### Topological Quantum Computing

It’s kinda weird hearing condensed matter physicists batting around phrases like “Modular Tensor Category” and “the Jones Polynomial.” But such is life. I’ve been talking a bit with the folks who are thinking about topological quantum computing, and that’s where their heads are at, these days.

#### Abelian Anyons

You’ll recall that the Fractional Quantized Hall Effect concerns strongly-interacting quasi-2-dimensional electron systems in a strong magnetic field at very low temperatures. The electrons form a partially-filled Landau level, and the bulk system has gap. (There are gapless excitations that propagate on the boundary of a droplet of Quantum Hall fluid.) For filling fraction $\nu = 1/m$, for $m$ an odd integer, Laughlin proposed that the multi-electron ground state wave function takes the form $\Psi_0(z_1,\dots,z_N) = \prod_{i\gt j} (z_i-z_j)^m e^{-\sum |z_i|^2/4l_0^2}$ where $l_0=\sqrt{hc/eB}$ is the magnetic length.

The quasiparticles are excitations above the gap, with fractional charges, $e/m$ and fractional statistics, $\theta = \pi/m$. The wave function in the presence of $n$ quasiparticles, at locations $w_1,\dots,w_n$, is
$\Psi_n = \prod_{i\lt j} (w_i-w_j)^{1/m} \prod_{i=1}^N\prod_{j=1}^n (z_i-w_j) \Psi_0$
Aside from the Gaussian factor (which can be understood as introduction of a uniform background charge to cancel the net charge of the insertions), this looks like nothing so much as the holomorphic part of a correlation function of primary fields in a $c=1$ CFT, specifically, the one at radius $R^2=k$. Here, the electrons are the holomorphic primary fields, $e^{i\sqrt{m} \phi(z)}$, which generate the extended chiral algebra^{1}, and the quasiparticles are the operators $e^{i \phi(z)/\sqrt{m}}$

#### Nonabelian Anyons

The next step was taken by Greg Moore and Nick Read in 1991. They wrote down a Laughlinesque wave function of the form $\Psi = \mathop{Pfaf}\left(\frac{1}{z_i-z_j}\right)\prod_{i\lt j} (z_i-z_j)^m e^{-\sum |z_i|^2/4l_0^2}$ which, for $m$ even, should correspond to a filling fraction, $\nu$, with even denominator. This, again, has an interpretation as a CFT correlator, this time in the tensor product of an Ising model and a $c=1$ CFT at radius $R^2=m$. The electron operator is $\psi e^{i\sqrt{m}\phi(z)}$. Quasiparticles are then created by $\sigma e^{i\phi(z)/2\sqrt{m}}$. These quasiparticles have interesting nonabelian braiding relations

In recent years, considerable evidence has emerged that the $\nu=5/2$ FQHE state is described by Moore-Read.

The obvious generalization generalization of these constructions is to the product of a $\mathbb{Z}_k$ parafermion and a free scalar (the Moore-Read case is $k=2$), as worked out by Read and Rezayi.

There’s some fleeting evidence for a $\nu = 12/5$ FQHE state, described by the $\mathbb{Z}_3$ Reed-Rezayi system.

#### Chern-Simons

Wave functions are all very well and good, but we would like to have an effective field theory. Since the bulk theory has a gap, the low energy effective theory is a topological one. For the abelian anyons described above, the answer has been known since my graduate student days. It’s an abelian Chern-Simons Theory, $S= \int \frac{k}{2\pi} a\wedge d a - \frac{1}{\pi} A \wedge d a - j_{\text{QP}} \wedge a$ Here, $A$ is the external electromagnetic field, which couples to the electromagnetic current (written here as 2-form), $j_{\text{EM}} = \frac{1}{\pi} d a$ The action is Gaussian, so one can actually integrate out $a$, and obtain the Hall conductance $\sigma_{x y} = \frac{1}{m} \frac{e^2}{h},\quad \sigma_{x x} =0$ The quasiparticles are heavy, and so appear as classical sources, represented by the 2-form $j_{\text{QP}}$.

The obvious generalization to the Moore-Read/Read-Rezayi systems is the $SU(2)_k$ Chern Simons theory, $S = \int \frac{k}{4\pi} Tr\left(a \wedge d a +\frac{2}{3} a\wedge a\wedge a\right) - \frac{1}{2\pi} A\wedge Tr\tau_3(d a + a\wedge a) - Tr (j_{\text{QP}}\wedge a)$

And, so I’ve been watching with bemusement as the condensed matter theorists start drawing knots on the blackboard and speaking animatedly about the Jones Polynomial, pentagon and hexagon identities, *etc*.

#### Quantum Computing

So why are the condensed matter theorists so interested in this subject, all of a sudden? Quantum Computing. Recall that a quantum computer is, basically, a system with a finite-dimensional Hilbert Space. We perform a computation by preparing the system in a certain state, make some manipulations, and then measure the final state of the system.

To build such a quantum computer, one wants a system with a finite number of low-lying states, separated by a gap from the rest of the (invariably infinite-dimensional) Hilbert space. That’s exactly what nonabelian FQHE systems provide us. The abelian systems have a unique ground state, even in the presence of $n$ quasiparticles. But the state of the nonabelian system is degenerate, with degeneracy equal to the number of conformal blocks of the associated $\mathbb{Z}_k$ parafermion correlation function. Moreover, we can manipulate the state of the system (do computations) by moving the quasiparticles around.

The braiding of the spin field in the Ising model is a bit too simple. But for $k\neq 2,4$, the Read-Rezayi system has been shown to be universal for quantum computation.

You can read a lot more about this in a wonderful review by Das Sarma *et al*.

^{1} We normalize the current $j = \frac{1}{\sqrt{m}} i\partial\phi$, so that the electron has unit charge.

^{2} Actually, this isn’t quite right. The Chern-Simons theory that we *want* is the one corresponding to the coset conformal field theory $(SU(2)_k\times U(1))/U(1)$. This is subtly different from the $SU(2)_k$ theory, and I’ve been trying to convince Steve Simon that this little fillip should be written down properly.

## Re: Topological Quantum Computing

A very nice introduction to the subject of topological quantum computation can be found in John Preskill’s lecture notes for the quantum computing class at Caltech (website: http://www.theory.caltech.edu/~preskill/ph229/)