Perturbation Gadgets: Arbitrary Energy Scales from a Single Strong Interaction

Abstract

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. In its low-energy subspace, the construction can approximate any normalized target Hamiltonian $H_{\mathrm{t}}={\sum }_{i=1}^N{h}_i$ with norm ratios $r={\max }_{i,j\in \{1,\dots ,N\}}\Vert h_i\Vert /\Vert h_j\Vert =O(\exp (\exp (\mathrm{poly}n)))$ to within relative precision $O(N^{-\delta })$. 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.

46 pages, significantly extended proofs

References in corpus (3)

• The power of quantum systems on a line
• Hamiltonian Quantum Cellular Automata in 1D
• The Feynman-Kitaev computer's clock: bias, gaps, idling and pulse tuning

Cited by in corpus (4)

• Translationally-Invariant Universal Quantum Hamiltonians in 1D
• Going Beyond Gadgets: The Importance of Scalability for Analogue Quantum Simulators
• Cost of Emulating a Small Quantum Annealing Problem in the Circuit-Model
• Perturbative gadgets for gate-based quantum computing: Non-recursive constructions without subspace restrictions