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Logical Operators and Derived Automorphisms of Tile Codes

Nikolas P. Breuckmann, Shin Ho Choe, Jens Niklas Eberhardt, Francisco Revson Fernandes Pereira, Vincent Steffan

TL;DR

The paper addresses understanding the logical operator structure and fault-tolerant manipulation of tile codes, a class of qLDPC codes with translational symmetry. It develops a cohesive framework combining homological algebra and algebraic geometry, showing tile codes admit a canonical symplectic basis of boundary-localized logicals and can be analyzed as higher sections of shifted Koszul complexes on $\mathbb{P}^1\times\mathbb{P}^1$. Central contributions include the explicit cellular-automaton construction of a $2D^2$-sized logical basis, the algebraic and geometric resolutions that realize tile codes as $R/(f,g)$ with $f,g$ governing stabilizers, and the introduction of derived automorphisms $T_x,T_y$ that implement fault-tolerant, CNOT-like actions via lattice extension/shrinkage. The results extend naturally to higher dimensions, yielding logical-dimension counts like $6D^3$ in 3D and $24D^4$ in 4D, and provide a principled path toward integrating tile codes into fault-tolerant quantum computation through derived symmetries and boundary-anchored logicals.

Abstract

The recently introduced tile codes are a promising alternative to surface codes, combining two-dimensional locality with higher encoding efficiency. While surface codes are well understood in terms of their logical operators and boundary behavior, much less is known about tile codes. In this work, we establish a natural and precise description of their logical operator space. We prove that, under mild assumptions, any tile code admits a canonical symplectic basis of logical operators supported along lattice boundaries, which can be generated efficiently by a simple cellular automaton with the number of update rules only depending on the non-locality of the tile code. Further, we develop algebraic and algebro-geometric frameworks for tile codes, by resolving them by translationally invariant Pauli stabilizer models and showing that they arise as derived sections of a Koszul complex on $\mathbb{P}^1 \times \mathbb{P}^1$. Finally, we introduce the concept of derived automorphisms for quantum codes. These are automorphism-like operations that can exist even for codes that do not have symmetries. We explain how derived automorphisms can be implemented for tile codes in a low-overhead and fault-tolerant manner by extending the lattice on one side and shrinking it on the other. While this operation is trivial for the surface code, it induces a product of logical CNOT gates on the encoded information. Our results provide new structural insights into tile codes and lay the groundwork for tile codes as building blocks for fault-tolerant quantum computation.

Logical Operators and Derived Automorphisms of Tile Codes

TL;DR

The paper addresses understanding the logical operator structure and fault-tolerant manipulation of tile codes, a class of qLDPC codes with translational symmetry. It develops a cohesive framework combining homological algebra and algebraic geometry, showing tile codes admit a canonical symplectic basis of boundary-localized logicals and can be analyzed as higher sections of shifted Koszul complexes on . Central contributions include the explicit cellular-automaton construction of a -sized logical basis, the algebraic and geometric resolutions that realize tile codes as with governing stabilizers, and the introduction of derived automorphisms that implement fault-tolerant, CNOT-like actions via lattice extension/shrinkage. The results extend naturally to higher dimensions, yielding logical-dimension counts like in 3D and in 4D, and provide a principled path toward integrating tile codes into fault-tolerant quantum computation through derived symmetries and boundary-anchored logicals.

Abstract

The recently introduced tile codes are a promising alternative to surface codes, combining two-dimensional locality with higher encoding efficiency. While surface codes are well understood in terms of their logical operators and boundary behavior, much less is known about tile codes. In this work, we establish a natural and precise description of their logical operator space. We prove that, under mild assumptions, any tile code admits a canonical symplectic basis of logical operators supported along lattice boundaries, which can be generated efficiently by a simple cellular automaton with the number of update rules only depending on the non-locality of the tile code. Further, we develop algebraic and algebro-geometric frameworks for tile codes, by resolving them by translationally invariant Pauli stabilizer models and showing that they arise as derived sections of a Koszul complex on . Finally, we introduce the concept of derived automorphisms for quantum codes. These are automorphism-like operations that can exist even for codes that do not have symmetries. We explain how derived automorphisms can be implemented for tile codes in a low-overhead and fault-tolerant manner by extending the lattice on one side and shrinking it on the other. While this operation is trivial for the surface code, it induces a product of logical CNOT gates on the encoded information. Our results provide new structural insights into tile codes and lay the groundwork for tile codes as building blocks for fault-tolerant quantum computation.

Paper Structure

This paper contains 19 sections, 16 theorems, 48 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Stabilizer CSS codes
  3. Logical operators and derived automorphisms of tile codes
  4. Stabilizer tiles on the 2D plane
  5. Tile codes
  6. Logical operators of tile codes
  7. Derived automorphisms of tile codes
  8. Homological algebra of tile codes
  9. Background on Koszul complexes
  10. Codes on infinite lattices as Koszul complexes
  11. Resolving tile codes by Koszul complexes
  12. Geometry of tile codes
  13. Line bundles and Koszul complexes on $\mathbb{P}^1 \times \mathbb{P}^1$
  14. Local Sections and codes on infinite lattices
  15. Tile codes as higher sections
  16. ...and 4 more sections

Key Result

Lemma 1

A non-empty product of stabilizers satisfying eq:stabilizertilecorners is never supported within a strip of height or width at most $D$.

Figures (3)

  • Figure 1: In (a), we show a tile code with a pair of logical operators $\bar{\operatorname{X}}$ and $\bar{\operatorname{Z}}$. The stabilizer tiles for this code are shown in (b). The logical $\bar{\operatorname{X}}$ operators can be constructed using the cellular automaton depicted in (c). In (a1) and (a2), we highlight how the logical operator can be extended using linear combinations of the rules in (c). From an algebraic perspective, the logical dimension of a tile code corresponds to the number of intersection points of the zero sets of the polynomials defining the stabilizer tiles; for an illustration, see (e).
  • Figure 2: An example of a derived automorphism for a tile code: We track the logical operator $\bar{\operatorname{X}}_1$ visualized in (a) where we stick to the labeling convention also used in \ref{['fig:main_figure']}. By extending the lattice to the left and measuring out qubits on the right as described in \ref{['thm: derivedautomorphismpedestrian']}, the logical operator $\bar{\operatorname{X}}_1$ gets mapped to the logical operator in (b). Multiplying by stabilizers to bring this operator in canonical form, we easily see that this logical operator is equivalent to $\bar{\operatorname{X}}_5$.
  • Figure 3: Koszul complexes and associated CSS codes supported on the four different quadrants.

Theorems & Definitions (36)

  • Lemma 1
  • Lemma 2
  • proof
  • Theorem 1
  • Lemma 3
  • proof
  • Corollary 1
  • proof
  • Corollary 2
  • proof
  • ...and 26 more