Perturbation Gadgets: Arbitrary Energy Scales from a Single Strong Interaction

Abstract

AbstractFundamentally, 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 $$\epsilon $$ϵ. 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 $$\Theta (1/N^{2+\delta })$$Θ(1/N2+δ), for a small parameter $$\delta >0$$δ>0, and for N terms in the target Hamiltonian $$\mathbf H_\mathrm {t}=\sum _{i=1}^N \mathbf h_i$$Ht=∑i=1Nhi to be simulated: in its low-energy subspace, our constructed system can approximate any such target Hamiltonian $$\mathbf H_t$$Ht with norm ratios $$r=\max _{i,j\in \{1,\ldots ,N\}}\Vert \mathbf h_i\Vert / \Vert \mathbf h_j \Vert ={{\,\mathrm{O}\,}}(\exp (\exp ({{\,\mathrm{poly}\,}}N)))$$r=maxi,j∈{1,…,N}‖hi‖/‖hj‖=O(exp(exp(polyN))) to within relative precision $${{\,\mathrm{O}\,}}(N^{-\delta })$$O(N-δ). 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.