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Theory of (Co)homological Invariants on Quantum LDPC Codes

Zimu Li, Yuguo Shao, Fuchuan Wei, Yiming Li, Zi-Wen Liu

Abstract

With recent breakthroughs in the construction of good qLDPC codes and nearly good qLTCs, the study of (co)homological invariants of quantum code complexes, which fundamentally underlie their logical operations, has become evidently important. In this work, we establish a systematic framework for mathematically analyzing these invariants across a broad spectrum of constructions, from HGP codes to sheaf codes, by synthesizing advanced math tools. We generalize the notion of canonical logical representatives from HGP codes to the sheaf code setting, resolving a long-standing challenge in explicitly characterizing sheaf codewords. Building on this foundation, we present the first comprehensive computation of cup products within the intricate framework of sheaf codes. Given Artin's primitive root conjecture which holds under the generalized Riemann hypothesis, we prove that $\tildeΘ(N)$ independent cup products can be supported on almost good qLDPC codes and qLTCs of length N, opening the possibility of achieving linearly many parallel, nontrivial, constant-depth multi-controlled-Z gates. Moreover, by interpreting sheaf codes as covering spaces of HGP codes via graph lifts, we propose a scheme that inductively generates families of both HGP and sheaf codes in an interlaced fashion from a constant-size HGP code. Notably, the induction preserves all (co)homological invariants of the initial code. This provides a general framework for lifting invariants or logical gates from small codes to infinite code families, and enables efficient verification of such features by checking on small instances. Our theory provides a substantive methodology for studying invariants in HGP codes and extends it to sheaf codes. In doing so, we reveal deep and unexpected connections between qLDPC codes and math, thereby laying the groundwork for future advances in quantum coding, fault tolerance, and physics.

Theory of (Co)homological Invariants on Quantum LDPC Codes

Abstract

With recent breakthroughs in the construction of good qLDPC codes and nearly good qLTCs, the study of (co)homological invariants of quantum code complexes, which fundamentally underlie their logical operations, has become evidently important. In this work, we establish a systematic framework for mathematically analyzing these invariants across a broad spectrum of constructions, from HGP codes to sheaf codes, by synthesizing advanced math tools. We generalize the notion of canonical logical representatives from HGP codes to the sheaf code setting, resolving a long-standing challenge in explicitly characterizing sheaf codewords. Building on this foundation, we present the first comprehensive computation of cup products within the intricate framework of sheaf codes. Given Artin's primitive root conjecture which holds under the generalized Riemann hypothesis, we prove that independent cup products can be supported on almost good qLDPC codes and qLTCs of length N, opening the possibility of achieving linearly many parallel, nontrivial, constant-depth multi-controlled-Z gates. Moreover, by interpreting sheaf codes as covering spaces of HGP codes via graph lifts, we propose a scheme that inductively generates families of both HGP and sheaf codes in an interlaced fashion from a constant-size HGP code. Notably, the induction preserves all (co)homological invariants of the initial code. This provides a general framework for lifting invariants or logical gates from small codes to infinite code families, and enables efficient verification of such features by checking on small instances. Our theory provides a substantive methodology for studying invariants in HGP codes and extends it to sheaf codes. In doing so, we reveal deep and unexpected connections between qLDPC codes and math, thereby laying the groundwork for future advances in quantum coding, fault tolerance, and physics.

Paper Structure

This paper contains 39 sections, 46 theorems, 269 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. Main results
  4. Key techniques
  5. Inductive lifting sequence.
  6. Canonical logical representatives and cup products on combinatorial complex.
  7. Polarized logical representatives on sheaf codes.
  8. Maximal ideals, Artin's conjecture and generalized Riemann hypothesis.
  9. Flat modules, hypergraph Turán problem and more.
  10. Discussion and outlook
  11. Acknowledgments.
  12. Preliminaries
  13. CSS codes, invariant polynomials and diagonal logical gates
  14. Expander graph families and Ramanujan graphs
  15. Presheaf and sheaf codes
  16. ...and 24 more sections

Key Result

Theorem 1.1

Given Artin's primitive root conjecture whose validity is implied by the generalized Riemann hypothesis Hooley1967, there are infinitely many primes $l$, $X'$ with size $O(\log l)$, and $l$-lift $X$ such that there exists a (nearly) good qLDPC code family defined on $X$ by the sheaf $\mathop{\mathrm

Figures (3)

  • Figure 1: A 3-cube $[g;a_1,a_2,a_3]$ with all its vertices (0-cubes), edges (1-cubes). For clarity, faces (2-cubes) are not given explicitly.
  • Figure 2: The function $S$ and $A$ on the subdivision of a 2-cube $[g;a_1,a_2]$.
  • Figure 3: The cup product in a 3-cube along the path of $[g;a_{1},0,0]$, $[a_{1} \cdot g;1,a_2,0]$ and $[a_1 a_2 \cdot g;1,1,a_3]$.

Theorems & Definitions (100)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • Definition 2.4
  • Definition 2.5
  • Theorem 2.6: Cheeger’s Inequality
  • Definition 2.7
  • ...and 90 more