribution

It introduces a new framework for constructing homological invariant forms called 'cupcap gates' that enable transversal logical multi-controlled-$Z$ gates on quantum codes.

Findings

Achieved transversal non-Clifford gates on quantum LDPC codes.

Developed a topological framework for constructing 'cupcap gates'.

Provided algebraic-topological proofs of gate nontriviality.

Abstract

We exhibit nontrivial transversal logical multi-controlled-$Z$ gates on $[\![N,\Theta(N),\tilde\Theta(N)]\!]$ quantum low-density parity-check codes and $[\![N,\Theta(N),\tilde\Theta(N)]\!]$ quantum locally testable codes with soundness $\tilde\Theta(1)$, combining nearly optimal code parameters with fault-tolerant non-Clifford gates for the first time. Remarkably, our proofs are almost entirely algebraic-topological, showing that such presumably intricate logical gates naturally arise as a fundamental topological phenomenon. We develop a general framework for constructing a rich new family of homological invariant forms which we call ''cupcap gates'' that induce transversal logical multi-controlled-$Z$ and, building on insights from [Li et al., arXiv:2603.25831], covering space methods to certify their nontriviality. The claimed almost-good code results follow immediately as examples.