Perturbative Gadgets at Arbitrary Orders

Adiabatic quantum algorithms are often most easily formulated using many-body interactions. However, experimentally available interactions are generally two-body. In 2004, Kempe, Kitaev, and Regev introduced perturbative gadgets, by which arbitrary three-body effective interactions can be obtained using Hamiltonians consisting only of two-body interactions. These three-body effective interactions arise from the third order in perturbation theory. Since their introduction, perturbative gadgets have become a standard tool in the theory of quantum computation. Here we construct generalized gadgets so that one can directly obtain arbitrary k-body effective interactions from two-body Hamiltonians. These effective interactions arise from the k order in perturbation theory. 1 Perturbative Gadgets Perturbative gadgets were introduced to construct a two-local Hamiltonian whose low energy effective Hamiltonian corresponds to a desired three-local Hamiltonian. They were originally developed by Kempe, Kitaev, and Regev in 2004 to prove the QMA-completeness of the 2-local Hamiltonian problem and to simulate 3-local adiabatic quantum computation using 2-local adiabatic quantum computation[4]. Perturbative gadgets have subsequently been used to simulate spatially nonlocal Hamiltonians using spatially local Hamiltonians[6], and to find a minimal set of set of interactions for universal adiabatic quantum computation[1]. It was also pointed out in [6] that perturbative gadgets can be used recursively to obtain k-local effective interactions using a 2-local Hamiltonian. Here we generalize perturbative gadgets to directly obtain arbitrary k-local effective interactions by a single application of k order perturbation theory. Our formulation is based on a perturbation expansion due to Bloch[2]. A k-local operator is one consisting of interactions between at most k qubits. A general k-local Hamiltonian on n qubits can always be expressed as a sum of r terms,