Abstract
We construct parent Hamiltonians involving only local 2-body interactions for a broad class of projected entangled pair states (PEPS). Making use of perturbation gadget techniques, we define a perturbative Hamiltonian acting on the virtual PEPS space with a finite order low energy effective Hamiltonian that is a gapped, frustration-free parent Hamiltonian for an encoded version of a desired PEPS. For topologically ordered PEPS, the ground space of the low energy effective Hamiltonian is shown to be in the same phase as the desired state to all orders of perturbation theory. An encoded parent Hamiltonian for the double semion string net ground state is explicitly constructed as a concrete example.
Highlights
Projected Entangled Pair States (PEPS) are a class of quantum states well suited for describing the ground states of interacting quantum many-body systems [1,2,3,4,5,6]
We argue that isometric matrix product operator (MPO)-injective PEPS [22] with trivial so-called generalized inverse satisfy the requirements of our construction; this class is known to include string-net ground states and (G, ω)-isometric PEPS
Our analysis demonstrates that the low energy effective Hamiltonian of our model is a parent Hamiltonian for the desired state, but it does not prove that this effective Hamiltonian is a good description of the low energy physics of our system
Summary
Projected Entangled Pair States (PEPS) are a class of quantum states well suited for describing the ground states of interacting quantum many-body systems [1,2,3,4,5,6]. Though these interactions act only within a finite sized region, there will still generally be a large number of qudits within this region For this reason these interactions may be challenging to implement experimentally, and it may be preferable to find an alternative parent Hamiltonian with interactions involving at most two neighbouring quantum systems (2-local interactions), whose ground state is a desired PEPS. We make use of stability results for topologically ordered states [25,26,27,28] to prove that the ground space of our effective Hamiltonian remains in the same phase to arbitrarily high order of perturbation theory For this reason, our results apply only to states with parent Hamiltonians satisfying the local topological quantum order conditions [27].
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