Anyonic Quantum Computing

Transcription

1 Anyonic Quantum Computing 1. TQFTs as effective theories of anyons 2. Anyonic models of quantum computing (anyon=particle=quasi-particle)

2 Topological quantum computation: 1984 Jones discovered his knot polynomial J(L,q) using his reps of the braid groups (1990 Fields Medal) 1988 Witten explained Jones evaluations using topological quantum field theories (TQFTs) (1990 Fields Medal) 1991 Welsh, Jaeger, Vertigan proved that the exact computation of {J(L,q), q=e 2π i/ r } of all links is # P hard for all r 1,2,3,4,6. When r=1,2,3,4,6, Jones evaluations are computable in poly-time Condensed matter physicists used CS TQFTs as effective theories for FQHE Moore, Read proposed non-abelian generalization of the FQHE based on CFT

3 ~ 1997 Freedman suggested TQFT computers. Kitaev proposed anyonic QC as an inherently fault-tolerant quantum computing model Freedman, Kitaev, Larsen, Wang: ---Computation based on TQFT is poly-equivalent to QCM ---Theory of anyons is essentially the theory of TQFTs Microsoft established Project Q at UCSB Nexus among quantum topology, quantum physics and quantum computing. Search for topological phases of matter

4 Two Kinds of Boundaries Given a topological system on Σ with anyons a 1, a 2, at z 1,z 2, and V(Σ)=ground states If Σ has boundaries S 1 i, then Σ [0,1] has boundaries: Temporal boundaries: S 1 i [0,1] Spacial boundaries: Σ 0,1 Witten: 1. V(Σ)= space of conformal blocks in WZW CFT 2.? Same CFT describes the edge theories of S 1 i [0,1]

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6 Trial Wavefunctions via CFT Trial ground state wavefucntions for n anyons a i with primary fields Φ ai at w i and N electrons at z i with primary field Φ el are given by conformal blocks Master Equation: Ψ GS (w 1,,w n ; z 1,,z N )= <Φ a1 (w 1 ),,Φ a2 (w n );Φ el (z 1 ), Φ el (z N );Φ bg > CFT = α F α 2

7 Moore-Read Conjecture The Berry phases from the exchange of anyons by adiabatic transport is the same as the analytic continuation of conformal blocks Abelian: Arovas, Wilczek, Schreifer SU(2): Blok-Wen, 1992, Read, 2004 In general, OPEN

8 TQFT Rep of Braid Groups Drinfeld-Kohno Theorem: Braiding of conformal blocks = TQFT reps of the braid groups If MR conjecture holds, then the Berry phases of adiabatically exchange anyons are the TQFT reps of the braid groups.

9 Atiyah s Axioms of (2+1)-TQFT: (TQFT w/o excitations and central charge=0) Surface Σ 2 vector space V(Σ) 3-manifold M 3 a vector Z(M 3 ) V( M 3 ) V( ) C V(Σ 1 Σ 2 ) V(Σ 1 ) V(Σ 2 ) V(Σ * ) V * (Σ) Z(Σ I)=Id V(Σ) Z(M 1 M 2 )=Z(M 1 ) Z(M 2 )

10 Topological Phases of Matter Suppose a gapped quantum system on an oriented surface Σ has anyons of types a 1, a 2,, a n localized at z 1,z 2,,z n, then the ground states of the system outside z 1,z 2,,z n form a Hilbert space V(Σ,a 1,a 2,,a n ). label

11 A diffeomorphism, e.g. braiding f: Σ\{z i } Σ\{z i } induces a unitary transformation V(f): V(Σ, a 1,,a n ) V(Σ,a 1,,a n ) If the collection {V(Σ, a 1,,a n ), V(f)} form a TQFT, then the quantum system is topological.

12 Algebraic Theory for Anyons A (unitary) ribbon tensor category: a finite label set {X 0 =1, X 1,,X n-1 } (anyon type representatives, and X m i X i ) with compatible Fusion rule: X i X h j i,jk X k Duality: 1 X X *, X * X 1 Braiding: c X,Y : X Y Y X Twist: θ X : X X e.g. G=finite abelian group, X i =g, g h=g h

13 Graphical Calculus Fusion rule: i k j Duality: i i * i * i Braiding: Twist: i j i j i

14 Trajectories of Anyons time Each line is labeled by a particle type and an arrow. Move forward=particle, move backward=anti-partilce Special trajectories are braids and links.

15 Quantum Dimensions For each particle type X i, d i = X i Global dimension: D 2 = i d i 2 d i d j = k h i,jk d k, Let (Q i ) a,b =h a i,b Then d i is the Perron-Frobenius eigenvalue of Q i. If d i <2, then d i =2cos (π/r) for some r>2. (Kronecker)

16 Twists Given a particle type X i, = θ i X i X i θ i is a root of unity (Vafa)

17 Central Charge Let p ± = i θ i± d i 2 Then p + p - =D 2 p + /D =e 2πi/8 c where c=c + - c - is the central charge of a boundary CFT mod 8

18 Classification Low rank TQFTs, rank=# of particle types: rank=2,3,4 (Belinschi, Stong, Rowell, W.) Abelian TQFTs: X i X j =X k fusion rules=abelian group, braid reps=1-dimensional (Ludwig, Read, W.) General case:? Fix rank, there are only finitely many TQFTs (True if fusion rules are fixed---ocneanu rigidity)

19 Read-Rezayi States ν=1/3 or 2/3 SU(2) at r=3 ν=5/2 SU(2) at r=4 ν=12/5 or 13/5 SU(2) at r=5 (Universal QC)

20 Jones-Kauffman SU(2)-Theory Pictorial formulation of the Jones/SU(2)-Chern-Simons theory by Kauffman (different for some levels). Fix r 3, and k=r-2, called the level The particle types are L={0,1,,r-2} and each is its own dual, a*=a Fusion rules: a b= c such that 1) a+b+c even 2) a+b c, b+c a, c+a b 3) a+b+c 2(r-2)

21 Given n particles in a disk at fixed positions, there will be a Hilbert space to describe the ground states of this quantum system. How to find a basis for such Hilbert spaces? Particle Fix a level k=r-2, a basis for the Hilbert space of n internal particles labeled by 1, and the outside boundary labeled by m can be chosen as follows: Any admissible label m

22 FQHE at ν=5/2 The effective theory is Ising TQFT (SU(2) 2 CS theory is different) Particle types are {1,σ,ψ} or {0,1,2} Fusion rules: or ψ σ 2 1+ψ, ψ 2 1, σ ψ ψ σ σ

23 5/2-Quantum Computer 4 σ s in a disk is C 2 ---qubit. 6 σ s C qubits. For 1-qubit gates, ρ: B 4 U(2) For 2-qubits gates, ρ: B 6 U(4) 1 1 /σ 1 1 1/σ 1/σ 1

24 For n qubits, consider the 2n+2 punctured disk D 2n+2 and ρ: B 2n+2 U(2 n ) Given a quantum circuit on n qubits: U L : (C 2 ) n (C 2 ) n Ideally to find a braid b B 2n+2 so that the following diagram commutes for any U L : No level=2 (C 2 ) n V(D 2n+2 ) U L ρ(b) (C 2 ) n V(D 2n+2 )

25 Fibonacci TQFT (FQHE at ν=12/5) SU(2) level 3 particle types are {0,1,2,3} or {1,φ,τ,ω} {1,τ} form a sub-theory, called the Fibonacci Fusion rules: τ 2 1+τ This is the same fusion rule for the Yang-Lee model, but Yang-Lee is not unitary. n anyons Ground states degeneracy =0,1,1,2,3,5,.. Fibonacci numbers F n-2

26 Fibonacci TQFT G 2 level=1 CFT, c=14/5 mod 8 Particle types: {1,τ} Quantum dimensions: {1,τ} Fusion rules: τ 2 =1+τ Braidings: =e3π i/5 Twists: =e4π i/5

27 Modular S-matrix: F-matrix: S=1/D τ -1 τ -1/2 1 τ τ -1 S τ =(e 3π i/1 0 ) F= τ -1/2 -τ -1 =τ 1/2 Theta symbol: θ =τ 3/2 Tetrahedron symbol: =-τ =-τ -1/2 Conformal block basis: 1 Any admissible label 1

28 Fibonacci Quantum Computer 4 τ s in a disk is C 2 ---qubit. 6 τ s C qubits+nc. For 1-qubit gates, ρ: B 4 U(2) For 2-qubits gates, ρ: B 6 U(4) U(5) 1 1 /τ 1 1 1/τ 1/τ 1

29 For n qubits, consider the 2n+2 punctured disk D 2n+2 and ρ: B 2n+2 U(F 2n ) Given a quantum circuit on n qubits: U L : (C 2 ) n (C 2 ) n Given a precision, find a braid b B 2n+2 so that the following diagram commutes up to this precision: (C 2 ) n V(D 2n+2 ) U L ρ(b) (C 2 ) n V(D 2n+2 )

30 Bonesteel et al

31 Error correction and fault tolerance will be essential in the operation of large scale quantum computers. The brute force approach to fault-tolerant quantum computing uses clever circuit design to overcome the deficiencies of quantum hardware. It works in principle, but achieving it in practice will be challenging. Topological quantum computing is a far more elegant approach, in which the hardware is intrinsically robust due to principles of local quantum physics (if operated at a temperature well below the mass gap). The topological approach also looks daunting from the perspective of current technology. But it is an attractive and promising long-term path toward realistic quantum computing. As a bonus, there are fascinating connections with deep issues in quantum many-body physics! John Preskill and mathematics.

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