Fermions and the Jordan-Wigner transform I: Introduction

I’m going to switch topics now, moving to the more physical topic of Fermions and the Jordan-Wigner transform. I’ve actually written papers using this stuff, but it’s only in writing these notes that I really feel like I’ve understood it well.

The series on expanders isn’t finished – I’ll be back with another installment in a day or two, and there will be several more installments after that.

General introductory note and prerequisites: This post is one in a series describing fermi algebras, and a powerful tool known as the Jordan-Wigner transform, which allows one to move back and forth between describing a system as a collection of qubits, and as a collection of fermions. The posts assume familiarity with elementary quantum mechanics, comfort with elementary linear algebra (but not advanced techniques), and a little familiarity with the basic nomenclature of quantum information science (qubits, the Pauli matrices).

Introduction

When you learn undergraduate quantum mechanics, it starts out being all about wavefunctions and Hamiltonians, finding energy eigenvalues and eigenstates, calculating measurement probabilities, and so on.

If your physics education was anything like mine, at some point a mysterious jump occurs. People teaching more advanced subjects, like quantum field theory, condensed matter physics, or quantum optics, start “imposing canonical commutation relations” on various field operators.

Any student quickly realizes that “imposing canonical commutation relations” is extremely important, but, speaking personally, at the time I found it quite mysterious exactly what people meant by this phrase. It’s only in the past few years that I’ve obtained a satisfactory understanding of how this works, and understood why I had such trouble in the first place.

These notes contain two parts. The first part is a short tutorial explaining the Fermionic canonical commutation relations (CCRs) from an elementary point of view: the different meanings they can have, both mathematical and physical, and what mathematical consequences they have. I concentrate more on the mathematical consequences than the physical in these notes, since having a good grasp of the former seems to make it relatively easy to appreciate the latter, but not so much vice versa. I may come back to the physical aspect in some later notes.

The second part of the notes describes a beautiful application of the Fermionic CCRs known as the Jordan-Wigner transform. This powerful tool allows us to map a system of interacting qubits onto an equivalent system of interacting Fermions, or, vice versa, to map a system of Fermions onto a system of qubits.

Why is this kind of mapping interesting? It’s interesting because it means that anything we understand about one type of system (e.g., Fermions) can be immediately applied to learn something about the other type of system (e.g., qubits).

I’ll describe an application of this idea, taking what appears to be a very complicated one-dimensional model of interacting spin-[tex]\frac 12[/tex] particles, and showing that it is equivalent to a simple model of non-interacting Fermions. This enables us to solve for the energy spectrum and eigenstates of the original Hamiltonian. This has, of course, intrinsic importance, since we’d like to understand such spin models – they’re important for a whole bundle of reasons, not the least of which is that they’re perhaps the simplest systems in which quantum phase transitions occur. But this example is only the tip of a much larger iceberg: the idea that the best way of understanding some physical systems may be to map those systems onto mathematically equivalent but physically quite different systems, whose properties we already understand. Physically, we say that we introduce a quasiparticle description of the original system, in order to simplify its understanding. This idea has been of critical importance in much of modern physics, including the understanding of superconductivity and the quantum Hall effect.

Another application of the Jordan-Wigner transform, which I won’t describe in detail here, but which might be of interest to quantum computing people, is to the quantum simulation of a system of Fermions. In particular, the Jordan-Wigner transform allows us to take a system of interacting Fermions, and map it onto an equivalent model of interacting spins, which can then, in principle, be simulated using standard techniques on a quantum computer. This enables us to use quantum computers to efficiently simulate systems of interacting Fermions. This is not a trivial problem, as can be seen from the following quote from Feynman, in his famous 1982 paper on quantum computing:

“[with Feynman’s proposed quantum computing device] could we 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.”

It wasn’t until 20 years later, and the work by Somma, Ortiz, Gubernatis, Knill and Laflamme (Physical Review A, 2002) that this problem was resolved, by making use of the Jordan-Wigner transform.

The next post will introduce the canonical commutation relations for fermions, and discuss something of their mathematical and physical significance.