Perturbation Gadgets: Arbitrary Energy Scales from a Single Strong Interaction

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Fundamentally, it is believed that interactions between physical objects are two-body. Perturbative gadgets are one way to break up an effective many-body coupling into pairwise interactions: a Hamiltonian with high interaction strength introduces a low-energy space in which the effective theory appears k-body and approximates a target Hamiltonian to within precision epsilon . One caveat of existing constructions is that the interaction strength generally scales exponentially in the locality of the terms to be approximated. In this work we propose a many-body Hamiltonian construction which introduces only a single separate energy scale of order Theta (1/N^{2+delta }), for a small parameter delta >0, and for N terms in the target Hamiltonian mathbf H_mathrm {t}=sum _{i=1}^N mathbf h_i to be simulated: in its low-energy subspace, our constructed system can approximate any such target Hamiltonian mathbf H_t with norm ratios r=max _{i,jin {1,ldots ,N}}Vert mathbf h_iVert / Vert mathbf h_j Vert ={{,mathrm{O},}}(exp (exp ({{,mathrm{poly},}}N))) to within relative precision {{,mathrm{O},}}(N^{-delta }). This comes at the expense of increasing the locality by at most one, and adding an at most poly-sized ancillary system for each coupling; interactions on the ancillary system are geometrically local, and can be translationally invariant. In order to prove this claim, we borrow a technique from high energy physics—where matter fields obtain effective properties (such as mass) from interactions with an exchange particle—and employ a tiling Hamiltonian to discard all cross-terms at higher expansion orders of a Feynman–Dyson series expansion. As an application, we discuss implications for QMA-hardness of the Local Hamiltonian problem, and argue that “almost” translational invariance—defined as arbitrarily small relative variations of the strength of the local terms—is as good as non-translational invariance in many of the constructions used throughout Hamiltonian complexity theory. We furthermore show that the choice of geared limit of many-body systems, where e.g. width and height of a lattice are taken to infinity in a specific relation, can have different complexity-theoretic implications: even for translationally invariant models, changing the geared limit can vary the hardness of finding the ground state energy with respect to a given promise gap from computationally trivial, to QMAEXP-, or even BQEXPSPACE-complete.