  Knill demonstrated a fault-tolerant quantum computation scheme based on
concatenated error-detecting codes and postselection with a simulated error
threshold of 3% over the depolarizing channel. %We design a two-dimensional
architecture for fault-tolerant quantum computation based on Knill's
postselection scheme. We show how to use Knill's postselection scheme in a
practical two-dimensional quantum architecture that we designed with the goal
to optimize the error correction properties, while satisfying important
architectural constraints. In our 2D architecture, one logical qubit is
embedded in a tile consisting of $5\times 5$ physical qubits. The movement of
these qubits is modeled as noisy SWAP gates and the only physical operations
that are allowed are local one- and two-qubit gates. We evaluate the practical
properties of our design, such as its error threshold, and compare it to the
concatenated Bacon-Shor code and the concatenated Steane code. Assuming that
all gates have the same error rates, we obtain a threshold of $3.06\times
10^{-4}$ in a local adversarial stochastic noise model, which is the highest
known error threshold for concatenated codes in 2D. We also present a Monte
Carlo simulation of the 2D architecture with depolarizing noise and we
calculate a pseudo-threshold of about 0.1%. With memory error rates one-tenth
of the worst gate error rates, the threshold for the adversarial noise model,
and the pseudo-threshold over depolarizing noise, are $4.06\times 10^{-4}$ and
0.2%, respectively. In a hypothetical technology where memory error rates are
negligible, these thresholds can be further increased by shrinking the tiles
into a $4\times 4$ layout.
