Download Topological Phases of Matter classification and application

Topological Quantum
Computation I.
Zhenghan Wang
Microsoft Station Q/UCSB
Guelph, June 17, 2014
Key “Post-Shor” Idea
Peter Shor
Shor’s Factoring Algorithm
To use topology to protect quantum information
Michael Freedman
Alexei Kitaev
2
Topological Quantum Computation
A revolutionary idea:
If a physical system were to have quantum topological
(necessarily nonlocal) degrees of freedom, which were
insensitive to local probes, then information contained
in them would be automatically protected against
errors caused by local interactions with the
environment.
This would be fault tolerance guaranteed by physics at
the hardware level, with no further need for quantum
error correction, i.e. topological protection.
Alexei Kitaev
3
Why does topology help?
• Topology is the part of geometry which
survives deformation/perturbation.
• Topological properties of quantum systems
are robust to perturbation/deformation.
4
• “Qubits” are just the modern name for a twolevel system and its composites---easily decohere
• Topology provides a solution to the fragility of
qubits---topological phases of matter
5
Topological Qubits
• Inherently fault-tolerant (logical) qubits:
information encoded in the ground states of
many anyons, and processed by braiding or
measuring anyons.
• Construction has begun: fractional quantum
Hall liquids, engineered quantum systems,...
Anyonic Quantum Systems
• Anyons: quasi-particles or topological
excitations in topological phases of matter--states of matter with topological order or
whose low energy physics are modeled by
TQFTs. Their statistics are more general than
bosons/fermions, can be even non-abelian.
• Models: Simple objects in unitary modular
categories.
Anyons in TPM
• Materializations of long ranged topological
order: for topological phases of matter, the
ground state manifold is an analogue of the
configuration space and the “first” excited
states are the phase space. (excitations form
an analogue of the tangent bundle)
• In 2D, to first approximation, long ranged
entanglement pattern is encoded by a unitary
modular category: D,S,T,…
TPM
TQC
TQFT/UMC
Topological phases of matter are TQFTs/UMCs in
Nature and hardware for the hypothetical
topological quantum computers.
Are we going to have non-abelian anyons?
Ground State Degeneracy I---TQFT
For a surface Y with anyons a1, a2, …, an localized at 1,…,n,
the (relative) ground states V(Y; a1, a2, …, an, l) of the system
“outside” 1,…,n form a Hilbert space depending on
boundary conditions l. This collection of ground state
manifolds V(Y; a1, a2, …, an, l) forms a TQFT.
Stable boundary conditions correspond to anyons.
label l
●●●●●●●●
Ground State Degeneracy II--Quantum Dimension
• A defining property of non-abelian anyons is
the ground state degeneracy: when n anyons
of type x are fixed in the 2-sphere, the ground
state degeneracy goes like 𝑑𝑥 𝑛 for some
positive real number 𝑑𝑥 >1---quantum
dimension. For abelian anyons, 𝑑𝑥 =1.
• Information will be encoded in this ground
state degeneracy.
Anyons in 2D
In R2, an exchange is of infinite order
Not
equal
Braids form groups Bn, then braiding anyons
leads to : Bn  U(V(a1, a2, …, an, l)) and the
ground state manifold is the rep space.
Anyon braidings form our “gate set”
Data for Anyon Systems
• Label set: a finite set L={a,b,c,…} of anyon
types with an involution and a trivial type
𝑐
• Fusion rules: {𝑁𝑎𝑏 , a,b,c∊L}. The fusion
rules determine when two anyons of types
a,b are fused, whether or not anyons of type
c appear, ie if 𝑁𝑎𝑏 𝑐 is ≥ 1 or =0. If 𝝨𝑐 𝑁𝑎𝑏 𝑐 >1,
𝑐
a,b have multi-fusion channels. If 𝑁𝑎𝑏 >1,
fusion of a, b has multiplicities.
• Others
Categorification of Fusion Rules
• When can a given fusion ring (a label set with
fusion rules) be categorified to an anyon
system, i.e. a unitary modular category? If so,
how many are there?
• Widely open. But we do know if we fix the
rank (the number of anyon types), there are
only finitely many anyon systems (Bruillard,
Ng, Rowell, W.)
Pointed Fusion Rules
• Let G be a finite abelian group. Then G is a
label set: each group element is an anyon type
𝑐
and 𝑁𝑎𝑏 =𝛿𝑎𝑏,𝑐 for a,b,c∊G. There are no
multi-fusion channels, so no multiplicites.
• An anyon (type) y is abelian if for any anyon
(type) x, there are no multi-fusion channels
for y and x. Otherwise, it is non-abelian.
Non-abelian Fusion Rules---Ising
Anyon types are L={1,,}
Quantum dimension {1, 𝟐,1}
TEE=-logD, D=2
Fusion rules:
2 1+,
2 1,
 

1 or 

Unitary Modular Category
Label set
Fusions
a
b
c
a
b
F-matrices (6-j)
d
a
d
b
R-matrix
a
b
=𝑅𝑐 𝑎𝑏
c
c
Compatibility: pentagons and hexagons
c
Count Ground State Degeneracy
• Given anyons 𝑎𝑖 , i=1,…,n with total charge l in
the disk, draw a trivalent tree:
𝑎1 𝑎2 𝑎3
𝑎𝑛
l
• Count how many admissible labeled trees.
They form a basis of the ground state mfd.
Fibonacci Anyon
Label set:
{1,}
Fusion rules: 2 1+ 
Quantum dimension: {1,𝜙}
TEE=-log 𝟐 + 𝜙
1
n anyons
Ground states degeneracy
=0,1,1,2,3,5,..
Fibonacci numbers Fn
How to Compute Braid Reps
• A basis of the ground state manifolds given by
admissible labelings of fusion trees---trivalent
graphs with a fixed total charge.
• Stack braids on top of a basis vector, and use
F-moves and R-moves to transform back to a
linear combination of the given tree basis.
Unitary Fusion Category
A quadruple (L,N,F,ԑ)
L---a finite label set,
e.g. a finite group G
,
N={𝑁𝑎 , a∈L},a  b=⊕𝑁𝑎,𝑏 𝑐 c
group multi. 𝑁𝑔,ℎ 𝑘 =𝛿𝑔ℎ,𝑘
F={𝐹 𝑎𝑏𝑐 𝑑 , a,b,c,d∈L},
ԑ𝑎 =1 or -1 (spherical structure),
𝐹 𝑎𝑏𝑐 𝑑 =𝛿𝑎𝑏𝑐,𝑑
ԑ𝑔 =1
a b c
m
a,b,c,d admissible
d
If (a,b,m) and (m,c,d)
admissible.
F satisfies the pentagons
Ising and Fibonacci Theory
L={1,σ,ψ},
L={1,𝜏}
𝑁1 =Id
0 1 0
0 0 1
𝑁𝜎 = 1 0 1 , 𝑁ψ = 0 1 0 ,
0 1 0
1 0 0
𝐹 𝜎𝜎𝜎 𝜎 = 1
2
1 1
,
1 −1
𝐹 𝜎ψ𝜎 ψ =𝐹 ψ𝜎ψ 𝜎 =-1,
𝑁𝜏 =
𝐹 𝜏𝜏𝜏 𝜏 =
0 1
, 𝜑 =golden ratio
1 1
𝜑 −1
−1
𝜑2
others F either 0 or 1
all ԑ𝑎 =1
−1
𝜑2
−𝜑 −1
R-Symbols for Ribbon Fusion Category
Given a unitary fusion category as (L,N,F,ԑ),
need extra R={𝑹𝒄 𝒂𝒃 ,a,b,c∈L} for a RFC.
a
a
b
b
𝑅𝑐 𝑎𝑏
1) L=a finite abelian group, N=group multiplication,
All R=0 or 1 for G
2) Ising: 𝑹𝟏 𝝈𝝈 =𝒆
−𝟐𝝅𝒊
𝟏𝟔
, 𝑹𝟏 𝝍𝝍=-1
𝟔𝝅𝒊
𝑹𝝈 𝝍𝝈 = 𝑹𝝍 𝝈𝝈 =-i, 𝑹𝝈 𝝈𝝍 =𝒆 𝟏𝟔 ;
3) Fib:
𝑹𝟏 𝝉𝝉 =𝒆
−𝟒𝝅𝒊
𝟓
, 𝑹𝝉 𝝉𝝉 =𝒆
𝟑𝝅𝒊
𝟓
c
c
Invariants for Anyon Tangles/Topological
Amplitudes of Quantum Processes
time
V(D2,a4,a1)
T
V(D2, a1,a2,a3,a4)
Curves are anyon trajectories. Each curve is labeled by
an anyon. Special cases include colored links and braids.
Fibonacci Quantum Computational Model
For n qubits, consider the 4n Fibonacci anyons,
braiding gates/circuits : B4n
U(𝑽𝟒𝒏 ),
●● ●● ●● ●● ●● ●●
𝑽𝟒𝒏 -ground state manifold of 4n Fibonacci anyons
with trivial total charge.
Its dimension is 𝑭𝟒𝒏−𝟐 --- the (4n-2)th Fib number.
It is known that the images of (B4n) are dense in SU(𝑽𝟒𝒏 ),
hence we have a universal quantum computing model.
(One issue is that 𝑭𝟒𝒏−𝟐 is rarely a power of some integer.)
𝑆𝑈(2)𝑘 Chern-Simons Anyon Systems
Fix k ≥ 1, called the level
The particle types are L={0,1,…,k} and each is its own dual, a*=a
Fusion rules:
a b=⊕c if
1)
2)
3)
c
a+b+c even
a+b≥c, b+c≥a, c+a≥b
a+b+c ≤ 2k
a
b
Ising Quantum Computational Model
4 Ising ’s in a disk for a qubit. 8 ’s for 2 qubits.
For 1-qubit gates,
: B4
U(2)
For 2-qubits gates, : B8
U(4)
●●●● ●●●●
● ● ● ●
1
Ising Braid Gates
1 0
e- i/8
e- i/8
0
i
(1-i)/2 (1+i)/2
(1+i)/2 (1-i)/2
 1  2  2  1
0 1
e- i/4
4 ’s
NOT Gate
1 0
Universality of Anyonic Computing Models
To be universal, it suffices to approximate every gate (up to a
phase) in a universal gate set.
For 𝑺𝑼(𝟐)𝒌 Chern-Simons anyon systems, we need dense
images for:
k(B4)  PSU and k(B8)  PSU
True if k  1,2,4 by a density theorem.
Then by Kitaev-Solvay Theorem that each unitary matrix can
be approximated efficiently.
Universal Braiding Gates
• Ising anyon  does not lead to universal braiding gates, but
Fib anyon  does
• Quantum dimension of Ising anyon  has quantum
dimension= 2, while Fib anyon  has quantum dimension
=( 5+1)/2---golden ratio
• Given an anyon type x, when does its braiding lead to
universal braiding gate sets ?
Conjecture: Not universal iff 𝑑𝑥2 ∊ Z, i.e., weakly integral.