Stabilizer subsystem codes with spatially local generators

Abstract

We derive new tradeoffs for reliable quantum information storage in a 2D local architecture based on subsystem quantum codes. Our results apply to stabilizer subsystem codes, that is, stabilizer codes in which part of the logical qubits does not encode any information. A stabilizer subsystem code can be specified by its gauge group - a subgroup of the Pauli group that includes the stabilizers and the logical operators on the unused logical qubits. We assume that the physical qubits are arranged on a two-dimensional grid and the gauge group has spatially local generators such that each generator acts only on a few qubits located close to each other. Our main result is an upper bound kd = O(n), where k is the number of encoded qubits, d is the minimal distance, and n is the number of physical qubits. In the special case when both gauge group and the stabilizer group have spatially local generators, we derive a stronger bound kd <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> = O(n) which is tight up to a constant factor.