[Paper Review] Quantum codes on a lattice with boundary

Sergey Bravyi, Alexei Kitaev|ArXiv.org|Nov 20, 1998

Quantum Computing Algorithms and ArchitectureComputer Science6 references412 citations

TL;DR

This paper introduces a new class of topological quantum codes on a 2D lattice with boundary, where qubits are placed on lattice edges and stabilizer generators act on vertices and faces. The logical qubits are encoded via relative homology classes of cycles on the lattice, and the code distance is determined by the shortest path connecting like-type boundaries, enabling fault-tolerant quantum computation with topological protection on open surfaces.

ABSTRACT

A new type of local-check additive quantum code is presented. Qubits are associated with edges of a 2-dimensional lattice whereas the stabilizer operators correspond to the faces and the vertices. The boundary of the lattice consists of alternating pieces with two different types of boundary conditions. Logical operators are described in terms of relative homology groups.

Motivation & Objective

• To extend topological quantum codes, such as the toric code, to lattices with boundaries while preserving topological protection.
• To define two distinct boundary types—x-boundary and z-boundary—that stabilize anyonic excitations differently, ensuring topological order remains robust.
• To establish a correspondence between logical operators and relative homology groups H₁(Q,V,Z₂) and H₁(Q,V*,Z₂), enabling logical qubit encoding on open surfaces.
• To characterize the code distance as the minimal length of a path connecting two boundaries of the same type, ensuring fault tolerance against local errors.
• To explain the physical origin of the two boundary types in terms of topological quantum order and anyonic statistics, showing only two stable boundary types exist under rigidity constraints.

Proposed method

• Assign qubits to edges of a 2D square lattice with alternating x- and z-boundaries, where boundary conditions are defined by the absence of certain stabilizer generators.
• Define stabilizer operators Aₛ (vertex operators) and Bₚ (face operators) as products of σˣ and σᶻ over edges in star and boundary of a face, respectively, with modified definitions for incomplete faces near boundaries.
• Use relative homology groups H₁(Q,V,Z₂) and H₁(Q,V*,Z₂) to classify logical operators, where V and V* denote x- and z-boundary components.
• Construct logical operators as Y([c],[c*]) = ∏ᵢ∈c σᶻᵢ ∏ⱼ∈c* σˣⱼ, with c and c* being 1-cycles on the primal and dual lattices, respectively.
• Ensure logical operators commute with all stabilizers and act nontrivially only when the relative homology class is nontrivial, preserving logical information.
• Derive the code distance d = min{min_{[c]≠0} |supp(c)|, min_{[c*]≠0} |supp(c*)|}, which corresponds to the shortest path connecting two boundaries of the same type.

Experimental results

Research questions

• RQ1 How can topological quantum codes be generalized from closed surfaces (like the torus) to surfaces with boundary while maintaining topological error correction?
• RQ2 What are the physical and topological constraints that determine the allowed types of boundary conditions in a topological quantum code?
• RQ3 How do logical qubits emerge in such codes, and what is the precise mathematical structure (e.g., homology groups) that classifies them?
• RQ4 What determines the code distance in a boundary code, and how does it relate to error correction capability?
• RQ5 Why are only two types of rigid boundaries—x-boundary and z-boundary—possible in a system with topological quantum order?

Key findings

• The number of logical qubits encoded in the code is given by dim H₁(Q,V,Z₂) = dim H₁(Q,V*,Z₂), which for a disk with k x-boundaries and k z-boundaries is k−1.
• The code distance d equals min{n+1, m+1} for an n×m lattice, and protects against up to ⌊(d−1)/2⌋ local errors.
• Logical operators are constructed as products of Pauli operators along cycles c and c* in the primal and dual lattices, with their action determined by relative homology classes.
• The two boundary types (x and z) correspond to the ability of electric and magnetic anyons to condense at the boundary, respectively, and are the only two stable boundary types under rigidity constraints.
• The construction generalizes beyond square lattices to any pair of mutually dual lattices on a surface with boundary split into x- and z-type components.
• The stability of the topological order near the boundary is explained by anyonic statistics: only bosonic anyons (electric or magnetic charges) can condense at boundaries, while fermionic composites cannot.