BULLETIN (New Series) OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 00, Number 0, Pages 000–000
S 0273-0979(XX)0000-0
MATHEMATICS OF TOPOLOGICAL QUANTUM COMPUTING
ERIC C. ROWELL AND ZHENGHAN WANG
Abstract. In topological quantum computation, information is encoded in
“knotted” quantum states, 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 has been harnessed to stabilize quantum
memory. In this survey, we discuss the conceptual development of this interdisciplinary field at the juncture of mathematics, physics and computer science.
Our focus is on basic notions, mathematical results, open problems, and future
directions related to and/or inspired by topological quantum computation.
1. Introduction
2πi
Can the Jones polynomial J(L, q) of oriented links L evaluated at q = e r , r =
1, 2, 3, ... be calculated by a computing machine quickly? The fact that the Jones
evaluation is #P -hard if r 6= 1, 2, 3, 4, 6 [24, 8, 26] is one of the origins of topological
quantum computing [5]. Freedman wrote in [5]: “Non-abelian topological quantum
field theories exhibit the mathematical features necessary to support a model capable of solving all #P problems, a computationally intractable class, in polynomial
time. Specifically, Witten [29] has identified expectation values in a certain SU(2)
field theory with values of the Jones polynomial [9] that are #P -hard [8]. This suggests that some physical system whose effective Lagrangian contains a non-abelian
topological term might be manipulated to serve as an analog computer capable of
solving NP or even #P hard problems in polynomial time. Defining such a system
and addressing the accuracy issues inherent in preparation and measurement is a
major unsolved problem.”
Another inspiration for TQC is fault-tolerant quantum computation by anyons
[10]. Kitaev wrote in [11]: “A more practical reason to look for anyons is their
potential use in quantum computing. In [10], I suggested that topologically ordered
states can serve as a physical analogue of error-correcting quantum codes. Thus,
anyonic systems can provide a realization of quantum memory that is protected
from decoherence. Some quantum gates can be implemented by braiding; this
implementation is exact and does not require explicit error-correction. Freedman
et al. [7] proved that for certain types of non-abelian anyons braiding enables one
to perform universal quantum computation. This scheme is usually referred to as
topological quantum computation (TQC).”
The resolution of Freedman’s problem [6], Kitaev’s idea of inherently faulttolerant quantum computation [10], and the existence of universal anyons by braidings alone [7] ushered in topological quantum computing [4].
2010 Mathematics Subject Classification. Primary: 18-02, 57-02, 81-02.
The authors thank Susan Frielander for soliciting this survey.
c
0000
(copyright holder)
1
2
ERIC C. ROWELL AND ZHENGHAN WANG
In TQC, information is encoded in “knotted” quantum states, 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 has been harnessed
to stabilize quantum memory. TQC has been driving an interaction between mathematics, physics, and computing science for the last two decades. In this survey,
we will convey some excitement of this interdisciplinary field.
We need physical systems that harbor non-abelian anyons to build a topological
quantum computer. In 1991, Moore and Read, and Wen proposed that non-abelian
anyons exist in certain fractional quantum Hall (FQH) liquids [12, 27]. Moore and
Read used conformal blocks of conformal field theories (CFTs) to model fractional
quantum Hall states, and Wen defined topologically ordered states, whose effective
theories are topological quantum field theories (TQFTs). Thus, (2 + 1)-TQFT and
(1 + 1)-CFT form the foundations of TQC. The algebraic input of a TQFT and
topological properties of a CFT are encoded by a beautiful algebraic structure—
modular tensor category, which will be simply called a modular category. Modular
categories were invented by Moore and Seiberg using tensors [13], and formulated
in a coordinate-free way by Turaev [22]. Thus, modular categories, algebraically
underpinning TQC, are algebraic models of anyon systems.
In 1999, Read and Rezayi [18] suggested a connection between FQH liquids at
k
with SU (2)k -Witten-Chern-Simons theories, mathfilling fractions ν = 2 + k+2
ematically Reshetikhin-Turaev TQFTs for k = 1, 2, 3, 4. In 2006, interferometer
experiments of FQH liquids were proposed that the Jones evaluation at q = i of
certain links directly appeared in the measurement of electrical current [1, 21]. In
2009, experimental data consistent with the prediction were published [28]. The
current most promising platform for TQC is nanowires and topological protection
has been experimental confirmed. Next milestone will be the experimental confirmation of non-abelian fusion rules and braidings of Majorana zero modes.
One afternoon during the early summer of 1935, Turing, while lying in a meadow
at Grantchester after a long run, had the inspiration to abstract a human being
working with pencils and papers to do calculation into “a mechanical process”—
now bearing his name Turing machine. In 1936, Turing published his paper “on
computable numbers” [23], in which data and program are unified in his universal
machine. Subsequently, computability becomes part of mathematics following independent work of Turing, Church, and Post. That a real number is computable
should be comparable to a number being algebraic. The absoluteness of computability is enshrined into the Church or Church-Turing thesis: anything computable by
a discrete-state-machine with finite means is computable by a Turing machine. No
serious challenge to Church thesis has ever appeared1. But in 1994, Shor’s quantum
algorithm for factoring integers [20] poses a serious challenge, not to the Church
thesis, but to the polynomial extension of the Church thesis: anything efficiently
computable can be efficiently computed by a Turing machine. While there seems
to be a unique notion of computability, it is likely that the notion of efficient computability would diverge.
Definiteness is an important feature of the classical world. Thus, computing
tasks can be formalized as Boolean functions. Definability is not the same as
computability because there exist plenty of non-computable numbers. As currently
formulated, quantum computing will not touch on computability. But classical
1hypercomputation
MATHEMATICS OF TOPOLOGICAL QUANTUM COMPUTING
3
notions can be smeared into quantum weirdness such as Schodinger’s cat. The
promise of quantum computing is a vast leap in the speed of processing classical
information using quantum medium.
The unit of information is bit. Qubits are the quantum embodiment of bits
using the two characteristic attributes of quantum mechanics—superposition and
entanglement. Qubits are computationally powerful, but notoriously fragile—the
outside world is constantly “looking” (measuring) the quantum system, which leads
to the decohence of the quantum states. Mathematically, a qubit is the abstraction
of all quantum systems with two-dimensional Hilbert spaces of states. We call
the two-dimensional Hilbert space C2 with preferred basis |0 > and |1 > a qubit.
Therefore, a qubit, utilizing superposition, can have any non-zero vector |ψ >∈
C2 as a state, but computability consideration will restrict us only to states that
can be algorithmically prepared. Experiment will further force us to work with
ony finite precision states. Einstein’s “spooky action at a distance”—quantum
⊗n
entanglement—is realized in states of multi-qubits (C2 ) , where a state |ψ > is
entangled if |ψ > cannot be written as a tensor product.
Topological quantum computing solves the fragility of the qubits at the hardware
level2 by using topological invariants of quantum systems. Information will be
encoded non-locally into topological invariants that spread into local quantities
just as the Euler characteristic spreads into local curvature by the Gauss-Bonnet
theorem. Nature does provide such topological invariants in topological phases
of matter such as the FQH liquids and topological insulators. The topological
invariant for TQC is the ground state degeneracy in topologically ordered states
with non-abelian anyons.
Anyons are topological quantum fields materialized as finite energy particle-like
excitations in topological phases of matter. Like particles, they can be moved, but
cannot be created or destroyed locally. There are two equivalent ways to model
anyon systems. We can focus on the ground state manfold V (Y ) of an anyonic
system on any possible space Y , and then the anyon system is modeled in low
energy by a (2 + 1)-TQFT. An alternative is to consider the fusion and braiding
structures of all elementary excitations in the plane. The anyon system is then
equivalently modeled by a unitary modular category. The two notions (2+1)-TQFT
and modular category are essentially the same [22]. Therefore, anyon systems can be
modelled either by TQFTs or unitary modular categories. In the modular category
model, an anyon X is a simple object that abstracts an irreducible representation
of some symmetry algebra.
There are many interesting open questions in TQC including the classification
of mathematical models of topological phases of matter such as TQFTs, CFTs and
MTCs, and the analysis of computational power of anyonic quantum computing
models. Classification of modular categories is achievable and interesting both in
mathematics and physics [19, 2, 3]. A recent result in this direction is a proof of the
rank-finiteness conjecture [2]. An interesting open question in the second direction
is the property F conjecture that the braid group representations afforded by a
simple object X of a modular category have finite images if and only if its squared
quantum dimension d2X is an integer [14]. Many open questions, for example the
theory for fermions and three spatial dimensions, will be discussed in the survey.
2shor
4
ERIC C. ROWELL AND ZHENGHAN WANG
Progress in building a useful quantum computer is accelerating in the last few
years. While it is hard to characterize our current computing power in terms
of a number of qubits, it is clear that a working quantum computer with one
hundred qubits would perform tasks that no classical computer can complete now.
Since TQC does not have a serious scaling issue, so when one topological qubit is
constructed, a powerful quantum computer is on the horizon.
The arenas of mathematics, computer science, and physics are Mind, Machine,
and Nature. We are at an important junction to see how the three worlds would
interact with each other. TQC is a tip of an iceberg that blurs the three. The
authors’ bet is on Nature, but we suspect that Nature has her eye on Machine. An
inevitable question that will soon confronts us is how are we going to evolve when
quantum computers become reality?
In this survey, we will focus on the mathematics of TQC as in [25]. For the
physical side, we recommend [15, 17, 16]. Three fundamental notions for TQC are:
modular category, (2 + 1)-TQFT, and topological phase of matter. Though closely
related, the definition of modular categories is universally accepted while TQFTs
vary significantly. For our applications, we emphasize two important principles
from physics: locality and unitarity. Locality follows from special relativity that
nothing including information can travel faster than light, whereas unitarity is a
consequence of quantum mechanics. We define unitary (2 + 1)-TQFTs adapting
the definition of Walker and Turaev. We propose a new mathematical definition
of 2D topological phases of matter using Hamiltonian formalism in section 4. The
notion of a topological phase might evolve further. As F. Crick said: “Tomorrow,
I may see (or be persuaded of) errors in my present thinking, but today I have to
do the best I can”. The content of the survey is as follows: In section 2, we give an
introduction to TQC and mathematical models of anyons. In section 3, we cover
abstract quantum mechanics and quantum computing. In section 4, first we formulate abstract condensed matter physics, then define topological phases of matter.
In section 5, we analyze the computational power of anyonic computing models.
Section 6 is a survey on the structure and classification of modular categories. In
section 7, we discuss various extensions and open problems. We conclude with two
eccentric research directions in section 8.
2. The ABC of TQC
We introduce three basic notions in TQC: Anyons, Braids, and Categories. One
feature of TQC is the use of graphical calculus. Space-time trajectories of anyons
will be represented by braids, and more general quantum processes such as creation/annihilation and fusion by tangles and tri-valent graphs. Algebraically, anyon
trajectories will be modelled by morphisms in certain unitary modular categories.
Computationally, we could regard a unitary modular category as a computing
model: the morphisms are Circuits for computation. After motivating the axioms of a modular category using anyons, we define this important model of anyon
systems and explain anyonic quantum computing using graphical calculus.
MATHEMATICS OF TOPOLOGICAL QUANTUM COMPUTING
5
2.1. Anyons and Braids. It is truly remarkable that all electrons, no matter
where, when, and how are found, are identical. Elementary particles3 are divided into bosons and fermions. Consider n quantum particles Xi , i = 1, ..., n,
in R3 at distinct locations ri , then their quantum state is given by a wave function
Ψ(r1 , ..., ri , ..., rj , ...rn ). Suppose we exchange Xi and Xj so that no two particles
collide during the exchange, then Ψ(r1 , ..., rj , ..., ri , ...rn ) = θΨ(r1 , ..., ri , ..., rj , ...rn )
for some complex number θ. If we repeat the exchange, the n particles will return
back exactly. Hence, θ2 = 1. Particles with θ = 1 are bosons, and θ = −1
fermions. It follows that if we perform any permutation σ of the n particles, then
Ψ(rσ(1) , ..., rσ(n) ) = π(σ)Ψ(r1 , ..., rn ), where π(σ) is the sign of the permutation
σ. Therefore, the statistics of quantum particles in R3 is a representation of the
permutation groups Sn to Z2 = {±1}.
Now we examine two implicit facts that are used in the above discussion. Firstly,
we assume that the final state of the n particles does not depend on the exchange
paths. This is because any two paths for the exchange are isotopic4. Technology has
made it realistic to consider particles confined in a plane. Then paths for exchange
form the braid groups Bn instead of the permutation groups Sn . Repeating the
argument above, we can only conclude that θ is on the unit circle U (1). Therefore,
θ potentially can be any phase θ ∈ U (1). Particles with any phase θ are dubbed
anyons by Wilczek.5 They are called abelian anyons because their statistics is a
representation of the braid groups Bn to the abelian group U (1).
The second fact used is that there is a unique state Ψ(r1 , ..., rn ) when the positions of the n particles are fixed. What happens if there are more than one linearly
independent states? If the n particles start with some state, they might come back
in a superposition. Suppose {ei }, i = 1, ..., m, is an orthonormal basis
Pm of all possible
states, then starting in ei , the n particles will return to a state j=1 Uij ej , where
(Uij )1≤i,j≤m is an m × m unitary matrix. The statistics of such particles could be
high dimensional representations of the braid groups Bn → U (m)6. When m > 1,
such anyons are called non-abelian.
Insert Braids
Space-time trajectories of n anyons form the n-strand braid group Bn . Mathematically, Bn is the the motion group of n points in the disk D2 given by the
presentation
D
E
Bn = σ1 , σ2 , . . . , σn−1 | σi σj = σj σi for |i − j| ≥ 2, σi σi+1 σi = σi+1 σi σi+1
The first relation is referred to as far commutativity and the second is the famous
braid relation. The terminologies become clear when one considers the graphical
representation of elements of the braid group. We have the following sequence of
3Elementary particles are elementary excitations of the vacuum, which explains why they are
identical. In TQC, anyons are elementary excitations of some quantum medium, so they are
called quasi-particles. The vacuum is also a very complicated quantum medium, so we will call
quasi-particles simply particles.
4This is the same as the fact there are no knots in R4 .
5Stability consideration restrict possible phases to θ = e2πis for s to be a rational number.
6We can ask why there cannot be elementary particles that realize higher dimensional representations of the permutation groups. Such statistics is called parastatistics. It can be argued
that parastistics does not lead fundamentally new physics as non-abelian statistics
6
ERIC C. ROWELL AND ZHENGHAN WANG
groups
1 → Pn → Bn → Sn → 1,
where Pn is called the n-strand pure braid group. Note that from this sequence,
any representation of Sn leads to a representation of Bn .
2.2. Category and Quantum Physics. There is a philosophical explanation why
category theory is a good language for quantum physics.7 In quantum physics, we
face the challenge that we cannot “see” what is happening. So we appeal to measurements and build our understanding from the responses to measuring devices.
A quantum particle is really defined by how it interacts with others. In category
theory, objects are usually the interested targets. An object A is determined by the
morphism sets Hom(A, X) for all X. Therefore, it is natural to treat objects as certain quantum states such as particles, and their morphisms as quantum processes.
Modular category model of anyons is such an example.
Another philosophical relation between category and quantum physics comes
from a similarity between quantization and categorification. A famous quote by E.
Nelson is: “first quantization is a mystery, but second quantization is a functor”.
The situation that we are interested in is very simple—the quantization of a finite
set S. First quantization is the process to go from a classical system to a quantum
system that is modeled by a Hilbert space of quantum states with a Hamiltonian.
In the case of a finite set S as the classical configuration space of a single particle,
then quantization is simply the linearization of S—the Hilbert space is just C[S]
spanned by the elements of S. Second quantization is the process to go from a
single particle Hilbert space to a multi-particle Hilbert space. For simplicity, if we
have a fermion with a single particle Hilbert space V of dimension= n, then the
multi-particle Fock space is just the exterior algebra ∧∗ V of dimension 2n . Hence,
second quantization is the functor from V to ∧∗ V . The process of de-quantization
is measurement: when we measure a physical observable O at state |Ψi, we arrive
at a normalized eigenvector ei of O with probability pi = hei |O|Ψi. A basis of
eigenstates from an observable consists of the stationary states that we obtain after
a measurement. Therefore, eigen-bases are what we “see”, therefore, real.
According to M.-M. Kapranov and V.-A. Voevodsky, the main principle in category theory is: “In any category it is unnatural and undesirable to speak about
equality of two objects”. The general idea of categorification is to weaken an equality to some version of isomorphism. Naively, we want to replace a natural number
n with a vector space of dimension n. Then the categorification of a finite set
S of n elements should be C[S] of dimension n. It follows that the equality of
two sets Si , i = 1, 2 should be relaxed to an isomorphism of the two vector spaces
C[Si ], i = 1, 2,. Isomorphisms between vector spaces may or may not be functorial as the isomorphisms between V and V ∗∗ or V ∗ demonstrated. So categories
are more physical than sets because the ability to instantaneously distinguish two
elements of any set is not physical.
2.3. Gedanken Experiments on Anyons. How are we going to model a universe of anyons following the laws of quantum physics, locality, and topological
invariance? An anyon system is much like the world of photons and electrons in
quantum electrodynamics. First we assume there are only finitely many anyon
7At a conference in 2003, Xiao-Gang Wen asked me what should be the right mathematical
framework to describe his topological order. I replied category theory.
MATHEMATICS OF TOPOLOGICAL QUANTUM COMPUTING
7
types in our universe, which form the label set L = {0, . . . , r − 1} of “labels” or
“anyon types”. Let Π = {Xi }i∈L be a representative sets of anyon types: one for
each anyon type. The ground state or “vacuum” X0 = 1 is always included as an
invisible anyon labelled by 0. It is important to distinguish anyons and their types,
so we will usually use upper case letters for physical anyons and the corresponding
lower cases for their types. An anyon X with type x has an anti-particle X ∗ with
type denoted by x̂. The ground state 1 is its own antiparticle so 0̂ = 0.
2.3.1. Ground State Degeneracy. For our Gedanken experiment, consider an oriented closed two dimensional space Y with some anyons {Ai } residing at {zi }. The
local information of positions zi will not change universal physical properties by
topological invariance, so we will usually drop them from our notations. Anyons
are point-like, but they do occupy some room. There are two equivalent ways to
picture them: either as a colored puncture in Y with a signed infinitesimal tangent vector or as an infinitesimal disk whose oriented boundary circle has a sign,
a based point and a color. By a color, we mean an anyon, not its type. We will
explain the sign and base point later and use the small disk with the induced orientation to picture an anyon. All quantum states of {Ai } at {zi } form a Hilbert
space L(Y ; Ai , zi ), which depends on local degree of freedom such as positions.
For topological phases, we are only interested in the low energy states. We assume there are well-defined lowest energy states, called the ground state manifold
V (Y ; Ai )8, which is the eigenspace corresponding to the smallest eigenvalue of the
Hamiltonian. Topological invariance means that only topological transformations
of Y lead to nontrivial operators on the Hilbert space of ground states V (Y ), which
is the topological degree of freedom in L(Y ; Ai , zi ). Locality implies that a state
in V (Y ; Ai ) can be constructed from states on local patches. Every surface has a
DAP decomposition: a decomposition of Y into disks, annuli, and pairs of pants.
There are two kinds of boundary circles in a DAP decomposition of Y : the new
cutting circles and the anyon boundaries. The new cutting circles should have some
boundary conditions ` that allow the reconstruction of the original quantum state.
A physical assumption is that the boundary conditions are anyons, which is a version of bulk-edge correspondence. So locality is encoded in a gluing formula of the
form:
V (Y ) ∼
= ⊕` V (Ycuts ; `).
Therefore, it suffices to know anyon states on disks, annuli, and pairs of pants
together with some general principles. The basic axioms start with:
• Empty Set Axiom: V (∅) ∼
=C
• Disk Axiom: V (D2 , Xi ) ∼
= Cδ0i
• Annulus Axiom: V (A, Xi , Xj ) ∼
= Cδiĵ
• Pair of Pants Axiom: V (P (Xi , Xj , Xk )) ∼
= CNijk for some integer Nijk ≥ 0.
The empty set axiom is a non-triviality axiom. The disk axiom means topological: since a disk has no topology, so it cannot support any nontrivial state
other than the ground state. We will see later that the disk axiom essentially
implies that topological ground state manifolds V (Y ) are error-correction codes:
V (Y ; Ai ) ⊂ L(Y ; Ai , zi ). The annulus axiom is also topological: an anyon cannot
8Physicist use mfd
8
ERIC C. ROWELL AND ZHENGHAN WANG
change its type across the annulus if we consider both boundaries as small disks for
anyons.
Other axioms follow from general quantum mechanics principles are (see section
3):
• Disjoint Union Axiom: The ground state Hilbert space on two disjoint
surfaces (Y1 , `1 ) and (Y2 , `2 ) is V (Y1 , `1 ) ⊗ V (Y2 , `2 ).
• Duality Axiom: V (Y, `)† ∼
= V (−Y , `∗ ), where −Y is the surface Y with the
opposite orientation and `∗ means applying anti-particle to each X ∈ `.
• Gluing Axiom: If a surface (Y, `) with boundary colors ` is obtained from
(Yg , Xi , Xi ∗ , `) by gluing the boundary circles labeled by Xi and Xi ∗ together, then
M
V (Y, `) ∼
V (Yg , Xi , Xi ∗ , `).
=
Xi ∈Π
One important consequence of the gluing and annulus axiom is that dim V (T 2 )
is |L|—the number of anyon types, where T 2 is the 2-dimensional torus.
c
Figure 1.
The labeled
surface associated with the
c
fusion channel Hab
.
a
b
2.4. Mathematical Models of Anyons. The first mathematical modeling of
anyon systems is through unitary topological modular functors underlying the
Jones polynomial []. We adapt the definition of modular functor as axiomatized
by Turaev9 [22], which is essentially the topological version of a modular category.
Therefore, an abstract anyon system can be either identified as a unitary modular functor topologically or a unitary modular category algebraically. There is a
subtle Frobenius-Schur indicator that complicates the axiomitization of a modular
functor. In the context of a modular category, it is then important to distinguish
between an anyon and its equivalent class, which is called by a variety of names
including a super-selection sector, a topological or anyon charge, an anyon type,
or simply a label. The distinction between an anyon and its type is emphasized in
[25]. Two anyons are equivalent in physics if they differ by local operators. In a
modular category, local operators correspond to the morhpisms that identify the
two simple objects.
2.4.1. Unitary Topological Modular Functor. The universe of an anyon system can
be formalized into a unitary strict fusion category first. Anyons and their composite
will be objects of a category. A composite of several anyons should be considered as
their product, called a tensor product. The ground state is a unit of the tensor product as we can attach it to any anyon without changing the topological properties.
Therefore, we have a special monoidal class in the sense of Turaev10 [22]. Another
9Our modular functor is what Turaev called fractional weak 2D modular functor
10Our definition differs from Turaev in two important differecnes: first our input data is a
strict fusion category rather than a monoidal class. Secondly we add an algebraic axiom to ensure
all mapping class group representations are within some number field
MATHEMATICS OF TOPOLOGICAL QUANTUM COMPUTING
9
important operation is a sum, which will be explained with fusion—measurement
for anyons later.
Recall from [?], a fusion category C over C is an abelian C-linear semisimple
rigid monoidal category with a simple unit object 1, finite-dimensional morphism
spaces and finitely many isomorphism classes of simple objects. A fusion category
is unitary if all morphism spaces are Hilbert spaces with certain properties.
Let ΠC be the set of isomorphism classes of simple objects of the fusion category
C. The set ΠC is called a label set in TQFTs, and the set of anyon types or
topological charges in anyonic models. The rank of C is the finite number r = |ΠC |,
and we denote the members of ΠC by {0, . . . , r − 1}. We simply write Vi for an
object in the isomorphism class i ∈ ΠC . By convention, the isomorphism class of 1
corresponds to 0 ∈ ΠC . The rigidity of C defines an involutive permutation i 7→ i∗
on ΠC which is given by Vi∗ ∼
= Vi∗ for all i ∈ ΠC .
For our definition of a UTMF, we define a projective version Hf.u.p of the category
of finite dimensional Hilbert spaces. The objects of Hf.u.p are finite dimensional
Hilbert spaces and morphisms are unitary maps up to phases. A projective functor
F : C → D is a map that satisfies all properties of a functor except that F preserves
composition of morphisms only to a phase, i.e. F (f g) = ξF (f )F (g) for some
ξ ∈ U (1).
Given a strict fusion category C, we define the category Bord(2,C) of colored surfaces with boundary conditions C: the objects of Bord(2,C) are compact oriented
surfaces Y with oriented, based, colored boundary circles. The boundary ∂Y of Y
consists of oriented circles with a base point, an orientation independent of the induced orientations from Y , and an object of C—a color. Morphisms of Bord(2,C) are
data preserving diffeomorphisms: a diffeomorphism which preserves the orientation
of Y , the based points, oreintations, and colors of boundary circles.
Definition 2.1. A topological modular functor F is a symmetric monoidal projective functor from Bord(2,C) → Hf.u.p satisfying some additional axioms.
Axioms for V :
(1) Empty surface axiom: V (∅) = C.
(2) Disk axiom: (
C if l is the trivial label,
V (B 2 ; l) ∼
where B 2 is a 2-disk.
=
0 otherwise,
(3) Annular axiom:
(
C if a ' b̂,
V (A; a, b) '
0 otherwise,
where A is an annulus and a, b ∈ Lstr are strict labels. Furthermore,
V (A; a, b) ∼
= C if a ∼
= b̂.
(4) Disjoint union axiom:
V (Y1 t Y2 ; λ1 ⊕ λ2 , l1 t l2 ) ∼
= V (Y1 ; λ1 , l1 ) ⊗ V (Y2 ; λ2 , l2 ).
The isomorphisms are associative, and compatible with the mapping class
group actions.
(5) Duality axiom:
V (−Y ; l) ∼
= V (Y ; ˆl)∗ . The isomorphisms are compatible with mapping
class group actions, orientation reversal, and the disjoint union axiom as
follows:
10
ERIC C. ROWELL AND ZHENGHAN WANG
(i): The isomorphisms V (Y ) → V (−Y )∗ and V (−Y ) → V (Y )∗ are
mutually adjoint.
(ii): Given f : (Y1 ; l1 ) → (Y2 ; l2 ) and letting f¯ : (−Y1 ; ˆl1 ) → (−Y2 ; ˆl2 ),
we have hx, yi = hV (f )x, V (f¯)yi, where x ∈ V (Y1 ; l1 ), y ∈ V (−Y1 ; ˆl1 ).
(iii): hα1 ⊗ α2 , β1 ⊗ β2 i = hα1 , β1 ihα2 , β2 i whenever
α1 ⊗ α2 ∈ V (Y1 t Y2 ) = V (Y1 ) ⊗ V (Y2 ),
β1 ⊗ β2 ∈ V (−Y1 t −Y2 ) = V (−Y1 ) ⊗ V (−Y2 ).
(6) Gluing axiom: Let Ygl be the extended surface obtained from gluing two
boundary components of an extended surface Y . Then
M
V (Ygl ) ∼
V (Y ; (l, ˆl)),
=
l∈L
where l, ˆl are strict labels for the glued boundary components. The isomorphism is associative and compatible with mapping class group actions.
Moreover, the isomorphism is compatible with duality as follows: Let
M
M
αj ∈ V (Ygl ; l) =
V (Y ; l, (j, ̂)),
j∈L
M
j∈L
βj ∈ V (−Ygl ; ˆl) =
j∈L
M
V (−Y ; ˆl, (j, ̂)).
j∈L
Then there is a nonzero real number sj for each label j such that
M
M X
αj ,
βj =
sj hαj , βj i.
j∈L
j∈L
j∈L
(7) Algebraic axiom:
There is a choice of basis for all representations that all matrix entries
are algebraic numbers.
Note that the axioms of a symmetric monoidal functor do not use the property
that a functor preserves the composition exactly.
2.4.2. Fusion rules as conservation laws.
2.4.3. Jones-Witten Theory. The two modular functors called SU (2)2 and Ising
theory constructed via Jones-Kauffman are different.
2.5. From Modular Functor to Modular Category.
2.5.1. Picture Calculus. A powerful tool to study categories with structures is
graphical calculus—a far reaching generalization of spin network. It pays to understand the subtleties of picture calculus using physical intuition. Basically all
categorical properties and structures in TQC make certain graphical calculus work
out.
Figure 2 The columns of the S-matrix can be seen to be simultaneous eigenvectors for the (commuting) fusion matrices {Na : a ∈ L}, so that S diagonalizes the
P S S Shatkr
k
where
Na . This leads to the famous Verlinde formula: Nij
= D12 r ir jr
S0r
D2 is an overall normalization constant.
MATHEMATICS OF TOPOLOGICAL QUANTUM COMPUTING
11
Figure 2. Creating particle-antiparticle pairs of types a and b.
2.5.2. Creation/Annihination and Fusion/Spltting. In this section, we consider the
realistic case of a disk D2 . Given an anyon system with anyon representatives
Π = {Xi }, i = 0, 1, ..., r − 1. We assume the outmost boundary of the disk is
colored by some anyon A∞ ∈ Π. Suppose there are n anyons Ai , i = 1, 2, ..., n,
in the interior. The dimension of V (D2 ; Ai , A∞ ) can be inductively found by the
gluing formula. If there are no anyons in the disk, then there will be no quantum
state in the disk unless A∞ = 1. We will choose a normalized state |0 > in
V (D2 ; 1) as the non-anyon state—the vacuum for the disk. Then we can create
pairs of anyons from |0 >. It is a fundamental property of anyons that a single
anyon cannot be created from the non-anyon state—some conservation law. Once
we create many pairs, we can braid them around. How would we measure the
resulting state? The most natural one is by fusion: we bring two anyons together
and see what other anyons would result. This process is represented by a Y if time
goes from top to bottom. If time goes up, then the same picture represents the
splitting of a single anyon into two. Using our picture of an anyon as a small disk,
we know Y really should be thickened a pairs of pants. If we label the tree circles
with all possible anyons Xi ∈ Π, we obtain a collection of non-negative integers
Nijk = Dim V (D2 ; Xi , Xj , Xk ). The collection of integers Nijk is called the fusion
rule.
Therefore, the elementary events for anyons are creation/annihilation and fusion/splitting.
2.5.3. Particle with a sign and a handle. Classical particles are perceived as points,
therefore their configuration spaces are topological spaces, which are generally manifolds. A defining feature of a quantum particle such as the electron is its quantum
spin. A deep quantum principle is the spin-statistics connection. It is common to
picture the spin of a particle through a 2π rotation as follows:
θa
a
a
where θa = e2πiha and ha ∈ Q is the topological twist. We must take care to
remember that the picture on the left acquires a phase θa when pulled straight.
particle with a handle–framing
Insert two pictures.
When we pull tight the string, it becomes the straight line. Hence, it seems that
spinning a particle has no observable consequence. A remedy for this misleading
picture is to use a ribbon. Then after we pull tight the ribbon, we see a full twist of
12
ERIC C. ROWELL AND ZHENGHAN WANG
one side around the other side. So indeed spinning a particle makes its trajectory
different from a straight line. One justification of the ribbon idea is as follows. A
quantum particle is an elementary excitation in a quantum system, so represented
by a non-zero vector in a Hilbert space. Even after we normalize the state vector,
there is still a phase ambiguity eiθ , θ ∈ [0, 2π), which parameterizes the standard
circle in the complex plane. So a semi-classical picture of the particle sitting at
the origin of the plane would be an arrow from the origin to a certain angle θ
(this arrow is really in the tangent space of the origin). Hence one picture model
of a quantum particle could be a small arrow. Visualizing a quantum particle as
an infinitesimal arrow leads to the ribbon picture of the worldline of the quantum
particle. When we twist the particle by 2π, the resulted full twist in the ribbon
encodes the spin-statistics connection.
An anyon in a surface need a label, so it will be an object in a category. It
has spin so it needs a ribbon. When do not know its history so we need a sigh to
indicate it comes from the past or going to the past. Therefore it is a signed object
in a strict fusion category for the graphical calculus. In the plane we can make a
convention that they will all go to the future.
2.5.4. Colored Ribbon Tangles and Graphs. It is convenient to denote such a vector
with the
by the skeleton of the corresponding space, i.e. the trivalent graph
three extremal vertices labeled by a, b and c and the degree three vertex labeled
c
to distinguish it from other states in Hab
. For example, we might choose a basis
c
c
for Hab , so that there are dim(Hab ) labels. Similarly, we use appropriately labeled
graphs Y to denote states in the dual space Hcab . It is tempting, and indeed can be
justified mathematically using the gluing axiom, to stack these graphs to represent
a cascade of splitting/fusion operations. For example, if we compose compatibly
labeled Y (input c, outputs a, b) and (inputs a, b, output c) the result is a vector
c ∼
in the 1-dimensional state space Hc0
= C. One typically choosing the bases so that
c
this pairing coincides with the inner product on the Hilbert space Hab
. A complete
treatment of this diagrammatic yoga of graphical calculus involving such pictures
can be found in [?, Appendix E] and [?, Section 4.2]. See also the discussion of the
topological twist below and Figure 3 for the picture associated with the braiding
operators. Some calculations of this form are illustrated in Figures ?? and ??.
As we explain above, we will use a category to model an anyon system. Anyons
are the main objects of interests, and just as electrons, we treat them as objects
with no internal structures. But physically although they do not have a definite
size, they do occupy a certain region as Pauli exclusion principle says. Therefore,
it is so natural that we think anyons as objects of some category.
Y
Y
2.5.5. Fusion tree basis. Fusion trees are wearing two hats: they are both operators
and states, which is a version of state-operator correspondence in conformal field
theory.
2.5.6. F-matrices, Pentagon, and Hexagons.
2.6. Modular Categories.
2.6.1. Braidings. A braiding c of a fusion category C is a natural family of isomorphisms cV,W : V ⊗ W → W ⊗ V in V and W of C which satisfy the hexagon
MATHEMATICS OF TOPOLOGICAL QUANTUM COMPUTING
13
axioms:
U ⊗ (V ⊗ W )
O
cU,V ⊗W
/ (V ⊗ W ) ⊗ U
α
α
V ⊗ (W ⊗ U )
O
(U ⊗ V ) ⊗ W
id ⊗c
c⊗id
(V ⊗ U ) ⊗ W
α
/ V ⊗ (U ⊗ W ) ,
(U ⊗ V ) ⊗ W
O
cU ⊗V,W
α−1
/ W ⊗ (U ⊗ V )
α−1
(W ⊗ U ) ⊗ V
O
U ⊗ (V ⊗ W )
id ⊗c
U ⊗ (W ⊗ V )
c⊗id
α−1
/ (U ⊗ W ) ⊗ V ,
for all U, V, W ∈ C where α is the associativity isomorphism of C (cf. [?]).
A braided fusion category is a pair (C, c) in which c is a braiding of the fusion
category C. We simply call C a braided fusion category if the underlying braiding
c is understood.
2.6.2. Spherical Fusion Categories. A pivotal structure of a fusion category C is an
isomorphism j : idC → (−)∗∗ of monoidal functors. One can respectively define the
left and the right pivotal traces of an endomorphism f : V → V in C as
db
∗
id ⊗j −1
id ⊗f
ev
V
V
trl (f ) = 1 −−−
→ V ∗ ⊗ V ∗∗ −−−−V−→ V ∗ ⊗ V −−−→ V ∗ ⊗ V −−→
1
db
f ⊗id
jV ⊗id
ev
∗
V
V
trr (f ) = 1 −−−→
V ⊗ V ∗ −−−→ V ⊗ V ∗ −−−−→ V ∗∗ ⊗ V ∗ −−−
→ 1.
Note that jV∗ = jV−1∗ (cf. [?, Prop. A.1]), and so we have trl (f ) = trr (f ∗ ). Since
1 is a simple object of C, both pivotal traces trl (f ) and trr (f ) can be identified with
some scalars in C. A pivotal structure on C is called spherical if the two pivotal
traces coincide for all endomorphisms f in C. In a spherical category, the pivotal
trace(s) will be denoted by ptr(f ).
For the purpose of this paper, a pivotal (resp. spherical) category (C, j) is a fusion
category C equipped with a pivotal (resp. spherical) structure j. We will denote
the pair (C, j) by C when there is no ambiguity. The left and the right pivotal
dimensions of V ∈ C are defined as dl (V ) = ptrl (idV ) and dr (V ) = ptrr (idV )
respectively.
2.6.3. Modular Categories. Following [?], a twist (or ribbon structure) of a braided
fusion category (C, c) is an C-linear automorphism, θ, of IdC which satisfies
θV ⊗W = (θV ⊗ θW ) ◦ cW,V ◦ cV,W ,
θV∗ = θV ∗
for V, W ∈ C. A braided fusion category equipped with a ribbon structure is
called a ribbon fusion or premodular category. A premodular category C is called a
modular category if the S-matrix of C, defined by
Sij = ptr(cVj ,Vi∗ ◦ cVi∗ ,Vj ) for i, j ∈ ΠC ,
is non-singular. Note that S is a symmetric matrix and that dr (Vi ) = S0i = Si0
for all i.
Picure of axioms
2.7. Further Properties and Basic Invariants.
14
ERIC C. ROWELL AND ZHENGHAN WANG
2.7.1. Grothendieck Ring and Dimensions. The Grothendieck ring K0 (C) of a fusion
category C is the Z-ring generated by ΠC with multiplication induced from ⊗. The
structure coefficients of K0 (C) are obtained from:
M
k
Vi ⊗ Vj ∼
Ni,j
Vk
=
k∈ΠC
k
Ni,j
where
= dim(HomC (Vk , Vi ⊗ Vj )). This family of non-negative integers
k
{Ni,j
}i,j,k∈ΠC is called the fusion rules of C.
In a braided fusion category K0 (C) is a commutative ring and the fusion rules
satisfy the symmetries:
(2.1)
∗
∗
j
k
k
k
Ni,j
= Nj,i
= Ni,k
∗ = Ni∗ ,j ∗ ,
0
Ni,j
= δi,j ∗
k
, is an integral matrix with
The fusion matrix Ni of Vi , defined by (Ni )k,j = Ni,j
non-negative entries. In the braided fusion setting these matrices are normal and
mutually commuting. The largest real eigenvalue of Ni is called the FrobeniusPerron dimension of Vi and is denoted by FPdim(Vi ). Moreover, FPdim can be
extended to a Z-ring homomorphism from K0 (C) to R and is the unique such
homomorphism that is positive (real-valued) on ΠC (see [?]). The Frobenius-Perron
dimension of C is defined as
X
FPdim(C) =
FPdim(Vi )2 .
i∈ΠC
Definition 2.2. A fusion category C is said to be
(1) weakly integral if FPdim(C) ∈ Z.
(2) integral if FPdim(Vj ) ∈ Z for all j ∈ ΠC .
(3) pointed if FPdim(Vj ) = 1 for all j ∈ ΠC .
Furthermore, if FPdim(V ) = 1, then V is invertible.
2.8. Dictionary of Modular Category and Anyon Systems. An anyon is a
simple object in a modular category—a statement first appeared in [].
2.9. (2+1)-TQFTs. Our TQFTs will have anomaly so they are formulated with
the applications to physics in mind. In the end, we make a dictionary for the
mathematical modeling of anyons.
The unitary oriented bordism category Bord(n + 1) has oriented n-manifolds as
objects and equivalence classes of oriented bordsim between them as morphisms.
We define a tensor category of Hilbert spaces up to projective equivalence. First
we define the category of finite dimensional Hilbert spaces: the objects are finite
dimensional Hilbert spaces and morphism are partial unitaries. The category of
Hilbert spaces with anomaly ξ ∈ U (1) is obtained from Hil by modifying the Hom
sets.
The notion of a monoidal category is a categorification of a monoid, while a
braided in particle symmetric monoidal category is a categification of an abelian
monoidal. There is now an excellent textbook on the subject to the [] for the
terminology.
Definition 2.3. A unitary (2+1)-TQFT is a triple (Z, V, M), where Z is a unitary
symmetric monoidal functor from Bord3 to pHil, and V is a unitary modular functor with monoidal class M from cBord2 to pHil. We will call Z a partition functor.
The partition functor Z and V are compatible in the sense that when oriented closed
MATHEMATICS OF TOPOLOGICAL QUANTUM COMPUTING
15
surfaces are included into Bord3 via the mapping cylinder construction, Z and V
are functorially isomorphic.
Theorem 2.4. Given a modular category C, there is a (2 + 1)-TQFT.
Given a modular functor, there is a modular category.
A subtlety if these two construction are inverse to each other disappear because
our input data for a modular functor is a strict unitary fusion category.
2.10. Anyonic Quantum Computing Models. The vacuum state corresponds
to a disk with boundary labeled by 0, which we can use as an invisible input or
output without changing the state. The creation of a particle-antiparticle from
the vacuum corresponds to a disk with boundary labeled by 0 and two interior
boundary circles labeled by a and â. This process translates to a linear (Hermitian)
0
operator on state spaces: ba : H00
→ H0aâ , with a corresponding (dual) annihilation
aâ
0
da : H0 → H00 . Indeed, we assume we can create (via some physical process)
any number of particle-antiparticle pairs, from which we obtain, from the vacuum
state, a state in H0a1 â1 ···an ân . This corresponds to initialization in the quantum
computational model. On the other hand, if we are given a system with state
m
vector in Hab11 ···b
···an we assume we can measure the total charge (i.e. the label) of a
pair of adjacent particles, perhaps by bringing them together and measuring the
energy. This is the measurement stage of the quantum computation, a Hermitian
operator represented graphically as composing with a fusion operator Y.
The time evolution of the space of states must be a unitary operator. In particular, a sequence of particle exchanges corresponding to a braid β induces a unitary
transformation |ψi 7→ Uβ |ψi. In the topological model of [?] these are (all of)11 the
quantum circuits. If we consider a simple case where a = â, then the state space
i
i+1
i
Figure 3. Interchanging the positions of two identical particles induces
a quantum gate–the image of σi under
a unitary representation on the state
space.
H0a,...,a corresponding to n particles of type a supports a unitary representation of
the braid group Bn via particle exchange, as illustrated in Figure 3. More generally,
we always obtain a unitary representation of the small group of pure braids consisting of those braids with each strand beginning and ending in the same position.
The computational strength of the model is hidden in this unitary representation of
Bn . A particle a is called non-abelian if the image of the Bn representation on the
state space of n type a particles is non-abelian (for some n). To have a reasonable
computational model this is a bare minimum. An anyon a is called (braiding) universal if any unitary operator can be approximately achieved as the image of some
11Recently some models employing partial measurement [?] have been explored, but for the
sake of simplicity we will only consider braiding operators as our quantum circuits.
16
ERIC C. ROWELL AND ZHENGHAN WANG
braid β acting on a state space of n type a particles via particle exchange (plus
some technical “no-leakage” condition that we ignore). The search for non-abelian
and universal anyons is a major thrust of experimental condensed matter physics.
To summarize the processes we assume are available: 1) we can create any
number of particle-antiparticle pairs, 2) we may exchange these particles to rotate
our initial state and 3) we may measure the particle type of any pair of neighboring
particles. One key is that after braiding the particles’ world lines, a neighboring
particle-antiparticle pair may have obtained a different total charge (besides 0, i.e.
the vacuum). To get meaningful information from this process we must repeat
the same process several times, taking a tally of the outputs (particle types). The
topological degrees of freedom ensure that slight variations in the process (e.g.
small deviations in the trajectory of a particle in space-time) do not influence the
output. The empirically computed probability distribution of output particle types
constitutes the result of the quantum computation.
Picture of TQC
2.11. Anyons In The Real World. FQH and Majorana zero mode
3. Quantum Computing
In this section we give an introduction to von Neumann’s axiomatization of quantum mechanics and quantum computing. Abstract quantum mechanics for finitely
dimensional systems is completely elementary, in striking contrast to quantum field
theory. More or less it is complex linear algebra in a different language, therefore
simply a physical way of thinking.
3.1. Computational Power of Physical Theories. Constructing machines is a
defining characteristic of human. Every new physical theory provides an opportunity to build new kinds of computing machines. Freedman articulated this idea in
[]: “As a generality, we propose that each physical theory supports computational
models whose power is limited by the physical theory. It is well known that classical physics supports a multitude of the implementation of the Turing machine”
[]. As quoted earlier, Freedman proposed that some computational model based on
TQFTs might be more powerful than quantum computing—the computing model
based on quantum mechanics. But when accuracy and measurement are carefully
analyzed, the computing model based on TQFTs are polynomially equivalent to
quantum computing. While this mathematical theorem suggests that any computing model implemented within quantum field theory is polynomial equivalent to
quantum computing, we do not have mathematical frameworks to prove hypercomputation is impossible within quantum framework. It is interesting to ponder about
the computational power of hypothetical quantum gravities.
3.2. Encoding and Computing Problems. To formalize concrete computing
problems such as factoring integers or evalutation of Jones polynomial into families
of Boolean maps, it is important that we encode the inputs such as integers and links
into bit strings in an intelligent way because encodings can change computational
complexities. If an integer N is encoded by unary strings, then dividing N by
primes is polynomial in the input length. Most reasonable encodings will lead to
the same complexity class, therefore, we will not discuss encodings further.
MATHEMATICS OF TOPOLOGICAL QUANTUM COMPUTING
17
Definition 3.1. A computing problem is a sequence of Boolean functions fn :
a(n)
Zn2 → Z2 . A computing problem with a(n) = 2 is called a decision problem. A
reversible computing problem is one such that a(n) = n and every fn is a permutation.
n
n
Let {0, 1}∗ = ∪∞
n=0 Z2 , where for n = 0, Z2 is the empty string. Then a comput∗
∗
ing problem is a map f : {0, 1} → {0, 1} . We consider classical physics as part of
quantum physics, and will embed classical computation into quantum computation
through reversible classical computing. Therefore, quantum computing can solve
the same class of computing problems potentially much faster.
3.3. Quantum Framework. Quantum mechanics is very different from quantum
field theory in that its principles have been formalized mathematically. Some basic
physical principles include for a quantum system, there is a definite state at each
moment, and its evolution from one moment to another is deterministic. Moreover,
a known state can be prepared and a known eveolution can be repeated as many
times as we need.
Quantum mechanics is a set of rules that predict the responses of the microscopic
world to our measuring devices. The most salient feature is the superposition of
different states. With the advent of quantum information science, another quantum
correlation comes to the center stage: entanglement–the characteristic attribute of
quantum mechanics according to Schodinger.
von Neumann’s axiomatization of quantum theory consists of four principles:
superposition, linear evolution, entanglement, and projective measurement. States
and linear evolutions are modelled by non-zero vectors and unitary transformations
in Hilbert spaces. The controversial measurement models well the un-controllable
disturbance of the quantum state by our measuring devices: energy from measuring
apparatus into the quantum system overwhelming the fragile state.
Two operations on Hilbert spaces that are used to describe superposition and
entanglement, respectively, are the direct sum and the tensor product. While interference is arguably more fundamental, a deeper understanding of quantum mechanics would come from the interplay of the two. The role for entanglement is
more pronounced for many-body quantum systems such as systems of 1011 electrons in condensed matter physics. Advancing our understanding of the role of
entanglement for both quantum computing and condensed matter physics lies at
the frontiers of current research.
More detailed axioms for quantum mechanics is as follows:
(1) State space: Just as in classical mechanics, a quantum system possesses a
state at any moment. There is a Hilbert space L describing all possible
states. Any nonzero vector |vi represents a state, and two nonzero vectors
|v1 i and |v2 i represent the same state iff |v1 i = λ|v2 i for some scalar λ 6= 0.
Hilbert space embodies the most salient feature of quantum mechanics—the
superposition principle.
Quantum computation uses ordinary finite-dimensional Hilbert space
Cm , whose states correspond to CP m−1 . Therefore information is stored
in state vectors, or more precisely, points on CP m−1 .
(2) Evolution: If a quantum system is governed by a Hamiltonian H, then its
state vector |ψi is evolved by solving the Schrödinger equation i~ ∂|ψi
∂t =
H|ψi. When the state space is finite-dimensional, the solution is |ψt i =
18
ERIC C. ROWELL AND ZHENGHAN WANG
i
i
e− ~ tH |ψ0 i for some initial state |ψ0 i. Since H is Hermitian, e− ~ tH is a
unitary transformation. Therefore we will just say states evolve by unitary
transformations.
In quantum computation, we apply unitary transformations to state vectors |ψi to process the information encoded in |ψi. Hence information
processing in quantum computation is multiplication by unitary matrices.
(3) Measurement: Measurement of a quantum system is given by a Hermitian
operator M such as the Hamiltonian (= total energy). Since M is Hermitian, its eigenvalues are real. If they are pairwise distinct, we say the measurement is complete. Given a complete measurement M with eigenvalues
{λi }, let {ei } be an orthonormal basis of eigenvectors of M corresponding
to {λi }.
PIf we measure M in a normalized state |ψi, which can be written as
|ψi = i ai |ei i, then the system will be in state |ei i with probability |ai |2
after the measurement. The basis {ei } consists of states that are classical
in a sense. This is called projective measurement and is our read-out for
quantum computation.
Measurement interrupts the deterministic unitary evolution and outputs
a random variable X : {ei } → {λi } with probability distribution p(X =
λi ) = |ai |2 , and hence is the source of the probabilistic nature of quantum
computation.
(4) Composite system: If two systems with Hilbert spaces L1 and L2 are
brought together, then the state space of the joint system is L1 ⊗ L2 . Composite systems have entangled states, which baffle many people, including
Einstein.
The construction of the Hilbert space L and the Hamiltonian H for a given
quantum system is in general difficult. If we start with a classical system, then a
procedure to arrive at L and H is called quantization. Sometimes we don’t even
have a classical system to begin with, and the same quantum system might have
different classical limits.
3.3.1. Entanglement. Superposition is only meaningful if we have a preferred basis,
ie. a direct sum decomposition of the Hilbert space. When the Hilbert space
of states has a tensor decomposition, then we can define an entangled state vs a
product state.
3.3.2. Error-correction code.
3.4. Gates, Circuits, and Universality. A gate set S is the elementary operations that we will carry out repeatedly to complete a computational task. Each
application of a gate is considered a single step, hence the number of gate applications in an algorithm represents consumed time, and is a complexity measure.
A gate set should be physically realizable and complicated enough to perform any
computation given enough time. It is not mathematically possible to define when a
gate set is physical as ultimately the answer comes from physical realization. Considering this physical constraint, we will require that all entries of gate matrices
are algebraic numbers when we define complexity classes depending
S∞ on a gate set.
Generally, a gate set S is any collection of unitary matrices in n=1 U(2n ). Our
choice is
S = {H, σz1/4 , CNOT}
MATHEMATICS OF TOPOLOGICAL QUANTUM COMPUTING
19
where
1
H=√
2
1
σz1/4 =
0

1
0
CNOT = 
0
0
1
1
1
−1
is the Hadamard matrix,
0
eπi/4
0
1
0
0
0
0
0
1
is called the

0
0

1
0
π
gate,
8
in the two-qubit basis |00i, |01i, |10i, |11i .
It is called controlled-NOT because the first qubit is the control bit, so that when
it is |0i, nothing is done to the second qubit, but when it is |1i, the NOT gate is
applied to the second qubit.
Definition 3.2.
(1) An n-qubit quantum circuit over a gate set S is a map UL : (C2 )⊗n →
(C2 )⊗n composed of finitely many matrices of the form Idp ⊗ g ⊗ Idq , where
g ∈ S and p, q can be 0.
(2) A gate set is universal if the collection of all n-qubit circuits forms a dense
subset of SU(2n ) for any n.
1/4
The gate set S = {H, σz , CNOT} will be called the standard gate set, which
we will use unless stated otherwise.
Theorem 3.3.
(1) The standard gate set is universal.
(2) Every matrix in U(2n ) can be efficiently approximated up to an overall phase
by a circuit over S.
3.5. Complexity Class BQP. The quantum computing model select a class of
efficiently computable functions called BQP.
NP incomplete problmes such graph minimal vector
Let CT be the class of computable functions CT and P the class of efficiently
computing functions by the Turing machines. Defining a computing model X is
the same as selecting a class of computable functions from CT , denoted as X P . The
class X P of efficiently computable functions codifies the computational power of
computing machines in X . Quantum computing selects a new class BQP–boundederror quantum polynomial time. In theoretical computer science, separation of
complexity classes is extremely hard as the millenium problem P vs N P problem
shows. It is generlly believed that the class BQP does not contain NP-complete
problmes. Therefore, good traget problems will be those NP problems which are
not known to be NP compete. Three candidates are factoring, graph iso., and
shortest vector.
The class P can be motivated by the following consideration. The volume is
P...as the size of a gate.
1/4
Definition 3.4. Let S be the gate set {H, σz , CNOT}. A problem f : {0, 1}∗ →
m(n)
{0, 1}∗ (represented by fn : Zn2 → Z2 ) is in BQP (i.e., can be solved efficiently
by a quantum computer) if ∃ polynomials a(n), g(n) : N → N satisfying n + a(n) =
20
ERIC C. ROWELL AND ZHENGHAN WANG
m(n) + g(n) and ∃ a classical efficient algorithm to output a map δ(n) : N → {0, 1}∗
describing a quantum circuit Uδ(n) over S of size O(poly(n)) such that:
X
Uδ(n) |x, 0a(n) i =
aI |Ii
I
X
3
|aI |2 ≥ ,
4
g(n)
where z ∈ Z2
|Ii=|f (x)zi
The a(n) qubits are ancillary working space, so we initialize an input |xi by
appending a(n) zeros and identify the resulting bit string as a basis vector in
(C2 )⊗(n+a(n)) . The g(n) qubits are garbage. The classical algorithm takes as input
the length n and returns a description of the quantum circuit Uδ(n) . For a given
|xi, the probability that the first m(n) bits of the output equal fn (x) is ≥ 34 .
The class BQP is independent of the choice of gate set as long as the gate set is
efficiently computable. The threshold 34 can be replaced by any constant between 12
and 1. In our definition of BQP, the quantum circuit Uδ(n) is uniform for all inputs
|xi of length n. In Shor’s algorithm, the quantum circuit for a fixed n depends on
the input |xi, but there is an efficient classical algorithm to convert Shor’s algorithm
into our formulation of BQP [?].
Classical computation moves the input x through sequences of bit string x0 before
we reach the answer bit string f (x). In quantum computing, the inut bit string x is
⊗n
represented as a basis quantum state |x >∈ (C2 ) —the Hilbert space of n-qubit
states. Then the computing is carried out by multiplying the initial state vector |x >
by unitary matrices from solving the Schodinger equation. The intermediate steps
are still deterministic, but now through quantum states which are superpositions of
basis states. The answer will be contained in the final state vector of the quantum
system—a superposition with exponential many terms. But there is an asymmetry
between the input and output because in order to find out the answer, we need to
measure the final quantum state. This quantum measurement leads to probability
by the von Neumann formulation of measurement.
Shor’s astonishing algorithm proved that factoring integers is in polynomial time
by quantum computing model, which launched quantum computing. But quantum
states are notoriously fragile. The fragility of qubits called decoherence prevents
us from a useful quantum computer. There is a solution of the problem using
error-correcting codes. But technology requirement is daunting if not impossible.
3.6. Quantum Algorithms.
3.6.1. Finding units.
3.6.2. Approximate topological invariants.
3.6.3. Simulating physics.
3.6.4. 100 qubits.
3.7. Qutrit and Qudit.
3.8. Open Problems.
3.8.1. CFTs and Topological String Theory.
MATHEMATICS OF TOPOLOGICAL QUANTUM COMPUTING
21
4. Quantum Matters
In 2003, Freedman, Nayak, and the second author organized the workshop topology in condensed matter physics in the American Institute of Mathematics. Afterwards, Freedman proposed the establishment of a Microsoft research institute
on the campus of UC Santa Barbara to build a topological quantum computer.
In 2005, Microsoft Station Q began with Freedman, Kitaev, Nayak, Walker and
the second author. Since then, Station Q has been instrumental in many exciting
developments in TQC as this section illustrates. We focus on algebraic and topological study of quantum many-body systems, while leave out the analytic study of
quantum man-body systems.
In this section, we will define 2D topological phases of matter mathematically.
Two guiding principles are locality and unitarity. In [], Freedman wrote:“Nature
has the habit of intruding on the prodigies of purest thought and encumbering
them with unpleasant embellishments. So it is astonishing when the chthonian
hammer of the engineering resonates precisely to the gossamer fluttering of theory.
Such a moment may soon be at hand in the practice and theory of quantum computation. The most compelling theoretical question, “localization,” is yielding an
answer which points the way to a solution of Quantum Computing’s (QC) most
daunting engineering problem: reaching the accuracy threshold for fault tolerant
computation.” Our definition of topological phases of matter is a form of localization of TQFTs. As we will see in the next section, the simulation of TQFTs by
quantum computer can be regarded as another localization of TQFTs. For braid
groups, localization has been intensively discussed in []. While a localization of a
TQFT is a step towards its physical realization, we are fully aware of the devil in
Nature’s “unpleasant embellishments”. In the end, we will discuss the bulk-edge
correspondence, which is the best example of AdS/CFT duality.
Hamiltonians in this article are all quantum Hamitonians, so they are Hermitian
matrices on Hilbert spaces.
4.1. Phases of Matter. Einstein’s relativity unveil the origin of ordinary matter.
Condensed matters come in various of forms: solid, liquid, and gas are all familiar
phases of matter12. By a more refined classification, each phase consists of many
different phases of matter. For example, within the crystalline solid phase, there are
many different crystals distinguished by their different lattice13 structures. All those
phases are classical in the sense they depend crucially on the temperature. More
mysterious and challenging to understand are quantum states of matter: phases
of matter at zero temperature (in reality very close to zero). The modeling and
classification of quantum phases of matter is an exciting current research area in
condensed matter physics and topological quantum computation. In recent years,
much progress has been made in a particular subfield: topological phases of matter
(TPMs). Besides their intrinsic scientific merits, another motivation comes from
the potential realization of fault-tolerance quantum computation using non-abelian
topological phases of matter.
12the words state and phase are used interchangeably for states of mater. Since we alos refer
to a wave functions as a quantum state, we will use phase more often.
13We mean lattices as in physics in the sense they are regular graphs, which are not lattices
in the mathematical sense because they are not necessarily subgroups of Rn for some n.
22
ERIC C. ROWELL AND ZHENGHAN WANG
Roughly speaking, a phase of matter is an equivalence class of quantum systems
that certain properties are the same within the equivalence class. The devil is in
the definition of the equivalence relation. Homotopy is a good example. All phase
information is presented by the so-called phase diagram: a diagram represents all
possible phases and some domain walls indicating the phases transitions. Quantum
systems in the same domain are the same, and the domain walls are where certain
physical quantities such as the ground state energy per particle become singular.
Matters are made of atoms and their arrangement patterns determine their properties. One important characteristic of their patterns is their symmetry. Liquids
have a continuous symmetry, but a solid has only a discrete symmetry broken down
from the continuous. In 3-spatial dimensions, all crystals symmetries are classed
into 230 space groups. A general theory for classical phases of matter and their
phase transitions is formulated by L. Landau. In Landau’s theory, phases of matter
are characterized by their symmetry groups, and phases transitions are characterized by symmetry breaking. It follows that group theory becomes an indispensable
tool in condensed matter physics. TPMs do not fit into the Landau paradigm,
and intensive effort in physics right now is to develop a post-Landau paradigm to
classify quantum phase of matter.
4.2. Quantum Temperature. Theoretical speaking, quantum phases of matter
are organized by quantum effect, not by energy. So ideally we would say they
are matters at zero temperature. But in relaity, zero temperature is not an option?
Then what is a quantum system? Particles normally have a characteristic oscillation
frequency depending on their environment. By quantum mechanics, they have
energy E = hν. If their environment has temperature T , then the thermal energy
per degree freedom is kT . Whether or not the particles behave as quantum particles
then depend on the compration of the two energy scales. If kT is much less than hν,
then the physics of particles are within the quantum realm. For quantum system
to be protected from noise, we system need to well isolated from the outside world.
Therefore, a particle with characteristic frequency ν have a quantum temperature
hν
. Classical or quantum is determined by the competition of T and T ∗ .
T ∗ = kT
Condensed matter physics is a study of phenomena in materials. The emergence
principle says universal properties arrives from the interaction or arrangement of
particles. Since atoms in solids are regular arrays, therefore lattices are real physical
existence. Therefore, we start with a Hilbert space which is a tensor product of a
small Hilbert space associated to each atom. The small Hilbert space comes with
a canonical basis and the large Hilbert space has the tensor basis, which represents
classical configuration of the physic systems. Therefore, lattices and basis are real
in quantum physics and should be taken seriously. Lattice in our discussion will
be triangulation of manifolds. In topological phases of matter, we are interested in
theories which are independent of triangulations, therefore has a continuous limit.
Such theories are rare.
How non-abelian anyon arised in condensed matter physics? Hamiltonian schema
and gap
4.3. Topological Qudit Liquids. Two fascinating macroscopic quantum phenomena are the high temperature superconductivity and the fractional quantum
Hall effect (FQHE). Both classes of quantum matters are related to topological
MATHEMATICS OF TOPOLOGICAL QUANTUM COMPUTING
23
qubit liquids14. While the quantum Hall liquids are universally accepted examples
of topological phases of matter, the role of topology in high temperature superconductors is still under debate. But the new state of matter “quantum spin liquid”
suggested for high temperature superconductors by P. Anderson is possibly realized
by a mineral: herbertsmithite.
Herbertsmithite is a mineral with the chemical formula Cu3 Zn(OH)6 Cl2 . The
unit cell has three layers of copper ions Cu2+ which form three perfect kagome
lattices in three parallel planes. The copper ion planes are separated by zinc and
chloride planes. Since the Cu2+ planes are weakly coupled and Cu2+ has a S = 21
magnetic moment, herbertsmithite can be modeled as a perfect S = 12 kagome
antiferromagnet with perturbations.
4.3.1. Fundamental Hamiltonian vs Model Hamiltonian. Theoretically, we can write
down the fundamental15 Hamiltonian for a herbertsmithite crystal and try to find its
ground state wave function. Then we would deduce its physical properties from the
ground state wave function. Unfortunately, this fundamental approach from first
principle physics cannot be implemented in almost all realistic systems in condensed
matter physics. Therefore, what we do instead in condensed matter physics is to
have an educated guess: to write down a model Hamiltonian and derive physical
properties of the state of matter from this model system. This emergent approach
has been extremely successful in our understanding of the fractional quantum Hall
effect.
The model Hamiltonian for herbertsmithite is the Heisenberg S = 12 kagome
antiferromagnet with perturbations:
X
H=J
Si · Sj + Hpert ,
hi,ji
where J is the exchange energy, and Si are the spin operators (we will discuss them
more carefully later). Experiments show that J = 170K ∼ 190K. The indices i, j
refer to the vertices of the lattice (also called sites in physics), and hi, ji means that
the sum is over all pairs of vertices that are nearest neighborhood of each other.
Competing Models.
There are many perturbations (small terms) to the Heisenberg spin exchange
Hamiltonian H. Three prominent ones are the anisotropy of the spin exchange
action, the next nearest neighbor interaction, and the disorder from impurities in
the zinc plane. Different perturbations lead to different potential spin liquid states:
numerical simulation using DMRG shows that next nearest neighborhood exchange
will stabilize a gapped spin liquid, while other models lead to gapless spin liquids.
These competing models can be organized into a phase diagram, which represents
the rich physics of the different phases when parameters of the model Hamiltonian
H change.
14The common term for them in physics is “spin liquids”, as we will call them some time
too. But the word “spin” implicitly implies an SU(2) symmetry, which is not present in general.
Therefore, we prefer “qubit liquids” or really “qudit liquids”, and use “spin liquids” only for those
with an SU(2) symmetry.
15By fundamental here, we mean the Hamiltonian comes from first physical principles such as
Coulomb’s law. Of course they are also model Hamiltonians. By model here, we mean that we
describe the system with some effective degrees of freedom, and keep only the most relevant part
of the interaction and treat everything else as small perturbations.
24
ERIC C. ROWELL AND ZHENGHAN WANG
4.4. Physical Quantization. Where do Hilbert spaces come from? Where do the
model Hamiltonians and observables come from? What are the principles to follow?
What is a phase diagram? What is a phase transition? What does it mean that a
model is exactly solvable or rigorously solvable?
When we have answers to all these questions, our subject will be so mature and
cease to be exciting. Fortunately, Nature would never fail to surprise us with her
new tricks and present us with new questions.
To quantize physically is to wear a Hat. Physicists have a powerful conceptual
method to quantize a classical observable: put a hat on the classical quantity. For
example, to quantize a classical particle with position x and momentum p, we define
two Hermitian operators x̂ and p̂, which satisfy the famous commutator [x̂, p̂] = i~.
The Hilbert space of states are spanned by eigenstates of x̂ denoted as |xi, where x is
the corresponding eigenvalue of the Hermition operator x̂. Similarly, we can use the
momentum p, but not both due to the uncertainty principle. The two quantizations
using x or p are related by the Fourier transform. Therefore, physical quantization
is easy: wear a hat. In general, a wave function Ψ(x) is really a spectral function
for the operator Ψ(x̂) at state |Ψ(x)i: Ψ(x̂)|Ψ(x)i = Ψ(x)|Ψ(x)i.
4.5. Quantum Phases of Matter.
4.5.1. Particle and Phase of Matter. Particles are just special quantum states,
which are localized exponentially in space. They are fuzzy spots in space modeled by Gaussian wave packets. While they are not points and occupy space of
non-zero size, it is hard to define any classical shape, size, or position.
In classical phases of matter, we start with pattern of atoms and study their
interaction. In quantum phases of matter, we study the pattern of wave packets.
Classification of Classic crystals and Group
4.6. Many-body Quantum Systems. Physical quantum field theories that are
made mathematically rigorous include conformal field theories (CFTs), topological
quantum field theories (TQFTs), and certain non-linear sigma models.
Unless stated otherwise, all our Hilbert spaces are finitely dimensional ones,
hence are isomorphic to Cd for some integer d. We will refer to Cd as a qudit
following the quantum information science language.
4.6.1. Linear algebra problems need quantum computers. To a first approximation,
the subject of quantum many-body systems is complex linear algebra in quantum
theory language. In practice, it is a linear algebra problem that even the most
powerful classical computer cannot handle. Memory is one major constraint. Stateof-the-art computational physicist can handle Hilbert spaces of dimension ≈ 272 .
But for a real quantum system, this dimension is very small: in quantum Hall
physics, there are about 1011 electrons per cm2 . So we will have thousands of
electrons per square micron. If we consider the Hilbert space for the electron spins,
we will deal with a Hilbert space of dimension ≈ 21000 . Therefore, it is almost an
impossible task to solve such a problem exactly. To gain understanding of such
problems, we have to rely on ingenious approximations or extrapolations from a
small number of electrons.
MATHEMATICS OF TOPOLOGICAL QUANTUM COMPUTING
25
One interesting open question is to understand thermodynamical properties of
quantum systems, which are quantum systems with Hilbert spaces as large as possible. A good approximation for arbitrarily large Hilbert spaces are infinite dimensional Hilbert spaces and their von Neumann algebras. Presumably, von Neumann
algebras are important for our understanding of quantum systems.
Definition 4.1.
(1) A many-body quantum system (MQS) is a triple (L, b, H),
where L is a Hilbert space with a distinguished orthonormal basis16 b =
{ei }, and H an Hermitian matrix regarded as an Hermitian operator on
L using b. The Hermitian operator H is called the Hamiltonian of the
quantum system, and its eigenvalues are the energy levels of the system.
The distinguished basis elements {ei } are classical states or configurations.
Conceptually, quantum mechanics is like a square root of probability
theory because amplitudes are square roots of probabilities.
(2) A quantum system on a graph Γ = (V, E) with local degrees of freedom
Cd is a mathematical quantum system whose Hilbert space L = ⊗e∈E Cd
with the tensor orthonormal bases from {ei } of Cd , where E are the edges
(bonds or links) of Γ.
We will say there is a qudit on each edge following quantum information
terminology.
A quantum system on a graph is a composite quantum system consisting
of identical subsystems of local qudits. We will use the Dirac notation to
represent the standard basis of Cd by ei = |i−1i, i = 1, · · · , d. When d = 2,
the basis elements of L are in one-one correspondence with bit-strings or
Z2 -chains of Γ.
Since we are interested in quantum phases of matter, we are not focusing on
a single quantum system. Rather we are interested in a collection of quantum
systems and their properties in some limit which is related to the physical idea of
thermodynamic or long-wave length/low energy limit. In most cases, our graphs
are the 1-skeleton of some triangulation of a manifold.
The most important Hermitian matrices are the Pauli matrices, which are spin= 21
operators. Pauli matrices wear two hats because they are also unitary.
Example 4.2. Let Γ be a kagome lattice on the torus, i.e., a kagome lattice in the
plane with periodic boundary condition. The Heisenberg Hamiltonian is
X
H=
σix σjx + σiy σjy + σiz σjz ,
<i,j>
where the Pauli matrices σkα , α = x, y, z, k = i, j acts on the k-qubit in the sense
σkα is the Pauli matrix σ α on the k tensor factor extended by Id to other factors.
Definition 4.3. Given a MQS (L, b, H). Let {λi , i = 0, 1, · · · } be the eigenvalues
of H ordered in an increasing order and Lλi the corresponding eigenspace.
(1) Lλ0 is called the ground state manifold17 and any state in Lλ0 is called a
ground state. Any state in its complement ⊕i>0 Lλi is called an excited
state. Usually we are only interested in the first excited states in Lλ1 , but
16The preferred basis is very important for later discussion of entanglement because it allows
locality to be defined. The role of basis is also essential for superposition to be meaningful.
17Physicists use the word manifold to mean multi-fold, but L
λ0 indeed is a simple manifold.
26
ERIC C. ROWELL AND ZHENGHAN WANG
sometimes also states in Lλ2 . Bases states in Lλi are called minimal excited
states.
(2) The partition function of a quantum system is Z = Tr(e−βH ).
(3) A quantum system is rigorously solvable if we can find bases of the ground
state manifold and minimal excitations for a few low excited state manifolds, and its partition function as an explicit analytic function of β. A
quantum system is exactly solvable if the exact answers as in rigorously
solvable models are given as physical theorems.
Minimal excitations vs elementary excitations vs particles
Entanglement is a property of a composite quantum system. If we have a
quantum system with several subsystems given by a collection of Hilbert spaces
{Li },N
i ∈ I, then the Hilbert space for the composite system is their tensor product
L = i∈I Li .
Definition
4.4. Given a composite quantum system with Hilbert space L =
N
L
,
then
a state is entangled with respect to this tensor decomposition into
i∈I i
subsystems if it is not a direct product of the form ⊗i∈I |ψi i for some |ψi i ∈ Li .
Definition 4.5. A unitary U is a symmetry of H if U HU † = H. An Hermitian
operator K is a symmetry of H if [K, H] = 0, and then eitK is a unitary symmetry
for any t. When U is a symmetry, then each energy eigenspace Lλi is further
decomposed into eiegnspaces of U . Those eigenvalues of U are called good quantum
numbers.
Definition 4.6.
(1) Correlation function: h0|O(r − r0 )|0i.
(2) Expectation value h0|O|0i is the expectation value for measuring O.
N
Definition 4.7.
(1) A MQS is on a control space Y if L = i∈I Li and there
is a map p : i → Y .
P
(2) A MQS is local if H =
Hi such that each Hi is of the form A ⊗ Id and
the support of A is bounded.
P
(3) A Hamiltonian is a sum of commuting local projectors (CLP) if H =
Hi
such that each Hi is a local projector and [Hi , Hj ] = 0 for all i, j.
4.7. Topological Phases of Matter. TPMs are phases of matter whose low energy physics and universal properties can be modeled well by TQFTs and their
enrichments. To characterize TPMs, we focus either on the ground states or their
first excited states. The ground state dependence on the topology of spaces where
the quantum system resides is organized into a TQFT, and the algebraic models
of elementary excitations of the TPMs are unitary modular categories. Topological phases of matter (TPMs) in nature include quantum Hall states–integral and
fractional—and the recently discovered topological insulators.
4.7.1. Pachner Poset of Triangulations and Thermodynamical Limit. Let Y be an
oriented n-manifold, possibly with boundary, and ∆i , i = 1, 2 two triangulations of
Y . Pachner theorem lists the a finite set of complete moves that we can change
from ∆1 to ∆2 . We will consider all triangulations of ∆i of Y as a poset. Define
the complexity c(∆) of a triangulation ∆ as the number of top n-simplices. Then
define the partial order as ∆1 ≤ ∆2 if c(∆1 ) ≤ c(∆2 ).
MATHEMATICS OF TOPOLOGICAL QUANTUM COMPUTING
27
4.7.2. Hamiltonian Schemas. When we specify a Hamiltonian, we are giving instructions on how to write down a class of mathematical quantum systems. We
will call such instructions for defining quantum systems as a Hamiltonian schema.
In this book, Hamiltonian schemas are given for triangulations of or lattices in the
space manifolds of a fixed dimension. Usually, we also need additional structures
on the manifolds or lattices. In real materials, the lattice points are atoms in the
material, so they form special lattices. In our consideration, we will consider every
triangulation of a manifold as a lattice, therefore our theories are very special: they
exist on every space manifold with an arbitrary triangulation. This requirement
is some strong locality: a theory id full determined by the Hamiltonian on a disk
region. An extreme example of lattice dependence is the 3D Haah code which is
defined only on translation invariant lattice on the three torus. Such theories are
not completely local.
Definition 4.8. A Hamiltonian schema is a uniform local rule to associate a MQS
to any triangulation of a surface, where local means the rule is determined by the
regular neighborhoods of simplices of each dimensions, and uniform means it suffices
to know the rules for one simplex of each dimension.
Example 4.9 (The toriccode schema). In the celebrated toric code, l = 2. Let
0
be the Pauli matrices. For each vertex v, define an
σx = ( 01 10 ), σz = 10 −1
N
2
operator Av on LΓ =
edges C , as a tensor product of σz ’s and identities: Av
acts on a qubit C2 as σz if the edge corresponding to C2 touches v, and as idC2
otherwise. Similarly, define a plaquette term Bp for each face p, as a tensor product
of σx ’s and identities: Bp acts on a qubit C2 by σx if the edge corresponding to C2
touches p, and as idC2 otherwise. We normalize the smallest eigenvalue (= lowest
energy) to zero. Hence the toric code Hamiltonian is
X I − Bp
X I − Av
+
H=
2
2
vertices
v
faces
p
If our lattices are arbitrary, the toric code Hamiltonian is not k-local for any
k because vertex valences in a graph can be arbitrarily large. In condensed matter physics, lattices describe particles such as atoms, and hence are not arbitrary.
Therefore it is reasonable, maybe even necessary, to restrict our discussion of HS’s
to certain types of lattices. In the toric code case, on the torus, we restrict to square
lattices. In general, we can restrict to any family of lattices with bounded valence
of both the original lattice Γ and its dual Γ̂. Then the toric code is a local HS, e.g.,
4-local for square lattices on T 2 . Given a Hamiltonian HΓ , we denote by V0 (Γ, Y )
the ground state manifold.
Example 4.10. Levin-Wen schema:
4.7.3. Spectral gap. A Hamiltonian schema (HS) is gapped or has a spectral gap if
the difference of energies λ1 −λ0 is bounded below by a non-zero constant for all the
quantum systems resulted from this HS. The spectral gap is difficult to establish
analytically and is crucial for application to quantum computing.
Definition 4.11. A Hamiltonian schema has a spectral gap if the spectrum of a
(j)
Hamiltonian has the following structure: there exist a set of eigenvalues of {λ0 , j =
1, 2, ..., k} and a constant ∆ such that all other eigenvalues λi ≥ c, i ≥ 1 and
(i)
(j)
|λ0 − λ0 | ≤ e−ξX .
28
ERIC C. ROWELL AND ZHENGHAN WANG
Conjecture 4.12. It is sufficient to establish gap for genus 0 and 1.
open question: how to define gapped on a disk?
4.7.4. Ground state functor.
Definition 4.13. A Hamiltonian schema is topological if the ground state functor
for closed oriented surfaces is equivalent to the part of a UTMF for closed surfaces.
Two topological Hamiltonian schemas are equivalent if their ground state functor
are equivalent as tenor functors. A 2D topological phase of matter is an equivalent
class of topological Hamiltonian schemas.
4.8. Stable Gapped Hamiltonian Schemas.
4.8.1. Connected Hamiltonian schemas. Two stable gapped Hamiltonian schemas
are connected if they can be connected to each other with the assistance of ancillas.
4.8.2. Stability.
Definition 4.14. A Hamiltonian schema is stable if ....
Pseudo-unitary FC vs Hermition Hamiltonian. Gapless???
Definition 4.15. A 2D topological phases of matter is a stably-connected equivalence class of stable gapped Hamiltonian schemas.
Two Hamiltonian schemas are in the same class if they can be connected by a
path of Hamitoniann schemas stably without closing the spectral gap.
The topological order in a topological phase of matter is the modular category
derived from the modular functor.
Remark: TPM can be defined using circuits and wave functions. should be
equivalent.
Elementary excitation form a unitary strict fusion category.
Conjecture 4.16. The ground state functor of a stable gapped Hamitlonian schemas
is part of a a UMTF.
4.8.3. Commuting local projectors. How to establish a gap is in a technique vacuum.
One obvious case is for Hamiltonians which are CLP.
Proposition 4.17. If H is a CLP then it has an energy gap and Ψ is a ground
state iff Hi Ψ = 0 for each i.
4.8.4. Elementary Excitations.
Conjecture 4.18. The elementary excitations of a TPM is a UMC.
4.8.5. Ground states and error correction. TQO1 is the disk axiom.
4.8.6. non-abelian statistics. TQC essay and PRA paper.
4.8.7. Long range entanglement.
4.8.8. Ground states determines.
4.8.9. Circuit definition.
4.9. Fractional Quantum Hall.
4.10. Topological order and TPM.
MATHEMATICS OF TOPOLOGICAL QUANTUM COMPUTING
29
4.11. Realization of TQFTs.
4.11.1. Kitaev and LW model. JW thesis
4.11.2. Hastings theorem. TQO1 and TQO2
4.11.3. Haldane for semions.
4.11.4. Kitaev mdoel for Ising.
4.12. Anyons and TPM.
4.12.1. Realization of anyons.
4.12.2. Bulk edge correspondence. Edge is the window to look into the bulk.
4.13. Topological symmetry.
4.14. Open Problems. No Hamiltonian realization of chiral modular categories.
5. Anyonic Quantum Computation
“Quantum computation is any computational model based upon the theoretical ability to manufacture, manipulate and measure quantum states”[4]. In TQC,
information is encoded in multi-anyon quantum states, and our goal is the construction of a large scale quantum computer based on braiding non-abelian anyons.
In this section, we cover the anyonic quantum computing models.
The nexus among TQC TPM and TQFT is summarized in the following picture.
To carry out quantum computation, we need quantum memories, quantum circuits, and protocols to write and read information to and from the quantum systems. In the anyonic quantum computing model, we first pick a non-abelian anyon
type, say x. Then information is stored in the ground state manifold Vn,x of n
type x anyons (for simplicity, we ignore the boundary conditions.) As n goes to
infinity, the dimension of Vn,x goes asymptotically as dnx , where dx is the quantum dimension of x. Since x is non-abelian, dx > 1. It follows that when n is
large enough, we can encode any number of qubits into some Vn,x . The ground
state manifold Vn,x is also a unitary representation of the n-strand braid group Bn ;
hence, unitary representation matrices serve as quantum circuits. An initial state
of computation is given by creating anyons from the ground state and measurement
is done by fusing anyons together to observe the possible outcomes. There are important subtleties regarding encoding qubits into Vn,x because their dimensions are
rarely powers of fixed integers. There is also the important question of whether the
braiding matrices alone will give rise to a universal gate set.
a blessing and curse of local dof.
In this section, we will use the anyon language. Recall that an anyon system is
a UMC and anyon is a simple object of the category.
30
ERIC C. ROWELL AND ZHENGHAN WANG
5.1. Topological Qudits. There are many choices to encode strings x ∈ Znd onto
topological degrees of freedom in ground states of many anyons. The explicit topological encoding is to use the so-called fusion channels of many anyons, so that
strings x correspond to fusion-tree basis elements. Earlier encoding in [] involves a
splitting of certain fusion channels and a reference bureau of standards, hence not
explicitly topological. Though the computation could still potentially be carried
out using only the topological degree of freedom, the relation of the two encodings
is not carefully analysed in the literature. We will follow the explicitly topological
encoding in [].
5.1.1. Dense vs Sparse encoding. The spare encoding is directly modelled on the
quantum circuit model, we choose a subspaces which are separated into qudits. In
the dense encoding, we directly encode qudits so there is a priori seperation into
qudits. There are protocols to go from one to the other by using measurements.
5.1.2. Topological gates. Straightforward topological gates are braiding gates. But
there are also topological resources that can be used to augmentation braidings
such as measurement and mapping classes in higher genus surfaces. We will mainly
focus on braiding gates and the simplest measurement: the measurement of total
charge of a group of anyons.
5.2. Braiding Gates and Universality.
Definition 5.1. Given an anyon x and an encoding of one qudit and two qudits,
the braiding gate set of an anyon x is the unitary matrices ρx (σi ).
5.2.1. Braiding universal.
Definition 5.2. An anyon is braiding universal if the afforded braid group representations are independently dense for Bn when n ≥ no .
5.2.2. Fibonacci quantum computer.
5.3. Resource-assisted Universality. measurement assisted, ancillary assisted,
topology-assisted,
5.3.1. Majorana qubits.
5.3.2. Metaplectic qutrits.
5.4. Density of TQFT Representations.
5.4.1. N-eigenvalue problems.
5.4.2. Distribution of Jones evaluations.
5.5. Topological Quantum Compiling.
5.5.1. Exact braiding gates.
5.5.2. Approximation braiding gates.
5.6. Simulations of TQFTs.
5.6.1. Hidden locality of TQFT.
5.6.2. Native computing problems.
MATHEMATICS OF TOPOLOGICAL QUANTUM COMPUTING
31
5.7. Open Problems.
5.7.1. Metaplectic.
6. On Modular Categories
Modular tensor categories or simply modular categories and their classification
are interesting in both mathematics and physics. Significant progress on their classification has been made during the last decade, and paved the way for a structure
theory. A fruitful analogy is to treat modular categories as quantum abelian finite
groups, and more generally fusion categories as quantum finite groups. A central
theme in tensor category theory is the extension of classical results in group theory
to modular categories such as the Cauchy and Landau theorems. In this section,
we survey the subject and discuss some open problems.
Modular tensor category first appeared as a collection of tensors in the study of
CFTs [], while modular categories were formulated to encode algebraically ReshetikhinTuraev TQFTs []. Modular categories turn out to be equivalent to modular tensor
categories []. A variety of mathematical subjects produce modular categories as
representation categories of algebraic structures such as quantum groups/Hopf algebras, vertex operator algebras, and von Neumann algebras.
A classification of modular categories contains a classification of finite groups in
the following sense. A finite group G can be reconstructed from its representation
category Rep(G) up to a choice of an order=2 central element z ∈ G. The Drinfeld
center Z(Rep(G)) of Rep(G) is always a modular category with Rep(G) as a subcategory. Therefore, the classification of modular categories is almost impossible
without some restrictions. One approach is to classify modular categories modulo
the classification of finite groups. Modular categories like Z(Rep(G)) are closely
related to finite groups. One characterization of this group-likeness is the notion of
a weakly-group theoretical modular category. The opposite of a group-like modular
category is one,
√ called anti-group, whose non-trivial simple objects have quantum
dimensions 6= m for any integer m. A general modular category is the product of
a group-like modular category and an anti-group one.
Our interest in modular categories comes from their application to TQC, where
unitary modular categories model anyon systems. Therefore, modular categories
form part of the mathematical foundation of TQC []. The book [] is an excellent
reference for the background materials, and the survey [] covered many of the
early results. All our linear categories are over the complex numbers C. Modular
categories are defined in section.
6.1. Constructions of Modular Categories. Modular categories are constructed
in a diverse variety of mathematics and physics subjects. The trivial modular category, denoted as Vec, is the category of finite dimensional vector spaces with the
standard tensor product.
6.1.1. Pointed modular categories. The simplest examples of modular categories
are those constructed from abelian finite groups with a non-degenerate quadratic
forms.
A function q : G → U (1) is a quadratic form if 1) q(−g) = q(g), and 2) the
q(g+h)
symmetric function s(g, h) = q(g)q(h)
is bi-multiplicative. The set of all quadratic
forms form a group under pointwise group multiplication.
32
ERIC C. ROWELL AND ZHENGHAN WANG
A pair (ω, b) is called a third abelian cocycle if ω : G × G × G → U (1) and
b : G × G → U (1) satisfy
b(g, h + k)b(g, h)−1 b(g, k)−1 = ω(g, h, k)ω(h, g, k)−1 ω(h, k, g),
b(g + h, k)b(g, k)−1 b(h, k)−1 = ω(g, h, k)−1 ω(g, k, h)ω(k, g, h)−1 .
3
The set of all third abelian cocyles form the group Zab
(G, U (1)) with respect to
pointwise multiplication. Two third abelian cocyles are equivalent if there exists a
group homomorphism f : G → G and a map φ : G × G → U (1) such that
φ(h, k)φ(gh , k)−1 φ(g, h + k)−1 φ(g, h)−1 = ω(g, h, k)ω(f (g), f (h), f (k)
b(g, h)b(f (g), f (h))−1 = φ(g, h)φ(g, h)−1 .
3
The equivalence classes of Zab
(G, U (1)) form the Eilenberg-McLane third abelian
3
cohomology group Hab (G, U (1)).
When the label set ΠC of a modular category C is a group under tensor product,
then C is fully encoded by the metric group (ΠC , θ), where θ is the topological
twist. Therefore, in this case a modular category is simply a finite abelian group
endowed with extra structures. To give the full structure of C from (ΠC , θ), we need
Eilenberg-Maclane abelian 3-cocyles.
Given a cocycle, we can construct a pointed braided fusion category.
3
(G, U (1)) is isomorphic to
Theorem 6.1.
(1) Hab
(2) The functor is an equivalence of braided fusion categories.
6.1.2. Drinfeld double and Drinfeld center. More complicated examples of modular
categories are the representation category of Drinfeld double of group algebras,
and its generalization to Drinfeld center or quantum double of spherical fusion
categories.
6.1.3. Conformal field theory. The notion of a modular category first appeared
in the axiomatization of a conformal field theory. The definition in is through a
collection tensor satisfying a collection of polynomial equations. A careful treatment
is in , where as a corollary we deduce the arithmetic property of modular categories.
If we regard a vertex operator algebra as a mathematical formulation of a chiral
CFT (χCFT), then the following theorem provides many examples of modular
categories.
6.1.4. TQFTs.
6.1.5. Quantum groups.
6.1.6. Vertex operator algerba.
6.1.7. Local conformal net.
6.1.8. Temperley-Lieb-Jones and skein theory.
6.1.9. Exotic Modular Categories.
6.2. Representation and Symmetry of Modular Categories.
6.2.1. Module categories.
6.2.2. Group action and Picard group.
6.2.3. Equivariantization and De-equivariantization.
6.2.4. Coring, and Gauging.
MATHEMATICS OF TOPOLOGICAL QUANTUM COMPUTING
33
6.3. Invariants of Modular Categories. To classify modular categories, we need
invariants for modular categories. The theorem that each modular category C leads
to a (2 + 1)-TQFT (V, Z) can be regarded as a pairing between modular categories
C and manifolds: (C, Y ) = V (Y ) regarded as a representation of the mapping class
group for a 2-manifold Y or (C, X) = Z(X) for a 3-manifold X, possibly with a link
L inside. Then for each fixed manifold, we obtain invariants of modular categories.
The most useful choices are the 2-torus T 2 and some links in the 3-sphere S 3 .
6.3.1. Modular data and ST -uniqueness conjecture. The invariant {da } for the unknot colored by
the label a ∈ ΠC is called the quantum dimension of the label. The
qP
2
number D =
a∈ΠC da is an important invariant of C. The invariant of the Hopf
link colored by a, b will be denoted as Sab . The link invariant of the unknot with a
right-handed kink colored by a is θa · da for some root of unity θa , which is called
the topological twist of the label a. The topological twists are encoded in a diagonal
matrix T = (δab θa ), a, b ∈ ΠC , and the S-matrix has entries Sab . The S-matrix and
T -matrix together lead to a projective representation of the modular group SL by
sending the generating matrices
0 −1
1 1
s=
,
t=
1 0
0 1
to S, T , respectively [?, ?]. Amazingly, the kernel of this projective representation
of C is always a congruence subgroup of SL [?]. The S-matrix determines the fusion
rules through the Verlinde formula, and the T -matrix is of finite order ord(T ) by
Vafa’s theorem [?]. Together, the pair (S, T ) is called the modular data of the
category C.
Definition 6.2. Let S, T ∈ GL(Cr ) and define constants dj := S0j , θj := Tjj ,
P
Pr−1
D2 := j d2j and p± = k=0 (S0,k )2 θk±1 . The pair (S, T ) is anadmissible modular
data of rank r if they satisfy the following conditions:
t
(1) dj ∈ R and S = S t with SS = D2 Id. Ti,j = δi,j θi with N := ord(T ) < ∞.
+
is a root of unity.
(2) (ST )3 = p+ S 2 , p+ p− = D2 and pp−
P
S
S
S
r−1
ka
k
(3) Ni,j
:= D12 a=0 ia Sja
∈ N for all 0 ≤ i, j, k ≤ (r − 1).
Pr−1 k 0a
0
(4) θi θj Sij = k=0 Ni∗ j dk θk where i∗ is the unique label such that Ni,i
∗ = 1.
n
P
r−1
k
(5) Define νn (k) := D12 i,j=0 Ni,j
di dj θθji . Then ν2 (k) = 0 if k 6= k ∗ and
ν2 (k) = ±1 if k = k ∗ . Moreover, νn (k) ∈ OQN for all n, k.
(6) FS ⊂ FT = QN , Gal(FS /Q) is isomorphic to an abelian subgroup of Sr
and Gal(FT /FS ) ∼
= (Z/2Z)k .
(7) The prime divisors of D2 and N coincide in QN .
6.3.2. Frobenius-Schur indicators. For a finite group G, the nth Frobenius-Schur
indicator
of a representation V over C with character χV is given by νn (V ) :=
P
1
n
g∈G χV (g ). The Frobenius-Schur Theorem asserts that the second FS-indicator
|G|
ν2 (V ) of an irreducible representation V must be 1, -1 or 0, which can be determined
by the existence of non-degenerate G-invariant symmetric (or skew-symmetric) bilinear form on V . Moreover, the indicator value 1, -1 or 0 indicates respectively
whether V is real, pseudo-real or complex.
Second FS-indicators for each primary field of a rational conformal field theory
were introduced by Bantay [?] as an expression in terms of modular data. This
34
ERIC C. ROWELL AND ZHENGHAN WANG
expression provides a formula for the 2nd FS-indicator of each simple object in a
modular category. Higher FS-indicators for pivotal categories were developed by
Ng and Schauenburg [?], and in particular, spherical fusion categories C over C [?].
The invariance of higher FS-indicators for Rep(H) [?] follows as a consequence of
their categorical development. In spherical fusion category C, there exists a minimal
positive integer N , called the FS-exponent of C, which satisfies νN (Xk ) = dk for
all k ∈ ΠC [?]. The FS-exponent of a spherical fusion category behaves, in many
ways, like the exponent of a finite group. In fact, the FS-exponent of Rep(G) for
any finite group G is equal to the exponent of G. These generalized FS-indicators
for modular categories uncover some new arithmetic properties which include the
congruence kernels of their projective representation of SL [loc. cit.] and the Galois
symmetry [?] conjectured by Coste and Gannon.
6.3.3. Galois symmetry. The following theorem of Galois symmetry of modular
categories is proved in [?, Thm. II (ii),(iii)]:
Theorem 6.3. Let C be a modular category of rank r, with T -matrix of order N .
Suppose (s, t) is a normalized modular pair of C. Set n = ord(t) and t = (δij ti ).
Then N | n | 12N and s, t ∈ GLr (Qn ). Moreover, for σ ∈ Gal(Qn /Q),
σ 2 (ti ) = tσ̂(i) .
6.3.4. Grading. Two groupd: pointed and Galois group
6.4. Congruence Subgroup Theorem.
Theorem 6.4. Let s = S/x and t = T /y be such that ρ : (s, t) → (s, t) defines a
representation of SL , and let n = ord(t). Then ker ρ is a congruence subgroup of
level n, N | n | 12N , and imρ ⊂ GL(r, Qn ).
6.5. Cauchy Theorem.
Theorem 6.5. If C is a spherical fusion category over C, the set of prime ideals
dividing the principal ideal generated by D2 in the Dedekind domain Z[e2πi/N ] is
identical to that of N = FSexp(C), where ζN is a primitive N th root of unity.
6.6. Rank-finiteness Theorem.
Theorem 6.6. There are only finitely many modular categories of fixed rank r, up
to equivalence.
6.7. Structure of Modular Categories.
6.7.1. Structure Conjecture for Modular Categories.
6.7.2. Group-like and anti-group decomposition. Group-likeness is reflected in the
following conjectures: every weakly-group theoretical modular categories can be
obtained by the gauging of a finite group action on a pointed modular categories
We conjecture then all its quantum dimensions are units of the cyclotomic field
Q[χ].
6.7.3. Prime decomposition.
6.8. Classification of Low Rank Modular Categories.
6.9. Morita Equivalence and Witt group.
MATHEMATICS OF TOPOLOGICAL QUANTUM COMPUTING
35
6.10. Some Open Problems.
6.10.1. 16 fold way.
6.10.2. WI and Property F. Question: if an integral, non-pointed, modular category contains a non-trivial symmetric fusion category.
Is weakly integral is the same as weakly group theoretical?
Can every weakly-integral be obtained from gauging a pointed modular category?
Known for group theoretical.
WI can be obtained from gauging abelian ones.
Anti-WI quantum dimensions are units and they are polynomial in rank. Al MC
are gauged of those with WI=pointed?
7. Extensions
7.1. Fermions. Disjoint union axioms does not hold
7.2. Defects.
7.3. Boundaries.
7.4. SPTs and SETs.
7.5. From MC to CFT. There is a boundary map ∂ : T QF T s → CF T s and an
inverse γ : CF T s → T QF T s.
7.5.1. Genus of CFTs.
7.6. 3+1.
7.6.1. TQFTs from G-crossed categories and spherical 2-fusion categories.
7.6.2. 3D lattice models.
7.6.3. Representation of motion groups. The group generated by σi and si for 1 ≤
i ≤ n − 1 is called the Loop Braid Group, LB n , defined abstractly as the group
satisfying:
Braid relations:
(R1) σi σi+1 σi = σi+1 σi σi+1
(R2) σi σj = σj σi if |i − j| > 1
Symmetric Group relations:
(S1) si si+1 si = si+1 si si+1
(S2) si sj = sj si if |i − j| > 1
(S3) s2i = 1
Mixed relations:
(M1) σi σi+1 si = si+1 σi σi+1
(M2) si si+1 σi = σi+1 si si+1
(M3) σi sj = sj σi if |i − j| > 1
This is a relatively new area of development, for which many questions and
research directions remain unexplored. A first mathematical step is to study the
unitary representations of the loop braid group, which is already underway [?, ?].
It might also be reasonable to consider other configurations, such loops bound
concentrically to an auxillary loop or knotted loops.
36
ERIC C. ROWELL AND ZHENGHAN WANG
Conjecture 7.1. The partition function of any unitary 3 + 1-TQFT is a homotopy
invariant.
8. Quantum Mathematics
The most rigourous creation of human mind is the mathematical world. Equally
impressive is our creation of the computing world. At this writing, Machine is
beating the best GO player in the world. Man, Machine, and Nature meet at TQC.
What will be the implication of TQC, if any, for the future of mathematics?
8.1. Quantum Logic. Logic seems to be empirical, then would quantum mechanics change logic? There are interesting research in quantum logics, and quantum
information provides another reason to return to this issue.
8.2. Quantum Artificial Intelligence. Machine learning has come a long way
since Turing’s paper. AlphaGo is an example for the amazing progress. The question “Can Machine think” is as fresh as in the 1930s?
8.3. Complexity Classes as Mathematical Axioms. Another direction from
TQC is Freedman’s suggestion of complexity classes as mathematical axioms. Some
interesting implications in topology from complexity theories can be found in.
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Department of Mathematics, Texas A&M University, College Station, TX 77843,
U.S.A.
E-mail address: rowell@math.tamu.edu
Microsoft Station Q and Dept of Mathematics, University of California, Santa Barbara, CA 93106-6105, U.S.A.
E-mail address: zhenghwa@microsoft.com, zhenghwa@math.ucsb.edu