Topological Quantum Computation I
Eric Rowell
October 2015
QuantumFest 2015
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.
Quantum Mechanics in Quantum Information
Basic Principles
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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.
Survey of Current Efforts
Recent Publicity
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
Hadamard gate: 1 1
H := √12
1 −1
Controlled
 Z:
1 0
0 1
CZ := 
0 0
0 0

0 0
0 0

1 0
0 −1
Quantum Circuits
Definition
A quantum circuit on S = {Ui } is:
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G1 · G2 · · · Gm ∈ U(Mn )
where Gj = IV⊗a ⊗ Ui ⊗ IV⊗b
Entangled GHZ state via Hadamard and controlled Z :
|0i
H
|0i
H
H
|0i
H
H
|000i+|111i
√
2
Given U ∈ U(Mn ), efficiently approximate:
|U − G1 · G2 · · · Gm | < .
Remarks on QCM
Remarks
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Typical physical realization: composite of n identical d-level
systems. E.g. d = 2: spin- 12 arrays.
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The setting of most quantum algorithms: e.g. Shor’s integer
factorization algorithm
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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
Definition
Topological phases of matter have:
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Ground State degeneracy (on a torus)
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Quantized geometric phases
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Long-range quantum entanglement
The Braid Group
A key role is played by the braid group Bn generated by σi :
1
i
i+1
n
...
1
...
i
i+1
n
i -1=
Multiplication :
=
Abstract Group Definition
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? R3 \ {zi } boring but R2 \ {zi } interesting!
=
Anyons cont.
C1
C2
Paths C1 and C2 cannot be continuously deformed into each other.
Topological Model
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).
Topology+Quantum Mechanics
Goal: surface+particles→ (ground) state space
a
b
c
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).
Heuristic
TQFT: a model for anyons on surfaces–topological
properties+quantum mechanics.
Topological Quantum Field Theory
Definition
A 3D TQFT assigns to any surface (+boundary data `) a Hilbert
space:
(M, `) → H(M, `).
Each boundary circle is labelled by i ∈ L a finite set of “colors”.
0 ∈ L is neutral. Orientation-reversing map: x → x̂.
Basic pieces
Any surface can be built from the following basic pieces:
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empty: H(∅) = C
(
C
2
disk: H(D ; i) =
0
i =0
else
(
C a = b̂
annulus: H(A; a, b) =
0 else
pants: P := D 2 \ {z1 , z2 } H(P; a, b, c) = CN(a,b,c)
Dictionary
TQFT
colors x ∈ L
0∈L
x̂
dim H(x, x̂) = 1
H(M, `)
|L|
N(a, b, c)
U ∈ U(H(M, `))
Physics
distinct anyons
vacuum type
antiparticle to x
indecomposable
state space on M
ground state degeneracy on torus
Fusion channels
Particle exchange
Fusion Channels
The dimension N(a, b, c) of:
a
b
c
represents the number of ways a and b fuse to c
neglected: 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 together
then
M
H(Mg , `) =
H(M, (`, x, x̂))
x∈L
Example: Fibonacci Theory
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L = {0, 1}


C
pants: H(P; a, b, c) = C


0
a=b=c
a + b + c ∈ 2Z
else
Define: Vki := H(D 2 \ {zi }ki=1 ; i, 1, · · · , 1)
qubits realized on V31 ∼
= C2 .
(
Fib(n − 2) i = 0
dim Vni =
Fib(n − 1) i = 1
Example: V60
V06=H
1
1
1
1
1
1
0
Example: V60
V06=H
1
1
1
1
1
1
j
k
0
Example: V60
V06=H
1
1
1
1
1
1
j
k
0
={j,k}H
H
1 1 1
j
H
1 1 1
j
k
k
= C2C2C  CCC=C4C
0
What do TQCs naturally compute?
Answer
(Approximations to) Link invariants!
Associated to x ∈ L is a link invariant InvL (x) approximated by the
corresponding Topological Model efficiently.
L
Prob(
)  x-t|InvL( )|
Fundamental Questions (next talk)
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Distinguish: indecomposable anyons
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Classify: (models) by number of colors.
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Detect: non-abelian anyons
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Detect: universal anyons
References
1. G. Collins, “Computing with quantum knots” Scientific
American, April 2006.
2. Chetan Nayak, Steven H. Simon, Ady Stern, Michael
Freedman, and Sankar Das Sarma, “Non-Abelian anyons and
topological quantum computation” Rev. Mod. Phys. 80,
1083 September 2008
3. J. Pachos “Introduction to Topological Quantum
Computation” Cambridge U. Press 2012.
4. Z. Wang “Topological Quantum Computation” Amer. Math.
Soc. 2008.