Simulation for the feature of non-Abelian anyons in quantum double model using

quantum state preparation

Jinhua Aug 14, 2012

Zheng-Wei Zhou(周正威） Key Lab of Quantum Information , CAS, USTC

In collaboration with:

Univ. of Sci. & Tech. of ChinaX.-W. Luo (罗希望 ) Y.-J. Han (韩永建 ) X.-X. Zhou (周幸祥 ) G.-C. Guo (郭光灿 )

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

qubits on links

plaque operators:

vertex operators:

Hamiltonian and ground states:

ground state has all

plaque operators:

vertex operators:

every energy level is 4-fold degenerate!!

plaquet operators:

vertex operators:

anti-commutes with two plaquet operators

excitation is above ground state

Excitations

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

Encode two qubitsinto the ground state

gap

Perturbation theory:

But for

Topological protection

Phase:

Abelian anyons

Hilbert space and linear operators

gLgL

gTgT

iG g

:H g g G

1

1,, , ,g g g gg z g

L z gz L z zg T z z T z z

，z

B: Introduction to quantum double model

Hamiltonian

0 1 1s p

H A s B p

1,g

g

A s A s pG

,eB p B s p

, ,g g gj star s

A s p A s L j s

1 1

, ,m

k

k

h h mh h h m

B s p T j p

…

[A(s),B(p)]=0

Ground state and excited states

A s GS GS B p GS GSFor 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.

, ( ) ( ) ( )g h h gD x B x A x

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.

The conjugacy class:

[ ] { : }N g g g

1[ ] { , }g g g G

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 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.

For an instance

Ribbon operator ,h gF r

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.

,h gF r

Topologically protected space

For the structure of Hilbert space with n quasiparticle excitations

It dose not have a tensor product structure.

To resolve this problem…

The base site (fixed)

connect the base site with other sites by nonintersecting ribbons

On quasiparticles：

Type and subtype

Topological state

Topologically protected space

the pure electric charge excitation

the pure magnetic charge excitation

1/2 ,

,, e z

z

R R z F r GS

1

1/2 ,

:

, u z

z z uz v

u v C F r GS

Braiding Non-Abelian anyons

112 1 2 1 2 1 1, ,R v v

12 1 22 1( )R RR R v

magnetic charge--- magnetic charge

magnetic charge--- electric charge

electric charge--- electric charge

Boson---Boson

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 electriccharges.

……

Infinity

× × ×× × ×

× × ×

× ×

× × × × ×

×

× × × × ×

×

Trivial local noise

Nonlocal noise braidingLow probability

Stabilization of topological protected space

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)

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.

In spite of the conceptual significance of anyons and their appeal for quantum computation applications, it is very difficult to study anyons experimentally.

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>

?

, ,g g gj star s

A s p A s L 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

All of the 2-qudit gate has this form: ,i

i

i i hsh

h h L j s

exp i i i iA BU i h h h h 2-qudit phase gate

Single qudit gate 6U

E) Physical Realization

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：

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).

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)