# Difference between Monte Carlo and Quantum Monte Carlo methods?

What are the differences between Classical Monte Carlo methods and Quantum Monte Carlo methods in condensed matter physics? If one want to study strongly correlated systems with Quantum Monte Carlo method, does he/she need to study Classical Monte Carlo method first? (just like if you want to study quantum mechanics you shall study classical mechanics first?)

• My understanding (although since I don't work in the field, I could be wrong) is that Monte Carlo algorithms are an extremely broad class of algorithms that use randomization, while thaere are a handful of fairly specific classes of algorithms called Quantum Monte Carlo methods, these are a subset of Classical Monte Carlo algorithms. There is no reason you should study the Classical Monte Carlo method before starting on one of the classes of Quantum Monte Carlo algorithms. – Peter Shor Mar 9 '13 at 17:28
• Let me expand on my comment a bit. Suppose you are interested in the Path Integral Quantum Monte Carlo method. One can view this as a classical Monte Carlo method applied to the evaluation of a path integral. It would be a good idea to study the specific classical Monte Carlo techniques that are used in this algorithm, but there are also zillions of classical Monte Carlo techniques which have nothing to do with this Quantum Monte Carlo method, and studying those might be wasting your time. – Peter Shor Mar 9 '13 at 19:48
• This might be one to send to Computational Science. – David Z Mar 10 '13 at 4:55

That depends on what you want to achieve and what literature you have access to. Monte Carlo (MC) and Quantum MC are based on the same methods. Usually textbooks describe MC first and then build on this to explain QMC. One good resource to start is this book: Thijssen: Computational Physics. It does not cover everything in depth but gives a good overview.

Another good resource is this course website. It's free and includes many many code examples.

From the way I learned about those things, I would answer your question with: yes, you should at least take a look at MC before you start with Quantum MC.

EDIT: I just found this resource as well. A good intro to Quantum MC.

Quantum statistical mechanics is just classical statistical mechanics in one additional dimension. This is because the path-integral formulation of quantum mechanics allows you to expand the partition function of your model (which is classically just a probability distribution) as a sum over "space-time" configurations of your degrees of freedom weighted with a path action.

Now, it is very important to understand that not every model, when expanded in the path integral formulation, has a real path action, and this means that the weight of a space-time configuration $\sim e^{-S}$ doesn't need to be a real, positive-definite number. If that's the case, then there is no notion of a probability distribution to sample, and you cannot use Monte Carlo methods. This is called the "sign problem."

If such a probability distribution does exist, then you can use Monte Carlo methods, which don't care what distribution you're trying to sample, just that it is, in fact, a probability distribution.

So the bottom line is, learn the path-integral formulation and classical Monte Carlo, because classical Monte Carlo is the only Monte Carlo there is.

There are some concepts which is quite confusion to starters, though the concepts themselves are quite simple:

Classical Monte Carlo method: simulating classical statistical model with Monte Carlo method on classical computer,i.e., the PC we use daily;

Quantum Monte Carlo method: simulating quantum statistical model with Monte Carlo method on classical computer;

Quantum simulation: studying a quantum system by setting up a real quantum system, usually with cold atoms.