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

Yiming Li, Zimu Li, Zi-Wen Liu

Abstract

We exhibit nontrivial transversal logical multi-controlled-$Z$ gates on $[\![N,Θ(N),\tildeΘ(N)]\!]$ quantum low-density parity-check codes and $[\![N,Θ(N),\tildeΘ(N)]\!]$ quantum locally testable codes with soundness $\tildeΘ(1)$, 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.

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

Abstract

We exhibit nontrivial transversal logical multi-controlled- gates on quantum low-density parity-check codes and quantum locally testable codes with soundness , 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- 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.

Paper Structure

This paper contains 17 sections, 18 theorems, 184 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main results.
  3. Discussion and outlook.
  4. Acknowledgments.
  5. Preliminaries
  6. Cell complexes and Alexandrov topology
  7. Sheaves and their operations
  8. Chain complexes and CSS codes
  9. Logical multi-controlled-$Z$ gates
  10. Cupcap gates on sheaf codes
  11. Cup and cap products on sheaved simplicial complexes
  12. Barycentric subdivision of cell complexes
  13. General definition of cup and cap products and cupcap gates
  14. Transversal non-Clifford gates on almost-good qLDPC codes and qLTCs
  15. Covering spaces and covering maps
  16. ...and 2 more sections

Key Result

Theorem 1.1

For any integer $r\geq 2$, there exist that support nontrivial transversal logical $\mathrm{C}^{r-1}Z$ gates. $\blacktriangleleft$$\blacktriangleleft$

Figures (1)

  • Figure :

Theorems & Definitions (51)

  • Theorem 1.1: Main
  • Definition 2.1: Regular cell complex
  • Definition 2.2: Cell poset
  • Definition 2.3: Alexandrov topology
  • Proposition 2.4
  • Definition 2.5: Sparse cell complex
  • Definition 2.6: Sheaf
  • Definition 2.7: Pullback sheaf
  • Definition 2.8: Tensor product of sheaves
  • Definition 2.9: External tensor product of sheaves
  • ...and 41 more