Confusion about Quantum Monte Carlo

Question

My question is about extracting observables from QMC methods, as described in this reference.

I understand the formal derivation of various QMC methods like Path Integral Monte Carlo. However, at the end of the day I'm still confused about on how to effectively use these techniques.

The basic idea of the derivation of Quantum MC methods is to discretise, via the Trotter approximation, an operator which may be either the density matrix or the time-evolution operator of a quantum system. We then obtain a classical system with an additional dimension which may be treated with MC methods.

Given that we can interpret $\beta$ in the quantum operator $e^{−\beta\hat{H}}$ both as an inverse temperature and an imaginary time, the aim of these algorithms should be to compute an approximation of this operator. Indeed, if we would directly measure quantities from the various configurations sampled along a simulation, in the "inverse temperature" case we would have samples respecting a probability density based on $\beta/M$, where $M$ is the number of discrete steps introduced in the Trotter decomposition. Instead, in the "imaginary time" case we would obtain samples at various discrete time-steps, thus getting averages along the time as well. We also wouldn't obtain quantities like $\langle\psi_t|\hat{A}|\psi_t\rangle$ at a given time $t$, with $\hat{A}$ some observable operator.

However, in my opinion the quantities we sample directly from this kind of simulations (taken from (5.34) of the document, page 35):

$$\bar{O} \equiv \langle \hat{O}(X) \rangle \equiv \frac{1}{N!} \sum_P \int O(X) \pi(X,P) dX $$

cannot be quantities related to the quantum system, given the additional dimension. Instead, the correct quantum quantities can be computed via formulas like (5.35), which contains in every sample a whole chain of $M$ simulated configurations:

$$\frac{E_{th}}{N}= \left\langle \frac{d}{2 \tau} - \frac{m}{2 (\hbar \tau)^2MN } \sum_{j=1}^{M} (\mathbf{R}_j -\mathbf{R}_{j+1})^2 + \frac{1}{MN} \sum_{j=1}^{M} V(\mathbf{R}_j) \right\rangle $$

Am I right that a series of QMC simulations is required to extract useful information about a given observable?

Answer 1

There is a lot of confusion in your question. The most important for me is that you miss that "naive" QMC which is Monte-Carlo calculation of integrals in some variational method and diffusion Monte-Carlo are different methods with different argumentation and derivation.

The main point however is about imaginary time. In diffusion Monte-Carlo imaginary time is a trick to convert time-independent Schroedinger equation into time-dependent diffusion-like equation which solution in the infinite "time" limit tends to a solution of original Schroedinger equation. That's it. Time in DQMC is not real.

Relatively good but simple explanation is given in Reviews of Modern Physics, 73, 33(2001).

P.S. By the way, what you mean by "Trotter approximation" in your question?

Answer 2

You're right that people use monte carlo techniques to calculate statistical averages (as opposed to time-resolved information) all the time. It's not necessarily true that this is what should be calculated: it depends on what kind of information you want. Maybe you have a time-dependent external forcing, for example, and want to see how the system evolves in response.