Perturbation Gadgets: Arbitrary Energy Scales from a Single Strong Interaction

In this work we propose a many-body Hamiltonian construction which introduces only a single separate energy scale of order Θ\(1/N^2+δ\), for a small parameter δ>0, and for N terms in the target Hamiltonian. In its low-energy subspace, the construction can approximate any normalized target Hamiltonian H_t=sum_i=1^N h_i with norm ratios r=max_{i,jin\{1,ldots,N\}}\|h_i\| / \| h_j \|=O\(exp(exp(poly n\))) to within relative precision O\(N^-δ\). This comes at the expense of increasing the locality by at most one, and adding an at most poly-sized ancilliary system for each coupling; interactions on the ancilliary system are geometrically local, and can be translationally-invariant. 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.