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!