The Mathematics of Topological Quantum Computing

Eric Rowell
Texas A&M University

Abstract: In topological quantum computing, information is encoded in “knotted” quantum states of topological phases of matter, thus being locked into topology to prevent decay. Topological precision has been confirmed in quantum Hall liquids by experiments to an accuracy of $10^{-10}$ and harnessed to stabilize quantum memory. In these lectures, we will discuss the conceptual development of this interdisciplinary field at the juncture of mathematics, physics and computer science. Our focus is on computing and physical motivations, basic mathematical notions and results, open problems and future directions related to and/or inspired by topological quantum computing.

Here are resources for the lectures, ordered roughly in difficulty/required dedication:

1. An Invitation to the Mathematics of Topological Quantum Computation, By Eric C. Rowell. ( Journal of Physics: Conference Series 698 (2016) 012012., Short survey, meant for Physics people that are not in condensed matter.)

2. Local unitary representations of the braid group and their applications to quantum computing. By Colleen Delaney, Eric C. Rowell, Zhenghan Wang. (Revista Colombiana de Matemáticas Lecture notes, it will be using these mostly. Also available in https://arxiv.org/abs/1604.06429. )

3. Mathematics of Topological Quantum Computing. By Eric C. Rowell and Zhenghan Wang. (BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY).

4. Topological Quantum Computation By Zhenghan Wang. ( CBMS Regional Conference Series in Mathematics, vol. 112.)

Lecture notes

Notes by Harshit Yadav

Lecture 1

Lecture 2

Lecture 3

Lecture 4

Problem session

In charge of Qing Zhang, Texas A&M

First lecture

1. Let $H=\frac{1}{\sqrt{2}}$ $\begin{pmatrix} 1 & 1\\ 1 &-1\end{pmatrix}$ and $T(a):=\begin{pmatrix} 1 & 0\\ 0 &a\end{pmatrix}$.

   (a) Show that if $a=i$ then $\langle H,T(a) \rangle$ is a finite group. (Hint: show that $H\cdot T(a)$ has order 3 modulo scalars, and deduce the relations $H^2=T(a)^4=(H\cdot T(a))^3=1$, projectively. This is an index 2 subgroup of the triangle group $\Delta(2,3,4)$, which is finite).

   (b) Show that if $a=e^{2\pi i/8}$ then $\langle H,T(a)\rangle$ is infinite. ( Hint: show that the eigenvalues of $H \cdot T(a)$ are not roots of unity. )

   (c) In the latter case, conclude that $\overline{\langle H,T(a)\rangle} \cap SU(2)=SU(2)$.

   (d) Challenging: show (perhaps using the theorem of Brylinski-Brylinski) that $H,T(e^{2\pi i/8}),\operatorname{CNOT}$ is universal.

   (e) Challenging: show that the group of $n$-qubit gates composed from $H,T(i),\operatorname{CNOT}$ is finite.

   (f) Assuming $a$ is a root of unity of finite order, when is $\langle H,T(a)\rangle$ finite?

2. Determine the dimension of the Hilbert spaces associated with the disk with trivial boundary label and $n$ identical punctures of type 1 for

   (a) the Fibonacci theory,

   (b) the Ising theory,

   (c) what is the asymptotics of the dimensions?

3. Determine the dimension of the Hilbert spaces associated with the $g$-holed torus for

   (a) the Fibonacci theory,

   (b) the Ising theory,

   (c) what are the asymptotics of the dimensions?

Online systems for computational discrete algebra

• Magma

• SageMath (Sage, GAP, Macauly2, Singular, Python).