  We propose an analog quantum simulator that uses ion traps to realize the
many-body electron-electron Coulomb interaction of an electron gas. This
proposal maps a system that is difficult to solve and control to an
experimentally-feasible setup that can be realized with current technologies.
Using a dilatation transform, we show that ion traps can efficiently simulate
electronic Coulomb interactions. No complexity overhead is added if only the
energy spectrum is desired, and only a simple unitary transform is needed on
the initial state otherwise. The runtime of the simulation is found to be much
shorter than the timescale of the corresponding electronic system, minimizing
susceptibility of the proposed quantum simulator to external noise and
decoherence. This proposal works in any number of dimensions, and could be used
to simulate different topological phases of electrons in graphene-like
structures, by using ions trapped in honeycomb lattices.
