Perturbation Gadgets: Arbitrary Energy Scales from a Single Strong Interaction

Abstract

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 . One caveat of existing constructions is that the interaction strength generally scales exponentially in the locality of the terms to be approximated, i.e. as Ω(1/ k ); if = 1/poly n in the system size n, as is necessary for e.g. QMA-hardness constructions, the energy differences become highly unphysical.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-i.e. all local terms of the simulator have either this norm, or one of O(1). In its low-energy subspace, we can approximate any normalized target Hamiltonian H t = N i=1 h i with norm ratios r = h i 2 / h j 2 = 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; the ancillas being qutrits for exponential scaling, and qudits for doubly exponential r; the interactions on the ancilliary 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 a tiling Hamiltonian to drop all cross terms at higher expansion orders, which simplifies the analysis of a traditional Feynman-Dyson series expansion.As an application, we discuss implications for QMA-hardness of the L H 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 QMA EXP -, or even BQEXPSPACE-complete.

Translationally Invariant Universal Quantum Hamiltonians in 1D

“…However, translationally invariant spin systems are common in condensed matter models of real-world materials, whereas models with precisely tuned interactions that differ from site to site are less realistic. It is known that QMAhardness of approximating the ground state energy to 1/ poly precision in the system size is a property of non-translationally invariant couplings, that prevails even when those couplings are arbitrarily close to identical [33,Cor. 21].…”

Translationally Invariant Universal Classical Hamiltonians

“…However, translationally-invariant spin systems are common in condensed matter models of real-world materials, whereas models with precisely-tuned interations that differ from site to site are less realistic. It is known that QMA-hardness of approximating the ground state energy to 1/poly precision in the system size is a property of non-translationally-invariant couplings, that prevails even when those couplings are arbitrarily close to identical [Bau19,Cor. 21].…”

Precision of quantum simulation of all-to-all coupling in a local architecture

“…A fermionic or spin chain near its critical point would have worked just as well. We note that other methods [10,13,14] for the study of perturbative gadgets can provide better performance guarantees than the Schrieffer-Wolff, but we choose to use it as its application is straightforward and it maintains the information about the basis change induced by the presence of the gadget.…”