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Topological Quantum Computation I Eric Rowell Supported by USA NSF grant DMS1108725 Joint with P. Bruillard, S.-H. Ng and Z. Wang arXiv:1310.7050 July 2014 Quantum Topology Outline Quantum Computation Models Quantum Circuit Model Topological Model Foundational Problems Mathematical Models Classification Problems What is a Quantum Computer? From [Freedman-Kitaev-Larsen-Wang ’03]: Definition Quantum Computation is any computational model based upon the theoretical ability to manufacture, manipulate and measure quantum states. Primer on Quantum Mechanics Basic Principles I I I I Superposition: a state is a vector in a Hilbert space |ψi ∈ H Eg. √12 (|e0 i + |e1 i) Schrödinger: Evolution of the system is unitary U ∈ U(H) |ψ(t)i = Ut |ψ(0)i Entanglement: Composite system state space is H1 ⊗ H2 Entangled: √12 (|e0 i ⊗ |f0 i + |e1 i ⊗ |f1 i) 6= |ψ1 i ⊗ |ψ2 i P Indeterminacy: Measuring |ψi = i ai |ei i gives |ei i with probability |ai |2 . |ei i eigenstates for some (Hermitian) observable M on H. Quantum Circuit Model Fix d∈ Z and let V = Cd . Definition The n-qudit state space is the n-fold tensor product: Mn = V ⊗ V ⊗ · · · ⊗ V . A quantum gate set is a collection S = {Ui } of unitary operators Ui ∈ U(Mni ) usually ni ≤ 4. Example 1 1 Hadamard gate: H := 1 −1 Controlled phase: P := Diag (1, 1, 1, −1). √1 2 Quantum Circuits Definition A quantum circuit on S = {Ui } is: I I I G1 · G2 · · · Gm ∈ U(Mn ) where Gj = IV⊗a ⊗ Ui ⊗ IV⊗b |0i H |0i H H |0i H H |000i+|111i √ 2 Given U ∈ U(Mn ), approximate |U − G1 · G2 · · · Gm | < (m a polynomial in , n). Remarks on QCM Remarks I Typical physical realization: composite of n identical d-level systems. E.g. d = 2: spin- 12 arrays. I The setting of most quantum algorithms: e.g. Shor’s integer factorization algorithm I Main nemesis: decoherence–errors due to interaction with surrounding material. Requires expensive error-correction... Topological Phases of Matter Definition Topological Quantum Computation (TQC) is a computational model built upon systems of topological phases. Fractional Quantum Hall Liquid 1011 electrons/cm2 T 9 mK quasi-particles Bz 10 Tesla The Braid Group A key role is played by the braid group: Definition Bn is generated by σi , i = 1, . . . , n − 1 satisfying: (R1) σi σi+1 σi = σi+1 σi σi+1 (R2) σi σj = σj σi if |i − j| > 1 Anyons For Point-like particles: I In R3 : bosons or fermions: ψ(z1 , z2 ) = ±ψ(z2 , z1 ) I Particle exchange reps. of symmetric group Sn I In R2 : anyons: ψ(z1 , z2 ) = e iθ ψ(z2 , z1 ) I Particle exchange reps. of braid group Bn I Why? π1 (R3 \ {zi }, x0 ) = 1 but π1 (R2 \ {zi }, x0 ) = Fn = Topological Model (non-adaptive) Computation Physics output measure (fusion) apply gates braid anyons initialize create anyons vacuum Mathematical Model for Anyons Problem Find a mathematical formulation for anyons (topological phases). Definition (Nayak, et al ’08) a system is in a topological phase if its low-energy effective field theory is a topological quantum field theory (TQFT). Fact (Most?) (2 + 1)-TQFTs come from the Reshetikhin-Turaev construction via modular categories. Some Axioms Definition A fusion category is an abelian monoidal category (C, ⊗, ⊕) that is: I C-linear: Hom(X , Y ) a f.d. vector space I finite rank: simple classes {X0 := 1, X1 , . . . , Xm−1 } L semisimple: X ∼ µi Xi = I I I i rigid: dual objects X ∗ , dX : X ∗ ⊗ X → 1, bX : 1 → X ⊗ X ∗ compatibility axioms... Braided and Pre-modular Categories Definition I A braided fusion category (BFC) has braiding isomorphisms: cX ,Y : X ⊗ Y → Y ⊗ X I I Gives ⊗(i−1) ⊗(n−i−1) ρX : Bn → Aut(X ⊗n ) via σi 7→ IX ⊗ cX ,X ⊗ IX In a spherical fusion category j : V ∼ = V ∗∗ Spherical BFC=pre-modular: has trC : End(Y ) → C, via twists θX : X ∼ =X Modular Categories Definition A Modular Category has: Sij := TrC (cij ∗ cj ∗ i ) non-degenerate: det(S) 6= 0. Remark Tij = δij θi encodes twists and ord(T ) < ∞ (Vafa’s Theorem) (ST )3 = kS 2 and S 4 = D 4 Id so (S, T ) gives a projective rep. of the modular group SL(2, Z). Example SU(2)`−2 I simple objects {X0 = 1, . . . , X`−2 } I Sij = I θj = e I X1 ⊗ Xk = Xk−1 ⊕ Xk+1 for 1 ≤ k ≤ ` − 3. (i+1)(j+1)π ) ` sin( π` ) πi(j 2 +2j) sin( 2` so ord(T ) = 2` Remark For ` − 2 odd, modular subcategory PSU(2)`−2 : simple objects Yj = X2j and ord(T ) = `. (` = 5 PSU(2)3 : Fibonacci). Topological Model Remarks I I I I state space for n particles of type X ∈ C: Hni := Hom(Xi , X ⊗n ) Physically, L gates are “particle exchanges” (braiding) acting on Hn := i Hni . X Mathematically, gates are {ϕX n (σi )} where ϕn : Bn → U(Hn ) Topological protection from decoherence i i+1 i What do TQCs compute? Answer (Approximations to) Link invariants! Associated to X ∈ C is a link invariant InvL (X ) approximated by the corresponding Topological Model efficiently. L Prob( ) x-t|InvL( )| Recent Publicity Algebraic Definition? For S, T ∈ C(r ,r ) define dj := S0j , θj := Tjj , D 2 := P p± := j dj2 θj±1 . (S, T ) admissible if: P j dj2 , t 1. S = S t , SS = D 2 Id, T diagonal, ord(T ) = N < ∞ N 2. (ST )3 = p+ S 2 , p+ p− = D 2 , pp−+ =1 P S S S 3. Nijk := a iaD 2jada ka ∈ N P 4. θi θj Sij = a Nik∗ j dk θk where Nii0∗ uniquely defines i ∗ . n P 5. νn (k) := D12 i,j Nijk di dj θθji satisfies: ν2 (k) ∈ {0, ±1} 6. Q(S) ⊂ Q(T ), AutQ (Q(S)) ⊂ Sr , AutQ(S) (Q(T )) ∼ = (Z2 )k . 7. Prime (ideal) divisors of hD 2 i and hNi coincide in Z[ζN ]. Conjecture: any such (S, T ) determines a modular category. Enumeration Problem Question (Physics) How many anyonic systems with exactly r indistinguishable, indecomposable anyons-types are there? Question (Mathematics) How many modular categories with rank r are there? Rank-Finiteness Theorem (Bruillard,Ng,R,Wang 2013) There are finitely many modular categories of a given rank r . History: I (2003) Conjectured by Wang, verified for rank = 2 (Ostrik). I (2005) Verified for: fusion categories with FPdim(C) ∈ N (Etingof, Nikshych and Ostrik), rank= 3 (Ostrik). I (2009-10) Verified for: rank = 4 (R,Stong,Wang), rank = 5 non-self-dual (R,Hong). First Reduction Definition The fusion rules of C are Nijk = dim Hom(Xi ⊗ Xj , Xk ), 0 ≤ i, j, k ≤ r − 1. Define matrices (Ni )kj = Nijk so Xi → Ni a rep. of K0 (C). Theorem (Ocneanu Rigidity) There are finitely many fusion categories with given K0 (C) (i.e. fusion rules {Nijk : 0 ≤ i, j, k ≤ r − 1}). Remark Also true for braided fusion categories [ENO], and pre-modular categories. Second Reduction Set di := max Spec(Ni ) and dim(C) := P j di2 . Lemma For any M > 0, there are finitely many rank = r fusion categories C with FPdim(C) < M. Proof. maxj,k Nijk ≤ di ≤ dim(C) for all i. For convenience, we Assume C is pseudo-unitary. From [Ng-Schauenburg 08]: V ∈ C a spherical fusion category. (n) k ) where The (n,k)-FS-indicator is ν(n,k) (V ) = TrC ( EV (n) k EV : Set ν(n,1) = νn . 7→ Define N = FSexp(C) := min{n : νn (Xk ) = dk for all k}. Lemma (NS) I I For C modular: 1 D2 I νn (Xk ) = I N = ord(T ) P i,j Nijk di dj n θi θj More generally, for C spherical: D 2 , di ∈ Z[ζN ] := ON , ζN primitive root of unity. (cyclotomic integers!) Set SC = {p ∈ Spec ON : p|hD 2 i ⊂ ON }. Example C = PSU(2)3 √ 5+ 5 2 I N = ord(T ) = 5 and D 2 = I hD 2 i = h1 − ζ5 i2 where ζ5 is a primitive 5th root of 1 I Thus: SC = {h1 − ζ5 i} For each r define: Sr := [ rank(C)=r SC Third Reduction Proposition (BNRW) |Sr | < ∞ implies rank-finiteness. Proof requires analytic number theory and Theorem (Etingof-Gelaki) In a modular category D2 (di )2 is an algebraic integer. Let K be a number field and S ⊂ Spec OK be finite. The (abelian group of) S-units is: Y × OK,S = {x ∈ K : hxi = pαp } p∈S × where αp ∈ Z. For SC as above, rank(OK,S ) = |SC | − 1 + ϕ(N)/2. C Third Reduction Theorem (Evertse 1984) There are finitely many solutions to 0 = 1 + x0 + · · · + xr −1 with × xi ∈ OK,S such that no sub-sum of 1 + x0 + · · · + xr −1 vanishes. Proof. (Of Proposition) Let K = Q(ζN ). 1. {(−D 2 , (d1 )2 , . . . , (dr −1 )2 ) : C modular} are solutions to 0 = 1 + x0 + · · · + xr −1 × 2. di ∈ OK,S as di | D 2 . r 3. Each (di )2 > 0 so no sub-sum vanishes. 4. If |Sr | < ∞, Evertse’s Theorem implies |{(−D 2 , (d1 )2 , . . . , (dr −1 )2 )}| < ∞. Fourth Reduction For N = FSexp(C) set MN = {p ∈ Spec ON : p|hNi}, and [ Mr = MC rank(C)=r Proposition |Mr | < ∞ implies rank-finiteness. Follows from previous and: Proposition (Cauchy Theorem for Spherical Fusion Categories) Sr = Mr , i.e. hD 2 i and hNi have the same prime divisors in ON . Remark and Sr ⊃ Mr is due to Etingof and Ng-Schauenburg proved Sr ⊂ Mr for integral categories. Why “Cauchy”? Example Let C = Rep(G ), G a finite group. Then FSexp(C) = exp(G ) and dim(C) = D 2 = |G |. Cauchy’s Theorem: Sr = {p : p | |G |} ⊂ {p : p | exp(G )} = Mr Final Step Lemma |Mr | < ∞. (Hence rank-finiteness follows). Proof. Enough to bound N = FSexp(C) < M(r ). Define Q(S) := Q[Sij ] and Q(T ) := Q[θi ]. I I I I I A = Gal(Q(S)/Q) < Sr (A abelian) Gal(Q(T )/Q) ∼ = Z∗ (cyclotomic extn.) N Gal(Q(T )/Q(S)) ∼ = (Z2 )k (Dong,Lin,Ng) 1 → (Z2 )k → Z∗N → A → 1 is exact. For p > 2, if p n kN then This bounds p and n. ϕ(p n ) 2 |r ! while if 2n kN then 2n−2 |r !. Next time... 1. Universality 2. Locality Thank you!