  The presence of symmetries, be they discrete or continuous, in a physical
system typically leads to a reduction in the problem to be solved. Here we
report that neither translational invariance nor rotational invariance reduce
the computational complexity of simulating Hamiltonian dynamics; the problem is
still BQP complete, and is believed to be hard on a classical computer. This is
achieved by designing a system to implement a Universal Quantum Interface, a
device which enables control of an entire computation through the control of a
fixed number of spins, and using it as a building-block to entirely remove the
need for control, except in the system initialisation. Finally, it is shown
that cooling such Hamiltonians to their ground states in the presence of random
magnetic fields solves a QMA-complete problem.
