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Topological Quantum Computation I. Zhenghan Wang Microsoft Station Q/UCSB Guelph, June 17, 2014 Key “Post-Shor” Idea Peter Shor Shor’s Factoring Algorithm To use topology to protect quantum information Michael Freedman Alexei Kitaev 2 Topological Quantum Computation A revolutionary idea: If a physical system were to have quantum topological (necessarily nonlocal) degrees of freedom, which were insensitive to local probes, then information contained in them would be automatically protected against errors caused by local interactions with the environment. This would be fault tolerance guaranteed by physics at the hardware level, with no further need for quantum error correction, i.e. topological protection. Alexei Kitaev 3 Why does topology help? • Topology is the part of geometry which survives deformation/perturbation. • Topological properties of quantum systems are robust to perturbation/deformation. 4 • “Qubits” are just the modern name for a twolevel system and its composites---easily decohere • Topology provides a solution to the fragility of qubits---topological phases of matter 5 Topological Qubits • Inherently fault-tolerant (logical) qubits: information encoded in the ground states of many anyons, and processed by braiding or measuring anyons. • Construction has begun: fractional quantum Hall liquids, engineered quantum systems,... Anyonic Quantum Systems • Anyons: quasi-particles or topological excitations in topological phases of matter--states of matter with topological order or whose low energy physics are modeled by TQFTs. Their statistics are more general than bosons/fermions, can be even non-abelian. • Models: Simple objects in unitary modular categories. Anyons in TPM • Materializations of long ranged topological order: for topological phases of matter, the ground state manifold is an analogue of the configuration space and the “first” excited states are the phase space. (excitations form an analogue of the tangent bundle) • In 2D, to first approximation, long ranged entanglement pattern is encoded by a unitary modular category: D,S,T,… TPM TQC TQFT/UMC Topological phases of matter are TQFTs/UMCs in Nature and hardware for the hypothetical topological quantum computers. Are we going to have non-abelian anyons? Ground State Degeneracy I---TQFT For a surface Y with anyons a1, a2, …, an localized at 1,…,n, the (relative) ground states V(Y; a1, a2, …, an, l) of the system “outside” 1,…,n form a Hilbert space depending on boundary conditions l. This collection of ground state manifolds V(Y; a1, a2, …, an, l) forms a TQFT. Stable boundary conditions correspond to anyons. label l ●●●●●●●● Ground State Degeneracy II--Quantum Dimension • A defining property of non-abelian anyons is the ground state degeneracy: when n anyons of type x are fixed in the 2-sphere, the ground state degeneracy goes like 𝑑𝑥 𝑛 for some positive real number 𝑑𝑥 >1---quantum dimension. For abelian anyons, 𝑑𝑥 =1. • Information will be encoded in this ground state degeneracy. Anyons in 2D In R2, an exchange is of infinite order Not equal Braids form groups Bn, then braiding anyons leads to : Bn U(V(a1, a2, …, an, l)) and the ground state manifold is the rep space. Anyon braidings form our “gate set” Data for Anyon Systems • Label set: a finite set L={a,b,c,…} of anyon types with an involution and a trivial type 𝑐 • Fusion rules: {𝑁𝑎𝑏 , a,b,c∊L}. The fusion rules determine when two anyons of types a,b are fused, whether or not anyons of type c appear, ie if 𝑁𝑎𝑏 𝑐 is ≥ 1 or =0. If 𝝨𝑐 𝑁𝑎𝑏 𝑐 >1, 𝑐 a,b have multi-fusion channels. If 𝑁𝑎𝑏 >1, fusion of a, b has multiplicities. • Others Categorification of Fusion Rules • When can a given fusion ring (a label set with fusion rules) be categorified to an anyon system, i.e. a unitary modular category? If so, how many are there? • Widely open. But we do know if we fix the rank (the number of anyon types), there are only finitely many anyon systems (Bruillard, Ng, Rowell, W.) Pointed Fusion Rules • Let G be a finite abelian group. Then G is a label set: each group element is an anyon type 𝑐 and 𝑁𝑎𝑏 =𝛿𝑎𝑏,𝑐 for a,b,c∊G. There are no multi-fusion channels, so no multiplicites. • An anyon (type) y is abelian if for any anyon (type) x, there are no multi-fusion channels for y and x. Otherwise, it is non-abelian. Non-abelian Fusion Rules---Ising Anyon types are L={1,,} Quantum dimension {1, 𝟐,1} TEE=-logD, D=2 Fusion rules: 2 1+, 2 1, 1 or Unitary Modular Category Label set Fusions a b c a b F-matrices (6-j) d a d b R-matrix a b =𝑅𝑐 𝑎𝑏 c c Compatibility: pentagons and hexagons c Count Ground State Degeneracy • Given anyons 𝑎𝑖 , i=1,…,n with total charge l in the disk, draw a trivalent tree: 𝑎1 𝑎2 𝑎3 𝑎𝑛 l • Count how many admissible labeled trees. They form a basis of the ground state mfd. Fibonacci Anyon Label set: {1,} Fusion rules: 2 1+ Quantum dimension: {1,𝜙} TEE=-log 𝟐 + 𝜙 1 n anyons Ground states degeneracy =0,1,1,2,3,5,.. Fibonacci numbers Fn How to Compute Braid Reps • A basis of the ground state manifolds given by admissible labelings of fusion trees---trivalent graphs with a fixed total charge. • Stack braids on top of a basis vector, and use F-moves and R-moves to transform back to a linear combination of the given tree basis. Unitary Fusion Category A quadruple (L,N,F,ԑ) L---a finite label set, e.g. a finite group G , N={𝑁𝑎 , a∈L},a b=⊕𝑁𝑎,𝑏 𝑐 c group multi. 𝑁𝑔,ℎ 𝑘 =𝛿𝑔ℎ,𝑘 F={𝐹 𝑎𝑏𝑐 𝑑 , a,b,c,d∈L}, ԑ𝑎 =1 or -1 (spherical structure), 𝐹 𝑎𝑏𝑐 𝑑 =𝛿𝑎𝑏𝑐,𝑑 ԑ𝑔 =1 a b c m a,b,c,d admissible d If (a,b,m) and (m,c,d) admissible. F satisfies the pentagons Ising and Fibonacci Theory L={1,σ,ψ}, L={1,𝜏} 𝑁1 =Id 0 1 0 0 0 1 𝑁𝜎 = 1 0 1 , 𝑁ψ = 0 1 0 , 0 1 0 1 0 0 𝐹 𝜎𝜎𝜎 𝜎 = 1 2 1 1 , 1 −1 𝐹 𝜎ψ𝜎 ψ =𝐹 ψ𝜎ψ 𝜎 =-1, 𝑁𝜏 = 𝐹 𝜏𝜏𝜏 𝜏 = 0 1 , 𝜑 =golden ratio 1 1 𝜑 −1 −1 𝜑2 others F either 0 or 1 all ԑ𝑎 =1 −1 𝜑2 −𝜑 −1 R-Symbols for Ribbon Fusion Category Given a unitary fusion category as (L,N,F,ԑ), need extra R={𝑹𝒄 𝒂𝒃 ,a,b,c∈L} for a RFC. a a b b 𝑅𝑐 𝑎𝑏 1) L=a finite abelian group, N=group multiplication, All R=0 or 1 for G 2) Ising: 𝑹𝟏 𝝈𝝈 =𝒆 −𝟐𝝅𝒊 𝟏𝟔 , 𝑹𝟏 𝝍𝝍=-1 𝟔𝝅𝒊 𝑹𝝈 𝝍𝝈 = 𝑹𝝍 𝝈𝝈 =-i, 𝑹𝝈 𝝈𝝍 =𝒆 𝟏𝟔 ; 3) Fib: 𝑹𝟏 𝝉𝝉 =𝒆 −𝟒𝝅𝒊 𝟓 , 𝑹𝝉 𝝉𝝉 =𝒆 𝟑𝝅𝒊 𝟓 c c Invariants for Anyon Tangles/Topological Amplitudes of Quantum Processes time V(D2,a4,a1) T V(D2, a1,a2,a3,a4) Curves are anyon trajectories. Each curve is labeled by an anyon. Special cases include colored links and braids. Fibonacci Quantum Computational Model For n qubits, consider the 4n Fibonacci anyons, braiding gates/circuits : B4n U(𝑽𝟒𝒏 ), ●● ●● ●● ●● ●● ●● 𝑽𝟒𝒏 -ground state manifold of 4n Fibonacci anyons with trivial total charge. Its dimension is 𝑭𝟒𝒏−𝟐 --- the (4n-2)th Fib number. It is known that the images of (B4n) are dense in SU(𝑽𝟒𝒏 ), hence we have a universal quantum computing model. (One issue is that 𝑭𝟒𝒏−𝟐 is rarely a power of some integer.) 𝑆𝑈(2)𝑘 Chern-Simons Anyon Systems Fix k ≥ 1, called the level The particle types are L={0,1,…,k} and each is its own dual, a*=a Fusion rules: a b=⊕c if 1) 2) 3) c a+b+c even a+b≥c, b+c≥a, c+a≥b a+b+c ≤ 2k a b Ising Quantum Computational Model 4 Ising ’s in a disk for a qubit. 8 ’s for 2 qubits. For 1-qubit gates, : B4 U(2) For 2-qubits gates, : B8 U(4) ●●●● ●●●● ● ● ● ● 1 Ising Braid Gates 1 0 e- i/8 e- i/8 0 i (1-i)/2 (1+i)/2 (1+i)/2 (1-i)/2 1 2 2 1 0 1 e- i/4 4 ’s NOT Gate 1 0 Universality of Anyonic Computing Models To be universal, it suffices to approximate every gate (up to a phase) in a universal gate set. For 𝑺𝑼(𝟐)𝒌 Chern-Simons anyon systems, we need dense images for: k(B4) PSU and k(B8) PSU True if k 1,2,4 by a density theorem. Then by Kitaev-Solvay Theorem that each unitary matrix can be approximated efficiently. Universal Braiding Gates • Ising anyon does not lead to universal braiding gates, but Fib anyon does • Quantum dimension of Ising anyon has quantum dimension= 2, while Fib anyon has quantum dimension =( 5+1)/2---golden ratio • Given an anyon type x, when does its braiding lead to universal braiding gate sets ? Conjecture: Not universal iff 𝑑𝑥2 ∊ Z, i.e., weakly integral.