Quantum simulations under translational symmetry

We investigate the power of quantum systems for the simulation of Hamiltonian time evolutions on a cubic lattice under the constraint of translational invariance. Given a set of translationally invariant local Hamiltonians and short-range interactions we determine time evolutions which can and those that cannot be simulated. Whereas for general spin systems no finite universal set of generating interactions is shown to exist, universality turns out to be generic for quadratic bosonic and fermionic nearest-neighbor interactions when supplemented by all translationally invariant on-site Hamiltonians.

Figure 1

Two-dimensional fermionic lattice with box $B_2$ (see proof of theorem 2). The interactions along $e_1$ and $e_2$ (solid arrows) are used to simulate the interactions in the direction $2(e_1+e_2)$ (dashed arrow).

Figure 2

Boxes $B_z$ and $B_{z+1}$. Any vector $v^{(z+1)}∊B_{z+1}$ can be decomposed into a sum of vectors from $B_z$, i.e., $v^{(z+1)}=p^{\left(z\right)}+q^{\left(z\right)}$, where $p^{\left(z\right)},q^{\left(z\right)}∊B_z$, such that $q^{\left(z\right)}-p^{\left(z\right)}∊B_z$ (see proofs of theorems 2 and 4).