Sign problem and stoquastic Hamiltonians

Question

What is the sign problem in quantum simulations and how do stoquastic Hamiltonians solve it? I tried searching for a good reference that explains this but explanations regarding what the sign problem is are very hand-wavy.

A related question, for stoquastic Hamiltonians are only off-diagonal terms zero or non-positive or are diagonal terms also zero and non-positive? Slide 2 here suggests all matrix terms are non-positive, but that means the diagonals have to all be zero, as a Hamiltonian is positive semi-definite and positive semi-definite matrices have non-negative diagonal entries.

Answer 1

Stoquastic Hamiltonians do not suffer from the "sign problem" since for any observable A $ \langle A \rangle = \frac{1}{Z} \cdot \text{Tr } Ae^{-\beta H} = \frac{1}{Z} \cdot \sum_c A(c)p(c) $ and all weights $ p(c) \geq 0 $.

A simple proof:

Define $ G = d I - H $, where $ d = \text{max}_i H_{ii} $. All matrix elements of $ G $ are non-negative and so this holds for $G^n, \forall n$.

\begin{align*} \langle A \rangle & = \frac{1}{Z} \cdot \text{Tr } A e^{-\beta H} \\ &= \frac{1}{Z} \cdot \text{Tr } A e^{-\beta (dI - G)} \\ &= \frac{e^{-\beta d}}{Z} \cdot \text{Tr } A e^{\beta G} \\ &= \frac{e^{-\beta d}}{Z} \cdot \sum_n \frac{\beta^n}{n!} \text{Tr }A G^n \\ &= \frac{e^{-\beta d}}{Z} \cdot \sum_{n} \sum_{x, y} \frac{\beta^n}{n!} \cdot \langle x|A|y \rangle \langle y|G^n|x \rangle \end{align*}

and all weights $ e^{-\beta d} \cdot \frac{\beta^n}{n!} \cdot \langle y|G^n|x \rangle $ are non-negative.

Answer 2

Stoquastic Hamiltonians have only non-positive off-diagonal terms, see for instance the abstract of this paper by Bravyi et al. The diagonal terms may be zero, but may also be stricly positive.

The sign problem is not restricted to only an appearance in quantum computing; it even stems from more general physics - check for instance this question and answer by user wsc on the physics stack exchange. The answer also links to this text by Troyer and Wiese.

My understanding is limited, but I know that it is closely correlated with Quantum Monte Carlo methods, which are methods of stochastic simulation of quantum mechanical systems that have been very effective, but only for Hamiltonians that do not suffer under the sign problem.