Simulation%20for%20the%20feature%20of%20non-Abelian%20anyons%20in%20quantum%20double%20model%20using%20quantum%20state%20preparation

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Simulation for the feature of non-Abelian anyons
in quantum double model using quantum state
preparation
Zheng-Wei Zhou(???) Key Lab of Quantum
Information , CAS, USTC In collaboration with
Univ. of Sci. Tech. of China X.-W. Luo (???)
Y.-J. Han (???) X.-X. Zhou (???) G.-C. Guo
(???)
Jinhua Aug 14, 2012
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Outline

• I. Some Backgrounds on Quantum Simulation
• II. Introduction to topological quantum computing
  based on
• Kitaevs group algebra (quantum double) model
• III. Simulation for the feature of non-Abelian
  anyons in quantum double model using quantum
  state preparation
• Summary

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I. Backgrounds on Quantum Simulation
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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)
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Why quantum simulation is important?
Answer 2 simulate and build new virtual quantum
materials.
Kitaevs models
topological quantum computing
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Physical Realizations for quantum simulation
Iulia Buluta and Franco Nori, Science 326,108
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II. Introduction to topological quantum computing
based on Kitaevs group algebra (quantum double)
model
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A Toric codes and the corresponding Hamiltonians
plaque operators
vertex operators
qubits on links
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Hamiltonian and ground states
plaque operators
vertex operators
ground state has all
every energy level is 4-fold degenerate!!
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Excitations
plaquet operators
vertex operators
anti-commutes with two plaquet operators
?
?
excitation is above ground state
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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)
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Topological qubit and operation
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Topological protection
Encode two qubits into the ground state
gap
Perturbation theory
But for
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Abelian anyons
Phase
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B Introduction to quantum double model
Hilbert space and linear operators
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Hamiltonian
A(s),B(p)0
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Ground state and excited states
For all s and p,
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.
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About excited states
Description Quantum Double D(G), which is a
quasitriangular Hopf algebra. Linear bases
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.
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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(Nµ) denotes a
unitary irrep of the centralizer of an arbitrary
element in the conjugacy µ and it labels the
electric charge.
The conjugacy class
The centralizer of the element µ
Once the types of the quasiparticles are
determined they never change. Besides the type,
every quasiparticle has a local degree of
freedom, the subtype.
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For an instance
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Ribbon operator
The ribbon operators 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.
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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.
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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 the pure
magnetic charge excitation
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Braiding Non-Abelian anyons
magnetic charge--- magnetic charge
magnetic charge--- electric charge
electric charge--- electric charge
Boson---Boson
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Fusion of anyons
The topologically protected space will become
small and the anyon with the new type will be
generated.
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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.
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Stabilization of topological protected space















Nonlocal noise braiding Low probability



Trivial local noise
Infinity
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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)
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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).
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A. Ground state preparation
?
egt
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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
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Realization of short ribbon operators
Key point to realize the projection operation

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Moving the anyonic excitation (I)
Mapping
1. perform the projection operation egtlte 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.
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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 .
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C) Fusion and topological state measurement
Braiding and fusion in terms of ribbon
transformations
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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.
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?
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Measure the topological states of the anyons by
using interference experiment.
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D) Demonstration of non-abelian statistics
Ground state
A pure electric charge anyon
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Demonstration for the fusion measurement
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E) Physical Realization
All of the 2-qudit gate has this form
Single qudit gate
2-qudit phase gate
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Summary

• We give a brief introduction to Kitaevs 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.

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Thanks for your attention
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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).