Topological Quantum Computation I

Eric Rowell

October 2015QuantumFest 2015

What is a Quantum Computer?

From [Freedman-Kitaev-Larsen-Wang ’03]:

DefinitionQuantum Computation is any computational model based uponthe theoretical ability to manufacture, manipulate and measurequantum states.

Quantum Mechanics in Quantum Information

Basic Principles

I Superposition: a state is a vector in a Hilbert space |ψ〉 ∈ HEg. 1√

2(|e0〉+ |e1〉)

I Schrodinger: Evolution of the system is unitary U ∈ U(H)|ψ(t)〉 = Ut |ψ(0)〉

I Entanglement: Composite system state space is H1 ⊗H2

Entangled: 1√2

(|e0〉 ⊗ |f0〉+ |e1〉 ⊗ |f1〉) 6= |ψ1〉 ⊗ |ψ2〉I Indeterminacy: Measuring |ψ〉 =

∑i ai |ei 〉 gives |ei 〉 with

probability |ai |2.|ei 〉 eigenstates for some (Hermitian) observable M on H.

Survey of Current Efforts

Recent Publicity

Quantum Circuit Model

Fix d∈ Z and let V = Cd .

DefinitionThe 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 operatorsUi ∈ U(Mni ) usually ni ≤ 4.

Example

Hadamard gate:

H := 1√2

(1 11 −1

) Controlled Z :

CZ :=

1 0 0 00 1 0 00 0 1 00 0 0 −1

Quantum Circuits

DefinitionA quantum circuit on S = {Ui} is:

I G1 · G2 · · ·Gm ∈ U(Mn)

I where Gj = I⊗aV ⊗ Ui ⊗ I⊗bV

Entangled GHZ state via Hadamard and controlled Z :

|0〉

|0〉

|0〉

H

H

H

H

H

|000〉+|111〉√2

Given U ∈ U(Mn), efficiently approximate:

|U − G1 · G2 · · ·Gm| < ε.

Remarks on QCM

Remarks

I Typical physical realization: composite of n identical d-levelsystems. E.g. d = 2: spin-12 arrays.

I The setting of most quantum algorithms: e.g. Shor’s integerfactorization algorithm

I Main nemesis: decoherence–errors due to interaction withsurrounding material. Requires expensive error-correction...

Topological Phases of Matter

DefinitionTopological Quantum Computation (TQC) is a computationalmodel built upon systems of topological phases.

Fractional Quantum Hall Liquid

1011 electrons/cm2

Bz 10 Tesla

quasi-particles

T 9 mK

DefinitionTopological phases of matter have:

I Ground State degeneracy (on a torus)

I Quantized geometric phases

I Long-range quantum entanglement

The Braid Group

A key role is played by the braid group Bn generated by σi :

. . . . . .

i1 i+1 n

1 i i+1 n

Multiplication :

=

i-1=

Abstract Group Definition

DefinitionBn is generated by σi , i = 1, . . . , n − 1 satisfying:

(R1) σiσi+1σi = σi+1σiσi+1

(R2) σiσj = σjσi 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 SnI In R2: anyons: ψ(z1, z2) = e iθψ(z2, z1)I Particle exchange reps. of braid group BnI Why? R3 \ {zi} boring but R2 \ {zi} interesting!

I

=

Anyons cont.

C1

C2

Paths C1 and C2 cannot be continuously deformed into each other.

Topological Model

initialize create anyons

apply gates braid anyons

output measure(fusion)

Computation Physics

vacuum

Mathematical Model for Anyons

ProblemFind a mathematical formulation for anyons (topological phases).

Topology+Quantum Mechanics

Goal: surface+particles→ (ground) state space

a

c

b

1

Definition (Nayak, et al ’08)

a system is in a topological phase if its low-energy effective fieldtheory is a topological quantum field theory (TQFT).

HeuristicTQFT: a model for anyons on surfaces–topologicalproperties+quantum mechanics.

Topological Quantum Field Theory

DefinitionA 3D TQFT assigns to any surface (+boundary data `) a Hilbertspace:

(M, `)→ H(M, `).

Each boundary circle © is labelled by i ∈ L a finite set of “colors”.0 ∈ L is neutral. Orientation-reversing map: x → x .

Basic pieces

Any surface can be built from the following basic pieces:

I empty: H(∅) = C

I disk: H(D2; i) =

{C i = 0

0 else

I annulus: H(A; a, b) =

{C a = b

0 else

I pants: P := D2 \ {z1, z2} H(P; a, b, c) = CN(a,b,c)

Dictionary

TQFT Physics

colors x ∈ L distinct anyons

0 ∈ L vacuum type

x antiparticle to x

dimH(x , x) = 1 indecomposable

H(M, `) state space on M

|L| ground state degeneracy on torus

N(a, b, c) Fusion channels

U ∈ U(H(M, `)) Particle exchange

Fusion Channels

The dimension N(a, b, c) of:

a

c

b

represents the number of ways a and b fuse to cneglected: orientation

Two more axioms

To determine H(M; {i}) use basic pieces and two axioms:

Axiom (Disjoint Sum=Entanglement)

H((M1, `1)∐

(M2, `2)) = H(M1, `1)⊗H(M2, `2)

Axiom (Gluing)

If Mg is obtained from gluing two boundary circles of M togetherthen

H(Mg , `) =⊕x∈LH(M, (`, x , x))

Example: Fibonacci Theory

I L = {0, 1}

I pants: H(P; a, b, c) =

C a = b = c

C a + b + c ∈ 2Z0 else

I Define: V ik := H(D2 \ {zi}ki=1; i , 1, · · · , 1)

I qubits realized on V 13∼= C2.

I dimV in =

{Fib(n − 2) i = 0

Fib(n − 1) i = 1

Example: V 06

1 1 11 11

0

V06=H

Example: V 06

V06=H

1 1 11 11

0

kj

Example: V 06

V06=H

1 1 11 11

0

kj

={j,k}H 1 1 1j

1 1 1k

j k0

H H

= C2C2C CCC=C4C

What do TQCs naturally compute?

Answer(Approximations to) Link invariants!Associated to x ∈ L is a link invariant InvL(x) approximated by thecorresponding Topological Model efficiently.

LProb( ) x-t|InvL( )|

Fundamental Questions (next talk)

I Distinguish: indecomposable anyons

I Classify: (models) by number of colors.

I Detect: non-abelian anyons

I Detect: universal anyons

References

1. G. Collins, “Computing with quantum knots” ScientificAmerican, April 2006.

2. Chetan Nayak, Steven H. Simon, Ady Stern, MichaelFreedman, and Sankar Das Sarma, “Non-Abelian anyons andtopological quantum computation” Rev. Mod. Phys. 80,1083 September 2008

3. J. Pachos “Introduction to Topological QuantumComputation” Cambridge U. Press 2012.

4. Z. Wang “Topological Quantum Computation” Amer. Math.Soc. 2008.