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Local stabilizer codes in three dimensions without string logical operators

Jeongwan Haah

TL;DR

This work constructs and analyzes 3D local stabilizer codes (cubic codes) with the goal of achieving self-correcting quantum memory by avoiding string-like logical operators. It introduces the concept of logical string segments to rigorously define string-like operators on discrete lattices and proves that several 3D cubic codes (Codes 1–4) are free of string logical operators while maintaining macroscopic code distance d ≥ L. The paper provides an explicit complete (up to symmetries) catalog of cubic codes, derives how many logical qubits are encoded on finite periodic lattices (k(L)) with empirical formulas, and discusses the existence of plane logical operators alongside the absence of string operators. It also addresses thermal stability, arguing that partial logical implementations incur boundary-energy costs and situates these results within the broader context of stabilizer codes and potential routes toward self-correcting memory, while noting that a full thermal-stability proof remains open.

Abstract

We suggest concrete models for self-correcting quantum memory by reporting examples of local stabilizer codes in 3D that have no string logical operators. Previously known local stabilizer codes in 3D all have string-like logical operators, which make the codes non-self-correcting. We introduce a notion of "logical string segments" to avoid difficulties in defining one dimensional objects in discrete lattices. We prove that every string-like logical operator of our code can be deformed to a disjoint union of short segments, and each segment is in the stabilizer group. The code has surface-like logical operators whose partial implementation has unsatisfied stabilizers along its boundary.

Local stabilizer codes in three dimensions without string logical operators

TL;DR

This work constructs and analyzes 3D local stabilizer codes (cubic codes) with the goal of achieving self-correcting quantum memory by avoiding string-like logical operators. It introduces the concept of logical string segments to rigorously define string-like operators on discrete lattices and proves that several 3D cubic codes (Codes 1–4) are free of string logical operators while maintaining macroscopic code distance d ≥ L. The paper provides an explicit complete (up to symmetries) catalog of cubic codes, derives how many logical qubits are encoded on finite periodic lattices (k(L)) with empirical formulas, and discusses the existence of plane logical operators alongside the absence of string operators. It also addresses thermal stability, arguing that partial logical implementations incur boundary-energy costs and situates these results within the broader context of stabilizer codes and potential routes toward self-correcting memory, while noting that a full thermal-stability proof remains open.

Abstract

We suggest concrete models for self-correcting quantum memory by reporting examples of local stabilizer codes in 3D that have no string logical operators. Previously known local stabilizer codes in 3D all have string-like logical operators, which make the codes non-self-correcting. We introduce a notion of "logical string segments" to avoid difficulties in defining one dimensional objects in discrete lattices. We prove that every string-like logical operator of our code can be deformed to a disjoint union of short segments, and each segment is in the stabilizer group. The code has surface-like logical operators whose partial implementation has unsatisfied stabilizers along its boundary.

Paper Structure

This paper contains 21 sections, 6 theorems, 70 equations, 12 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Complete list of cubic codes
  3. Commutation relations of corner operators
  4. Macroscopic code distance
  5. Logical string segments
  6. Reduction to flat segments
  7. Confusing constraints
  8. Inconsistent quasi-period
  9. Number of encoded qubits and logical operators in finite periodic lattices
  10. Number of encoded qubits
  11. Plane logical operators
  12. Thermal stability
  13. Discussion
  14. Proof of Theorem \ref{['thm:Code1234are_free_of_strings']} for Code 3, 4
  15. Cubic codes with string logical operators
  16. ...and 6 more sections

Key Result

Lemma 1

Given a commutation relation $\omega$ of Pauli operators, all realizations $P$ of $\omega$ using minimum number of qubits are equivalent up to symplectic transformations.

Figures (12)

  • Figure 1: Stabilizer Generators for non-CSS(top) and CSS(bottom) cubic codes. Throughout the paper we fix the coordinate system as shown.
  • Figure 2: Stabilizer generators for non-CSS Code 0, and CSS Code 1, 2, 3, and 4. They all have code distance $\ge L$ (Theorem \ref{['thm:macroscopic_code_distance']}). The bottom four are free of string logical operators. See Section \ref{['section:logical_string_segments']}.
  • Figure 3: Proof of macroscopic code distance. Deformation of a finite logical operator is depicted for each Code. For Code 1,2,3,4, the logical operator of $X$-type is considered.
  • Figure 4: Deformations of $X$-type logical string segments of Code 1 and 2. The small cubes and the filled dots are the anchors. The good edges for erasing are easily identified from the figures.
  • Figure 5: Deformation of flat logical string segement $\zeta^{(1)}_y$. The bottom two figures are implied by 'confusing constraints'. If $\zeta^{(1)}_y$ is long compared to its width, it is equivalent to a disconnected one.
  • ...and 7 more figures

Theorems & Definitions (15)

  • Lemma 1
  • proof
  • Theorem 1
  • Lemma 2
  • Definition 1
  • proof
  • Definition 2
  • Definition 3
  • Definition 4
  • Theorem 2
  • ...and 5 more