  We consider measurement-based quantum computation using the state of a
spin-lattice system in equilibrium with a thermal bath and free to evolve under
its own Hamiltonian. Any single qubit measurements disturb the system from
equilibrium and, with adaptive measurements performed at a finite rate, the
resulting dynamics reduces the fidelity of the computation. We show that it is
possible to describe the loss in fidelity by a single quantum operation on the
encoded quantum state that is independent of the measurement history. To
achieve this simple description, we choose a particular form of spin-boson
coupling to describe the interaction with the environment, and perform
measurements periodically at a natural rate determined by the energy gap of the
system. We found that an optimal cooling exists, which is a trade-off between
keeping the system cool enough that the resource state remains close to the
ground state, but also isolated enough that the cooling does not strongly
interfere with the dynamics of the computation. For a sufficiently low
temperature we obtain a fault-tolerant threshold for the couplings to the
environment.
