TITLE: Quantum PCP Theorem
QUESTION [9 upvotes]: Although I think I know the answers to these, I'd just like to collect them all in one place.
What is the quantum PCP theorem, what implications does its proof have for simulation of Hamiltonians and is following Irit Dinur's reproof of the classical version the best/only current mode of attack (and if so why?) What is the sort of math/physics/theoretical CS background needed to approach this problem?

REPLY [3 votes]: One more comment:
The quantum PCP conjecture implies that calculating the energy of the Gibbs state of a quantum system in constant temperature is QMA-Complete. This answers the question: 

what implications does its proof have for simulation of Hamiltonians?

Given a Hamiltonian $H$, the Gibbs state is defined by:
$$\rho_{\mathrm{gibbs}} = \frac{1}{Z}\exp(-H/T)$$
where the normalization factor $Z=\operatorname{Tr}(\exp(-H/T))$.
It is known that the Gibbs state minimizes the free energy (H - TS):
$$\rho_{\mathrm{gibbs}}=\operatorname{argmin}_{\rho \succeq 0, \operatorname{Tr}(\rho)=1} \operatorname{Tr}(H\rho) + T \cdot \operatorname{Tr}(\rho \ln \rho)$$
Let $\rho_{\min}$ be one of the pure ground states of $H$, with energy $E_0$. Since $\rho_{\min}$ is a pure state, $Tr(\rho \ln \rho)=0$.
Therefore, 
$$\operatorname{Tr}(H\rho_{\mathrm{gibbs}}) + T \cdot Tr(\rho_{\mathrm{gibbs}} \ln \rho_{\mathrm{gibbs}}) \leq \operatorname{Tr}(H \rho_{min}) = E_0.$$
Since $\operatorname{Tr}(\rho \ln \rho) \leq n \ln d$, where $n$ is the number of qudits (of dimension $d$ in the systems,
$$\operatorname{Tr}(H\rho_{gibbs}) \leq E_0 + T n \ln d .$$
By choosing $T = O\left(\frac{\epsilon}{\ln d}\right)$ one gets an additive approximation of $\epsilon n \leq \epsilon m$ to the ground energy, which is $QMA$-hard by the quantum PCP conjecture ($m$ is the number of terms in the Hamiltonian, therefore it is at least linear in the number of qubits in the system).
I believe this result is a folklore, and clearly also holds in the classical setting. I learned about it from the following paper:
Brandão, F. G., & Harrow, A. W. (2013, June). Product-state approximations to quantum ground states. In Proceedings of the 45th annual ACM symposium on Symposium on theory of computing (pp. 871-880). ACM.