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 $epsilonllgamma$ 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.
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