The Jordan-Wigner transform is an amazing tool. It let’s you move back and forth between two seemingly very different ways of describing a physical system, either as a collection of qubits, or as a collection of fermions. To give you an idea of the power of the Jordan-Wigner transform, in his famous 1982 paper on quantum computing, Richard Feynman wrote the following:
could we [use a quantum computer to] imitate every quantum mechanical system which is discrete and has a finite number of degrees of freedom? I know, almost certainly, that we could do that for any quantum mechanical system which involves Bose particles. I’m not sure whether Fermi particles could be described by such a system. So I leave that open.
As shown in the notes, once you understand the Jordan-Wigner transform, the answer to Feynman’s question is obvious: yes, we can use quantum computers to simulate systems of fermions. The reason is that the Jordan-Wigner transforms lets us view the fermi system as a system of qubits which is easy to simulate using standard simulation techniques. Obviously, the point here isn’t that Feynman was silly: it’s that tools like the Jordan-Wigner transform can make formerly hard things very simple.
The notes assume familiarity with elementary quantum mechanics, comfort with elementary linear algebr, and a little familiarity with the basic nomenclature of quantum information science (qubits, the Pauli matrices).
I’m releasing the notes under a Creative Commons Attribution license (CC BY 3.0). That means anyone can copy, distribute, transmit and adapt/remix the work, provided my contribution is attributed. The notes could be used, for example, to help flesh out Wikipedia’s article about the Jordan-Wigner transform. Or perhaps they could usefully be adapted into course notes, or part of a review article.