  We show that any classical two-way communication protocol with shared
randomness that can approximately simulate the result of applying an arbitrary
measurement (held by one party) to a quantum state of $n$ qubits (held by
another), up to constant accuracy, must transmit at least $\Omega(2^n)$ bits.
This lower bound is optimal and matches the complexity of a simple protocol
based on discretisation using an $\epsilon$-net. The proof is based on a lower
bound on the classical communication complexity of a distributed variant of the
Fourier sampling problem. We obtain two optimal quantum-classical separations
as easy corollaries. First, a sampling problem which can be solved with one
quantum query to the input, but which requires $\Omega(N)$ classical queries
for an input of size $N$. Second, a nonlocal task which can be solved using $n$
Bell pairs, but for which any approximate classical solution must communicate
$\Omega(2^n)$ bits.
