  $ \newcommand{\Xlin}{\mathcal{X}} \newcommand{\Zlin}{\mathcal{Z}}
\newcommand{\C}{\mathbb{C}} $We give a quantum multiprover interactive proof
system for the local Hamiltonian problem in which there is a constant number of
provers, questions are classical of length polynomial in the number of qubits,
and answers are of constant length. The main novelty of our protocol is that
the gap between completeness and soundness is directly proportional to the
promise gap on the (normalized) ground state energy of the Hamiltonian. This
result can be interpreted as a concrete step towards a quantum PCP theorem
giving entangled-prover interactive proof systems for QMA-complete problems.
  The key ingredient is a quantum version of the classical linearity test of
Blum, Luby, and Rubinfeld, where the function $f:\{0,1\}^n\to\{0,1\}$ is
replaced by a pair of functions $\Xlin, \Zlin:\{0,1\}^n\to \text{Obs}_d(\C)$,
the set of $d$-dimensional Hermitian matrices that square to identity. The test
enforces that (i) each function is exactly linear,
$\Xlin(a)\Xlin(b)=\Xlin(a+b)$ and $\Zlin(a) \Zlin(b)=\Zlin(a+b)$, and (ii) the
two functions are approximately complementary, $\Xlin(a)\Zlin(b)\approx
(-1)^{a\cdot b} \Zlin(b)\Xlin(a)$.
