  Perturbative gadgets are used to construct a quantum Hamiltonian whose
low-energy subspace approximates a given quantum $k$-body Hamiltonian up to an
absolute error $\epsilon$. Typically, gadget constructions involve terms with
large interaction strengths of order $\text{poly}(\epsilon^{-1})$. Here we
present a 2-body gadget construction and prove that it approximates a target
many-body Hamiltonian of interaction strength $\gamma = O(1)$ up to absolute
error $\epsilon\ll\gamma$ using interactions of strength $O(\epsilon)$ instead
of the usual inverse polynomial in $\epsilon$. A key component in our proof is
a new condition for the convergence of the perturbation series, allowing our
gadget construction to be applied in parallel on multiple many-body terms. We
also show how to apply this gadget construction for approximating 3- and
$k$-body Hamiltonians.
  The price we pay for using much weaker interactions is a large overhead in
the number of ancillary qubits, and the number of interaction terms per
particle, both of which scale as $O(\text{poly}(\epsilon^{-1}))$. Our
strong-from-weak gadgets have their primary application in complexity theory
(QMA hardness of restricted Hamiltonians, a generalized area law
counterexample, gap amplification), but could also motivate practical
implementations with many weak interactions simulating a much stronger quantum
many-body interaction.
