local constraints, each involving a constant number of qubits.  In a sense, then, this paper completes a ‘trilogy’ of which [@aar:adv; @aar:learn] were the first two installments.

Here is a formal statement of our result.

\[mainthm0\] Let $c, \varepsilon>0$, and let $\rho$ be any $n$-qubit quantum state.  Then there exists a $2$-local Hamiltonian $H$ on $\operatorname*{poly}\left(n, \frac{1}{\varepsilon}  \right)  $ qubits with unique ground state $\left\vert \psi\right\rangle \left\langle
\psi\right\vert$, and a transformation $C\longrightarrow C^{\prime}$ of quantum circuits, computable in time $\operatorname*{poly}\left( n, 1 / \varepsilon  \right)$ given $H$, such that the following holds: $\left\vert C^{\prime}\left(  \left\vert \psi\right\rangle \left\langle \psi\right\vert \right)  -C\left(  \rho\right)  \right\vert \leq\varepsilon$ for any measurement $C$ definable by a quantum circuit of size $n^c$.  (Here $C\left(  \rho\right)$ is the probability that $C$ accepts $\rho$.)

In other words, the ground states of local Hamiltonians are ‘universal quantum states’ in a very non-obvious sense.  For example, suppose you own a quantum software store, which sells quantum states $\rho
$ that can be fed as input to quantum computers.  Then our result says that *ground states of local Hamiltonians are the only kind of state you ever need to stock*.  What makes this surprising is that being a good piece of quantum 