  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 the 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.
