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A Classification of Invertible Stabilizer Codes

Roman Geiko, Georgii Shuklin

TL;DR

The paper develops a rigorous framework to classify invertible translation-invariant stabilizer codes by reformulating stabilizer codes as lagrangians in an algebraic setting based on $(R,S)$-modules and linking forms. It generalizes Pauli stabilizers to twisted quasi-Pauli groups and uses relative L-theory to compute equivalence classes, identifying the three-dimensional case with the Witt group of 2D abelian topological orders. The authors derive a complete determination of the reduced groups $ ext{E}_d$ and connect them to a generalized Eilenberg-MacLane spectrum $ ext{E}$ that encodes the cohomological structure of invertible stabilizer phases, suggesting a homotopical classification aligned with Kitaev’s conjecture. They also articulate a comprehensive equivalence framework incorporating stacking, condensation, and finite-depth quantum circuits, paving the way for connecting invertible stabilizer codes to QCA and topological order data on lattices.

Abstract

We develop a framework for the classification of invertible translation-invariant stabilizer codes modulo condensation and stabilization with simple codes. We introduce generalizations of the Pauli groups of local unitaries for quantum systems of qudits on cubic lattices and analyze stabilizer Hamiltonians whose terms are chosen from these groups. We define invertible stabilizer codes to be ground states of stabilizer Hamiltonians with trivial topological charges and completely classify them in any spatial dimension in terms of relative L-theory groups. In particular, we show that the group of equivalence classes of such codes in three spatial dimensions is isomorphic to the Witt group of abelian topological orders in two spatial dimensions. Additionally, we propose the spectrum of the relative L-theory as a representative of the generalized cohomology theory corresponding to the invertible stabilizer states.

A Classification of Invertible Stabilizer Codes

TL;DR

The paper develops a rigorous framework to classify invertible translation-invariant stabilizer codes by reformulating stabilizer codes as lagrangians in an algebraic setting based on -modules and linking forms. It generalizes Pauli stabilizers to twisted quasi-Pauli groups and uses relative L-theory to compute equivalence classes, identifying the three-dimensional case with the Witt group of 2D abelian topological orders. The authors derive a complete determination of the reduced groups and connect them to a generalized Eilenberg-MacLane spectrum that encodes the cohomological structure of invertible stabilizer phases, suggesting a homotopical classification aligned with Kitaev’s conjecture. They also articulate a comprehensive equivalence framework incorporating stacking, condensation, and finite-depth quantum circuits, paving the way for connecting invertible stabilizer codes to QCA and topological order data on lattices.

Abstract

We develop a framework for the classification of invertible translation-invariant stabilizer codes modulo condensation and stabilization with simple codes. We introduce generalizations of the Pauli groups of local unitaries for quantum systems of qudits on cubic lattices and analyze stabilizer Hamiltonians whose terms are chosen from these groups. We define invertible stabilizer codes to be ground states of stabilizer Hamiltonians with trivial topological charges and completely classify them in any spatial dimension in terms of relative L-theory groups. In particular, we show that the group of equivalence classes of such codes in three spatial dimensions is isomorphic to the Witt group of abelian topological orders in two spatial dimensions. Additionally, we propose the spectrum of the relative L-theory as a representative of the generalized cohomology theory corresponding to the invertible stabilizer states.

Paper Structure

This paper contains 22 sections, 3 theorems, 92 equations.

Table of Contents

  1. Introduction
  2. $(R,S)$-modules
  3. Compactification
  4. Topological charges
  5. Linking forms and formations
  6. Linking forms
  7. Linking formations
  8. L-groups
  9. Spectra
  10. Groups of unitaries
  11. Pauli group
  12. Lattice Pauli group
  13. Local bases and $(R,S)$-modules
  14. Quasi-Pauli groups and standard anti-hermitian linking forms over $(R,S)$
  15. Twisted quasi-Pauli groups
  16. ...and 7 more sections

Key Result

Proposition 2.3

Let $M=\operatorname{coker} \partial$ be an $(R,S)$-module and let $M_{\ell}=M\otimes_{R}R_{\ell}$. Then, the number of elements $|M_{\ell}|$ in $M_{\ell}$ is given byWe note that the determinant is defined up to the units of $R$. Throughout the paper, we choose a representative which is a positive

Theorems & Definitions (15)

  • Definition 2.1: Ranicki1981ExactSI, p.181
  • Remark 2.2
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Definition 3.1: Ranicki1981ExactSI, p. 181
  • Definition 3.2: Ranicki1981ExactSI, p.223
  • Remark 3.3
  • Definition 3.4
  • ...and 5 more