Jiannis K. Pachos. Introduction. Berlin, September 2013

Transcription

1 Jiannis K. Pachos Introduction Berlin, September 203

2 Introduction Quantum Computation is the quest for:» neat quantum evolutions» new quantum algorithms Why? 2D Topological Quantum Systems: How? ) Continuum gases: FQHE Chern-Simons 2) Spin lattices: Quantum Double Models Topological Degeneracy Anyons Charge Fractionalization Effective gauge theories H! [Wilczek, Freedman, Wen, Bais, Wang, Kitaev, ]

3 Two dimensional systems Anyons Dynamically trivial (H=0). Only statistics. 3D Bosons Fermions 2D Anyons Anyons: vortices with flux & charge (fractional). Aharonov-Bohm effect Berry Phase.

4 Topological Degeneracy System with degenerate ground states where: The degeneracy is protected by topology (genus). Degenerate states are not locally distinguishable. encode information in degenerate subspace Index Theorem: n fermionic zero energy modes => degenerate ground states [Atiyah, Singer (963)]

5 Topological Quantum Systems Anyonic Statistics Effective Fractional Charge Gauge Theory Topological Degeneracy

6 Assume we can: Anyon Properties Create identifiable anyons vacuum pair creation vacuum Braid anyons Statistical evolution: braid representation B time Fuse anyons e.g. Fusion Hilbert space:

7 Assume we can: Anyon Properties Create identifiable anyons vacuum pair creation Braid anyons Statistical evolution: braid representation B time Fuse anyons e.g. Fusion Hilbert space:

8 The braid group Bn The braid group Bn has elements b, b2,, bn- that satisfy: Pictorially:

9 Braiding and Fusion properties The action of braiding of two anyons depends on their fusion outcome: E.g. for c: fermion then R c ab=i is possible =R c ab c c Changing the order of fusion is non-trivial: a b c a b c a b a b i d d j

10 Construction of Anyonic Models. Take a certain number of different anyons, a, b,... the vacuum () and one or more non-trivial particles 2. Define fusion rules between them xa=a, axb=c+d+..., axa=+... The vacuum acts trivially. Each particle has an anti-particle (might be itself or not). - Abelian anyons axb=c - Non-Abelian anyons axb=c+d+...

11 Construction of Anyonic Models 3. The F and B matrices are determined from the Pentagon and Hexagon identities a b 5 a b e c e d c d 5 5 5

12 Construction of Anyonic Models 3. The F and B matrices are determined from the Pentagon and Hexagon identities b b 2 3 a c a c 4 4

13 Ising Anyons Consider the particles:, and ψ Fusion rules: x=+ψ, ψxψ=, xψ= ψ All these states span the fusion Hilbert space

14 Ising Anyons Consider the particles:, and ψ Fusion rules: x=+ψ, ψxψ=, xψ= ψ 2 n/2 increase in dim of Hilbert space ψ ψ d2=2 d4=4 d6=8 d8=6

15 Ising Anyons Consider the particles:, and ψ Fusion rules: x=+ψ, ψxψ=, xψ= From 5-gon and 6-gon identities we have: Rotation of basis states = + ψ = - ψ ψ

16 Ising Anyons Braiding = ψ 4 4 = ψ 4 4 = ψ Clifford group: non-universal!

17 Ising Anyons Qubit initialization: ψ State 0> State > Measurement: Outcome of pairwise fusion, or ψ Gates: Clifford group. Non-universal! Can be employed as a quantum memory. One needs a phase gate: employ interactions between anyons.

18 Fibonacci Anyons Consider anyons with labels or with the fusion properties: x=, x =, x =+ d = d 2 =2 d 3 =3 d 4 =5 d 5 =8 Fibonacci sequence! Dimension of Hilbert space Golden mean

19 Fibonacci Anyons and QC Qubit encoding: Evolving a qubit: a b =R c ab c c State 0> State > a b =Σ j F i j Unitaries B and F are dense in SU(2). [Freedman, Larsen, Wang, CMP 228, 77 (2002)] i j

20 Fibonacci Anyons and QC Qubit encoding: Evolving a qubit: a b a b i c =R c ab j c i =Σ j F j i j Unitaries B and F are dense in SU(2). Extends to SU(dn) when n anyons are employed.

21 Fibonacci Anyons and QC Qubit encoding: i j CNOT Unitaries B and F are dense in SU(2). Extends to SU(dn) when n anyons are employed.

22 Conclusions Topological Quantum Computation promises to overcome the problem of decoherence and errors in the most direct way. There is lots of work to be done to make anyons work for us. Is it worth it? Aesthetics says YES!