  Fault-tolerant quantum computation demands significant resources: large
numbers of physical qubits must be checked for errors repeatedly to protect
quantum data as logic gates are implemented in the presence of noise. We
demonstrate that an approach based on the color code can lead to considerable
reductions in the resource overheads compared with conventional methods, while
remaining compatible with a two-dimensional layout. We propose a lattice
surgery scheme that exploits the rich structure of the color-code phase to
perform arbitrary pairs of commuting logical Pauli measurements in parallel
while keeping the space cost low. Compared to lattice surgery schemes based on
the surface code with the same code distance, our approach yields about a
$3\times$ improvement in the space-time overhead, obtained from a combination
of a $1.5\times$ improvement in spatial overhead together with a $2\times$
speedup due to the parallelisation of commuting logical measurements. Even when
taking into account the color code's lower error threshold using current
decoders, the overhead is reduced by 10\% at a physical error rate of $10^{-3}$
and by 50\% at $10^{-4}$.
