N ov 1 99 8 Quantum codes on a lattice with boundary . ∗

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. Since Shor’s discovery of the quantum error correcting codes [1], a large number of examples have been constructed. Most of them belong to the class of additive codes [2]. More specifically, codewords of an additive code form a common eigenspace of several commuting stabilizer operators, each of which is a product of Pauli matrices acting on different qubits. A peculiar property of toric codes [3, 4, 5] is that the stabilizer operators are local: each of them involves only 4 qubits, each qubit is involved only in 4 stabilizer operators, while the code distance goes to infinity. (The number 4 is not a matter of principle; it could be any constant). Furthermore, this locality is geometric while the codeword subspace and error correction properties are related to the topology of the torus. Operators acting on codewords are associated with 1-dimensional homology and cohomology classes of the torus (with Z2 coefficients). Similar codes can be defined for lattices on an arbitrary closed 2-D surface. In this paper we extend this definition to surfaces with boundary. A similar construction has been proposed by M. Freedman and D. Meyer [6]. Let us briefly recall the definition and the properties of the toric codes. In a toric code, qubits are associated with edges of an n× n square lattice on the torus T . To each vertex s and each face p we assign a stabilizer operator of the form: