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 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 )

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 qubits into the ground state gap Perturbation theory: But for Topological protection

Phase: Abelian anyons

Hilbert space and linear operators B: Introduction to quantum double model

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

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.

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: 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 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.

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

Braiding Non-Abelian anyons 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 S 3, universal quantum computation is also possible if one includes some operations using electric charges.

…… Infinity × × × × × × × ×× × × × ×× × × × Trivial local noise Nonlocal noise braiding Low 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 ， （ 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, (2007); Phys. Rev. Lett. 102, (2009). Phys. Rev. Lett. 101, (2008); New J. Phys. 11, (2009); New J. Phys. 12, (2010).

A. Ground state preparation |e> ?

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: 2-qudit phase gate Single qudit gate 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, (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 ， （ 2011)