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Duality theorem and topological properties in local stabilizer codes

Beni Yoshida, Massachusetts Institute of Technology

(Session 8 : Saturday from 5:00-5:30)

Abstract. Beni Yoshida
Massachusetts Institute of Technology

Abstract.

Topological codes offer the possibility of a naturally fault-tolerant quantum memory, and significant progress has been made with theoretical constructions in four dimensions. However, recent work (Bravyi and Terhal; Kay and Colbeck) has ruled out the possibility of such memories in two-dimensions, and left an open question about three-dimensional topological codes. Specifically, Bravyi and Terhal show that the distance of geometrically local stabilizer codes in a D-dimensional lattice of volume L^D is bounded above by O(L^{D - 1}).

Here, we present a new approach to the problem, which sharpens these bounds, by limiting consideration to topological codes whose stabilizers are geometrically local and have translational symmetry. Using this physically reasonable assumption, and assuming that the number of logical qubits is a small number which is independent of the lattice size, we find that the logical qubits must obey a duality theorem, whereby each logical qubit may be described by a pair of weight O(L^a) and O(L^{D - a}) logical operators. This gives a full set of relations between all logical operators. It follows from this theorem that the distance of such codes is bounded above by O(Lⁿ) for 2n- and (2n+1)-dimensional lattices.

This non-trivial duality clearly distinguishes systems of even and odd dimension. One surprising consequence is that for certain definitions of topological protection, encodings are possible only in systems of even dimension. This is consistent with a fact in topological quantum field theory, that the quantum Hall effect can occur only in systems of even dimension. We illustrate the implications of this observation by showing that on a two-dimensional Bravais lattice with a small number of encoded qubits, all the logical operators have O(L) weight, such that all the logical qubits are topologically protected from local errors. This also allows us to directly relate the number of encoded qubits with the topological entropy, providing insights which will be useful in designing gapped Hamiltonians with topological properties which may be useful for quantum memories.