Fast initial qubit dephasing and the influence of substrate dimensions on error correction rates

Keeping single-qubit quantum coherence above some threshold value not far below unity is a prerequisite for fault-tolerant quantum error correction (QEC). We study the initial dephasing of solid-state qubits in the independent-boson model, which describes well recent experiments on quantum dot (QD) excitons both in bulk and in substrates of reduced geometry such as nanotubes. Using explicit expressions for the exact coherence dynamics, a minimal QEC rate is identified in terms of error threshold, temperature, and qubit-environment coupling strength. This allows us to systematically study the benefit of a current trend towards substrates with reduced dimensions.