topological quantum computation irowell/mtrrytectalk1.pdf · quantum mechanics in quantum...
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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.
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
=
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
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.