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The presence of symmetries, be they discrete or continuous, in a physicalsystem typically leads to a reduction in the problem to be solved. Here wereport that neither translational invariance nor rotational invariance reducethe computational complexity of simulating Hamiltonian dynamics; the problem isstill BQP complete, and is believed to be hard on a classical computer. This isachieved by designing a system to implement a Universal Quantum Interface, adevice which enables control of an entire computation through the control of afixed number of spins, and using it as a building-block to entirely remove theneed for control, except in the system initialisation. Finally, it is shownthat cooling such Hamiltonians to their ground states in the presence of randommagnetic fields solves a QMA-complete problem.

The Computational Power of Symmetric Hamiltonians