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Homological origin of transversal implementability of logical diagonal gates in quantum CSS codes

Junichi Haruna

TL;DR

The paper establishes a homological framework for transversal logical diagonal gates in quantum CSS codes, showing that the refinement of discrete Z-rotations from angle $\pi/2^{m-1}$ to $\pi/2^{m}$ is governed by the Bockstein homomorphism $\beta_m: H_1(C;\mathbb{Z}_{2^m}) \to H_0(C;\mathbb{Z}_2)$. A necessary-and-sufficient condition for refinement is $\beta_m([\theta]) = [0]$, making the obstruction central to transversal implementability. Under a stronger commutativity condition modulo $2^{m+1}$ and linear independence of the $X$-stabilizers, $\beta_m$ vanishes for all classes, yielding universal refinement of all logical $Z$ rotations at level $m$. The work also discusses non-uniqueness of parity-check matrices and introduces the broader chain-complex lifting problem, suggesting a deeper topological structure underlying fault-tolerant transversal gates and linking to existing divisibility-based criteria. This provides a canonical, topological perspective that unifies prior algebraic conditions and points to future directions in a formal theory of transversal structures in quantum error correction.

Abstract

Transversal Pauli Z rotations provide a natural route to fault-tolerant logical diagonal gates in quantum CSS codes, yet their capability is fundamentally constrained. In this work, we formulate the refinement problem of realizing a logical diagonal gate by a transversal implementation with a finer discrete rotation angle and show that its solvability is completely characterized by the Bockstein homomorphism in homology theory. Furthermore, we prove that the linear independence of the X-stabilizer generators together with the commutativity condition modulo a power of two ensures the existence of transversal implementations of all logical Pauli Z rotations with discrete angles in general CSS codes. Our results identify a canonical homological obstruction governing transversal implementability and provide a conceptual foundation for a formal theory of transversal structures in quantum error correction.

Homological origin of transversal implementability of logical diagonal gates in quantum CSS codes

TL;DR

The paper establishes a homological framework for transversal logical diagonal gates in quantum CSS codes, showing that the refinement of discrete Z-rotations from angle to is governed by the Bockstein homomorphism . A necessary-and-sufficient condition for refinement is , making the obstruction central to transversal implementability. Under a stronger commutativity condition modulo and linear independence of the -stabilizers, vanishes for all classes, yielding universal refinement of all logical rotations at level . The work also discusses non-uniqueness of parity-check matrices and introduces the broader chain-complex lifting problem, suggesting a deeper topological structure underlying fault-tolerant transversal gates and linking to existing divisibility-based criteria. This provides a canonical, topological perspective that unifies prior algebraic conditions and points to future directions in a formal theory of transversal structures in quantum error correction.

Abstract

Transversal Pauli Z rotations provide a natural route to fault-tolerant logical diagonal gates in quantum CSS codes, yet their capability is fundamentally constrained. In this work, we formulate the refinement problem of realizing a logical diagonal gate by a transversal implementation with a finer discrete rotation angle and show that its solvability is completely characterized by the Bockstein homomorphism in homology theory. Furthermore, we prove that the linear independence of the X-stabilizer generators together with the commutativity condition modulo a power of two ensures the existence of transversal implementations of all logical Pauli Z rotations with discrete angles in general CSS codes. Our results identify a canonical homological obstruction governing transversal implementability and provide a conceptual foundation for a formal theory of transversal structures in quantum error correction.
Paper Structure (13 sections, 5 theorems, 34 equations)

This paper contains 13 sections, 5 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. preliminaries
  3. CSS codes as chain complexes
  4. Transversal Pauli Z rotations
  5. Commutativity modulo powers of two
  6. Transversal logical diagonal gates and Bockstein homomorphism
  7. Coefficient lifting problem: finding transversal logical square roots
  8. Existence of solution and Bockstein homomorphism
  9. Main result
  10. Discussion
  11. Beyond coefficient lifting: non-uniqueness of parity-check matrices
  12. Interpretation and relation to previous criteria
  13. Conclusion and outlook

Key Result

Theorem 1

Consider a quantum CSS code whose parity-check matrices $H_X$ and $H_Z$ satisfy the commutativity condition modulo $2^{m+1}$ ($H_XH_Z^T = 0 \pmod{2^{m+1}}$) with an integer $m \geq 1$. Let $U(\theta/2^{m-1})$ with $\theta \in \mathbb{Z}_{2^m}^n$ denote a transversal Pauli $Z$ rotation defined as For a given $[\theta] \in H_1(C;\mathbb{Z}_{2^m})$ and its representative $\theta \in \ker{H}_X \subse

Theorems & Definitions (10)

  • Definition 1: Coefficient lifting problem
  • Definition 2: Bockstein homomorphism
  • Theorem 1: Bockstein obstruction
  • proof
  • Corollary 1: Multi-level lifting
  • Proposition 1: Inner product preservation
  • Proposition 2: Linearly independent $X$-stabilizers
  • proof
  • Corollary 2: Transversal implementability of logical $Z$ rotations
  • Definition 3: Chain complex lifting problem