Transcript A. Ground state preparation

Simulation for the feature of non-Abelian
anyons in quantum double model using
quantum state preparation
Zheng-Wei Zhou(周正威）
Key Lab of Quantum Information , CAS, USTC
In collaboration with:
Univ. of Sci. & Tech. of China
X.-W. Luo (罗希望)
Y.-J. Han (韩永建)
X.-X. Zhou (周幸祥)
G.-C. Guo (郭光灿)
Jinhua Aug 14, 2012
Outline

I. Some Backgrounds on Quantum Simulation

II. Introduction to topological quantum computing based on
Kitaev’s group algebra (quantum double) model

III. Simulation for the feature of non-Abelian anyons in
quantum double model using quantum state preparation

Summary
I. Backgrounds on Quantum Simulation
“Nature isn't classical, and
if you want to make a
simulation of Nature, you'd
better make it quantum
mechanical, and it's a
wonderful problem,
because it doesn't look so
easy.” (Richard Feynman)
Why quantum simulation is important?
Answer 2: simulate and build new virtual quantum materials.
Kitaev’s models
topological quantum
computing
Physical Realizations for quantum simulation
Iulia Buluta and Franco Nori, Science 326,108
II. Introduction to topological quantum
computing based on Kitaev’s group
algebra (quantum double) model
A: Toric codes and the corresponding Hamiltonians
plaque operators:
vertex operators:
qubits on links
Hamiltonian and ground states:
plaque operators:
vertex operators:
ground state has all
every energy level is 4-fold degenerate!!
Excitations
plaquet operators:
vertex operators:
anti-commutes with two plaquet operators
 
excitation is
above ground state
excitations particles come in pairs (particle/antiparticle)
at end of “error” chains
two types of particles, X-type (live on vertices of dual lattice)
Z-type (live on vertices of the lattice)
Topological qubit and operation
Topological protection
Encode two qubits
into the ground state
gap
Perturbation theory:
But for
Abelian anyons
Phase:
B: Introduction to quantum double model
Hilbert space and linear operators
G   gi 
H   g : g  G
g

L
g

g

L T
g

T
Lg z  gz , Lg z  zg 1 , Tg z   g , z z , Tg z   g 1，z z
Hamiltonian
H 0   1  A  s     1  B  p  
s
p
1
A s 
G
 A  s, p 
g
g
B  p   Be  s, p 

Ag  s, p   Ag  s  
Bh  s, p  
jstar  s 
Lg  j, s 
k
 T  j
h1 …hk  h
m 1
hm
m
, p
[A(s),B(p)]=0
Ground state and excited states
For all s and p,
A  s  GS  GS
B  p  GS  GS
The excited states involve some violations of these conditions.
Excitations are particle-like
living on vertices or faces, or
both, where the ground state
conditions are violated.
A combination of a vertex and
an adjacent face will be called
a site.
About excited states
Description: Quantum Double D(G), which is a quasitriangular
Hopf algebra.
Linear bases: Dg ,h ( x)  Bh ( x) Ag ( x)
Quasiparticle excitations in this system can be created by
ribbon operators：
For a system with n quasi-particles, one can use
to
denote the quasiparticles’ Hilbert space. By investigating
how local operators
act on this
Hilbert space, one can define types and subtypes of these
quasiparticles according to their internal states.
The types of the quasiparticles
the irreducible representations of D(G)
These representations are labeled
where [μ] denotes a conjugacy class of G which labels the
magnetic charge. R(N[μ]) denotes a unitary irrep of the
centralizer of an arbitrary element in the conjugacy [μ] and
it labels the electric charge.
The conjugacy class:
[  ]  {g  g 1 , g  G}
The centralizer of the element μ : N[  ]  {g : g    g}
Once the types of the quasiparticles are determined they
never change. Besides the type, every quasiparticle has a
local degree of freedom, the subtype.
For an instance
Ribbon operator
F
h, g
r 
The ribbon operators F h, g  r  commute with every projector
A(s) and B(p), except when (s,p) is on either end of the ribbon.
Therefore, the ribbon operator creates excitations on both
ends of the ribbon.
Topologically protected space
For the structure of Hilbert space with n quasiparticle excitations
To resolve this problem…
It dose not have
a tensor product
structure.
Topologically protected space
The base site (fixed)
connect the base site with
other sites by nonintersecting
ribbons
On quasiparticles：
Type and subtype
Topological state
the pure electric charge excitation
 ,  R
1/2
e, z
R
z
F


  ,  r  GS
z
the pure magnetic charge excitation
u, v  C
1/2

z:z 1uz  v
F u , z  r  GS
Braiding Non-Abelian anyons
magnetic charge--- magnetic charge
R12  1 , v2   1 2 11 , v1
magnetic charge--- electric charge
R12 
1
R
2
 R ( )  R
1
v
electric charge--- electric charge
Boson---Boson
2
Fusion of anyons
The topologically protected space will become small and the
anyon with the new type will be generated.
On universal quantum computation
Mochon proved two important facts:
firstly, that by working with magnetic charge anyons alone
from non-solvable, non-nilpotent groups, universal quantum
computation is possible.
secondly, that for some groups that are solvable but not
nilpotent, in particular S3, universal quantum computation is
also possible if one includes some operations using electric
charges.
Stabilization of topological protected space
× × ×
×
× ×
×
×
……
×
××
××
×
× ××
× ××
Nonlocal noise
 braiding
Low probability
× ××
Trivial local noise
Infinity
III. Simulation for the feature of non-Abelian
anyons in quantum double model using
quantum state preparation
Simulation of non-Abelian anyons using ribbon operators connected to a common
base site，Xi-Wang Luo, Yong-Jian Han, Guang-Can Guo, Xingxiang Zhou，and
Zheng-Wei Zhou，Phys. Rev. A 84，052314 （2011)
In spite of the conceptual significance of anyons and their
appeal for quantum computation applications, it is very difficult
to study anyons experimentally.
Key point:
to generate dynamically the ground state and the
excitations of Kitaev model Hamiltonian instead of
direct physical realization for many body Hamiltonian
and corresponding ground state cooling.
Here, we will prepare and manipulate the quantum
states in the topologically protected space of Kitaev
model to simulate the feature of non-Abelian anyons.
References:
Phys. Rev. Lett. 98, 150404 (2007); Phys. Rev. Lett. 102, 030502 (2009).
Phys. Rev. Lett. 101, 260501 (2008); New J. Phys. 11, 053009 (2009);
New J. Phys. 12, 053011 (2010).
A. Ground state preparation
A  s  GS  GS
?
B  p  GS  GS
|e>
Ag  s, p   Ag  s  

jstar  s 
Lg  j, s 
B) Anyon creation and braiding
Ribbon
operator：
By applying the superposition ribbon operator
arbitrary topological states of a given type can be created.
the pure magnetic
charge excitation :
Realization of short ribbon operators
Key point: to realize the projection
operation:
+
Moving the anyonic excitation (I)
Mapping:
1. perform the projection operation |e><e| on the qudit on edge [s_1,s_2]
2. apply the symmetrized gauge transformation A(s_1) at vertex s_1
to erase redundant excitation at site x_1.
Moving the anyonic excitation (II)
1. map the flux at site x_2 to the
ancillary qudit at p_1 by the controlled
operation:
2. apply the controlled unitary operation:
to move the flux from site x_2 to site x_3.
3. disentangle the ancillary qudit p_1 from the system by first swapping
ancilla p_1 and p_2 and then applying
.
C) Fusion and topological state measurement
Braiding and fusion in terms of ribbon transformations
Realize the projection ribbon operator on the vacuum quantum
number state
(reason: For TQC, the only measurement we need is to detect whether
there is a quasi-particle left or whether two anyons have vacuum
quantum numbers when they fuse.)
In principle, projection operators corresponding to other fusion channels
can be realized in a similar way.
?
Measure the topological states of the anyons by using interference
experiment.
D) Demonstration of non-abelian statistics
Ground state
A pure electric charge anyon
Demonstration for the fusion measurement
E) Physical Realization
All of the 2-qudit gate has this form:
h
i s
hi  Lhi  j, s 
hi
Single qudit gate
U  6
2-qudit phase gate U  exp  i hi
A
hi  hi
B
hi

Summary


We give a brief introduction to Kitaev’s quantum double
model.
We exhibit that the ground state of quantum double model
can be prepared in an artificial many-body physical
system. we show that the feature of non-Abelian anyons in
quantum double model can be dynamically simulated in a
physical system by evolving the ground state of the model.
We also give the smallest scale of a system that is
sufficient for proof-of-principle demonstration of our
scheme.
References：
Simulation of non-Abelian anyons using ribbon operators connected to a
common base site，Xi-Wang Luo, Yong-Jian Han, Guang-Can Guo, Xingxiang
Zhou，and Zheng-Wei Zhou，Phys. Rev. A 84，052314 （2011)
Integrated photonic qubit quantum computing on a superconducting chip,
Lianghu Du, Yong Hu, Zheng-Wei Zhou, Guang-Can Guo, and Xingxiang Zhou, New. J.
Phys. 12, 063015 (2010).