What are the types of Quantum Monte Carlo?

Stochastic Series Expansion (SSE) Monte Carlo

Theory: SSE is a finite-temperature, discrete-time technique that works well for quantum spin problems (e.g. Heisenberg model) and other lattice Hamiltonians in any number of dimensions. The method works by expanding the partition function in a Taylor series
$$tag{1} Z = mathrm{Tr}[ rho] = mathrm{Tr}[e^{-beta H}] = mathrm{Tr} sum limits_{alpha_0} sum limits_{n=0}^infty leftlangle alpha_0 left| frac{(-beta H)^n}{n!} right| alpha_0 rightrangle $$ and then inserting a complete set of states in some basis between each term in the Taylor expansion
$$tag{2} Z = sum limits_n frac{(-beta)^n}{n!} sum limits_{alpha_0} sum limits_{alpha_1} ... sum limits_{alpha_{n-1}} langle alpha_0 | H | alpha_1 rangle langle alpha_1 | H | alpha_2 rangle ... langle alpha_{n-1} | H | alpha_0 rangle $$

The resulting matrix elements $langle alpha_{i} | H | alpha_{i+1} rangle$ are usually simple to evaluate.

Algorithm: The goal of the algorithm is to sample the sum in the previous equation. This is sum in an extremely high-dimensional space, perfect for importance sampling Monte Carlo. In general, the program stores a MC configuration as a starting state $alpha_0$ and the list of local operators that act upon that state (the operator string). There are many different methods of updates, but they generally consist of two steps:

1. A diagonal update that adds and removes diagonal matrix elements, sampling the order ($n$) of the Taylor expansion.
2. An off-diagonal update which leaves $n$ fixed and samples the configurations ${alpha}$. These are commonly called operator loop updates and there are many different algorithms for different situations.

Use case: SSE is efficient for lattice Hamiltonians at finite temperature (although $T=0$ can be reached as a limit). To measure any quantity that is diagonal in the working basis is each, since the measurements can be directly computed from the SSE configurations, e.g. $$tag{3} langle S^z rangle = mathrm{Tr} left[ S^z e^{-beta H} right] $$ If an observable is off-diagonal in the simulation basis, but appears in the Hamiltonian, it can sometimes be calculated using some clever formula, e.g. $$tag{4} langle E rangle = -frac{langle n rangle }{beta}$$ $$tag{5} langle C_v rangle = -frac{langle n(n-1) rangle }{beta } $$

Computing arbitrary off-diagonal observables often requires complicated secondary sampling procedures.

References The main source for learning about SSE is this big review article by Sandvik: arXiv:1101.3281. There is also an excellent recorded lecture by Roger Melko from the 2010 Boulder Summer School (the link to the videos is right below the title).