1
Samuel J. Lomonaco, Jr.Samuel J. Lomonaco, Jr.Dept. of Comp. Dept. of Comp. SciSci. & Electrical Engineering. & Electrical Engineering
University of Maryland Baltimore CountyUniversity of Maryland Baltimore CountyBaltimore, MD 21250Baltimore, MD 21250
Email: Email: [email protected]@UMBC.EDUWebPageWebPage: : http://http://www.csee.umbc.edu/~lomonacowww.csee.umbc.edu/~lomonaco
Quantum ComputingQuantum Computing
OverviewOverviewFour TalksFour Talks
•• A Rosetta Stone for Quantum ComputationA Rosetta Stone for Quantum Computation
•• Three Quantum AlgorithmsThree Quantum Algorithms
•• Quantum Hidden Subgroup AlgorithmsQuantum Hidden Subgroup Algorithms
•• An Entangled Tale of Quantum EntanglementAn Entangled Tale of Quantum Entanglement
ElementaryElementary
AdvancedAdvanced
Samuel J. Lomonaco, Jr.Dept. of Comp. Sci. & Electrical Engineering
University of Maryland Baltimore CountyBaltimore, MD 21250
Email: [email protected]: http://www.csee.umbc.edu/~lomonaco
Continuous Quantum Continuous Quantum Hidden Subgroup Hidden Subgroup
AlgorithmsAlgorithms
Defense Advanced Research Projects Agency (DARPA) &Defense Advanced Research Projects Agency (DARPA) &Air Force Research Laboratory, Air Force Materiel Command, USAFAir Force Research Laboratory, Air Force Materiel Command, USAF
Agreement Number F30602Agreement Number F30602--0101--22--05220522
Lecture 3Lecture 3
This work is in collaboration withThis work is in collaboration with
Louis H. KauffmanLouis H. Kauffman
• The Defense Advance Research ProjectsAgency (DARPA) & Air Force Research
Laboratory (AFRL), Air Force Materiel Command,USAF Agreement Number F30602-01-2-0522.
• The National Institute for Standards and Technology (NIST)
• The Mathematical Sciences Research Institute (MSRI).
• The L-O-O-P Fund.LL--OO--OO--PP
This work is supported by:This work is supported by:
2
Existing Quantum AlgorithmsExisting Quantum Algorithms
•• Hidden Subgroup Algorithms Hidden Subgroup Algorithms –– ShorShor--Like AlgorithmsLike Algorithms
•• Amplitude amplification Amplitude amplification –– GroverGrover--Like AlgorithmsLike Algorithms
•• Quantum Algorithms Simulating Quantum SystemsQuantum Algorithms Simulating Quantum Systems
•• SipserSipser’’ss AlgorithmAlgorithm
•• Adiabatic AlgorithmsAdiabatic Algorithms ??????
Some Existing Some Existing HSAHSA’’ss
•• Hidden subgroup algorithmsHidden subgroup algorithms
DeutschDeutsch--JozsaJozsa
SimonSimon
ShorShor
LegendreLegendre symbol symbol
HallgenHallgen
Various NonVarious Non--abelabel. Algorithms. Algorithms
OthersOthers
We will now discuss the following Six We will now discuss the following Six HSAHSA’’ss
Continuous Continuous ShorShor on on
Wandering Wandering ShorShor
Lift of Lift of ShorShor toto
HSA on CircleHSA on Circle
Dual Dual ShorShor HSAHSA
HSA for Functional IntegralsHSA for Functional Integrals•• Lomonaco & Kauffman,Lomonaco & Kauffman, Continuous Quantum Hidden Continuous Quantum Hidden Subgroup Algorithms,Subgroup Algorithms,http://xxx.lanl.gov/abs/quanthttp://xxx.lanl.gov/abs/quant--ph/0304084ph/0304084
•• Lomonaco & Kauffman,Lomonaco & Kauffman, Quantum Hidden Subgroup Quantum Hidden Subgroup Algorithms:Algorithms: A Mathematical Perspective,A Mathematical Perspective, AMS, AMS, CONM/305, (2002), 139CONM/305, (2002), 139--202.202.http://xxx.lanl.gov/abs/quanthttp://xxx.lanl.gov/abs/quant--ph/0201095ph/0201095
•• Lomonaco & Kauffman,Lomonaco & Kauffman, A Continuous Variable A Continuous Variable ShorShorAlgorithmAlgorithm, , http://xxx.lanl.gov/abs/quanthttp://xxx.lanl.gov/abs/quant--ph/0210141ph/0210141
These Six Algorithms Can Be Found These Six Algorithms Can Be Found in the Following Three Papersin the Following Three Papers
Hidden Subgroup Hidden Subgroup AlgorithmsAlgorithms •• A subgroup of , andA subgroup of , and
•• An injectionAn injection
s. t. the diagrams. t. the diagram
is commutative.is commutative. /
A S
A Kϕ
ϕ
ν ι→
↑
HiddenHidden SubgroupSubgroupStructureStructure
Def.Def. A Map is said to have A Map is said to have hiddenhiddensubgroupsubgroup structurestructure if there exist if there exist
: A Sϕ →
Kϕ A: /A K Sϕ ϕι →
ϕ
AmbientAmbientGroupGroup
TargetTargetSetSet
HiddenHiddenSubgroupSubgroup
Set of RightSet of RightCosetsCosets
Hidden NaturalHidden NaturalSurjectionSurjection
3
/
A S
A Kϕ
ϕ
ν ι→
↑
HiddenHidden SubgroupSubgroup StructureStructure (Cont.)(Cont.)
ϕ
If is an If is an invariantinvariant subgroupsubgroup of , then of , then
is a group, and is an is a group, and is an epimorphismepimorphism
Kϕ A
/H A Kϕ ϕ=: /A A Kϕν →
Hidden QuotientHidden QuotientGroupGroup
HiddenHiddenEpimorphismEpimorphism
KitaevKitaev observed that finding the period observed that finding the period is equivalent to finding the subgroup , is equivalent to finding the subgroup , i.e., the kernel of .i.e., the kernel of .
P ⊂Z Z
mod
modn
N
n a N
ϕ →Z Z
P
ϕ
ShorShor’’ss Quantum factoring algorithm Quantum factoring algorithm reduces the task of factoring an integer reduces the task of factoring an integer
to the task of finding the period to the task of finding the period of a function of a function
PN
Origin of QHS AlgorithmsOrigin of QHS Algorithms
ShorShor FactoringFactoring
SimonSimon
DeutschDeutsch--JozsaJozsa
HiddenHidden SubgpSubgpAmbientAmbient GpGpQuantumQuantum AlgorithmAlgorithmA
2
0K ϕ
=
2 2 2⊕ ⊕ ⊕
2
K Pϕ =
2Kϕ ≅
Quantum Hidden Subgroup AlgorithmsQuantum Hidden Subgroup Algorithms
Kϕ
The The HiddenHidden SubgroupSubgroup ProblemProblem ((HSPHSP))
Given a mapGiven a map
with hidden subgroup structure, determine with hidden subgroup structure, determine the hidden subgroup of the ambient the hidden subgroup of the ambient group . An algorithm solving this group . An algorithm solving this problem is called a problem is called a hiddenhidden subgroupsubgroupalgorithmalgorithm ((HSAHSA))
: A Sϕ →
KϕA
•• Lomonaco & Kauffman,Lomonaco & Kauffman, Quantum Hidden Subgroup Quantum Hidden Subgroup Algorithms:Algorithms: A Mathematical Perspective,A Mathematical Perspective, AMS, AMS, CONM/305, (2002).CONM/305, (2002). http://xxx.lanl.gov/abs/quanthttp://xxx.lanl.gov/abs/quant--ph/0201095ph/0201095
The First of the Three PapersThe First of the Three Papers The Quantum Hidden Subgroup Paper The Quantum Hidden Subgroup Paper Shows how to create aShows how to create a
MetaMeta AlgorithmAlgorithm
4
•• Autos before Henry FordAutos before Henry Ford
An AnalogyAn Analogy
•• Autos after Henry FordAutos after Henry Ford
Quantum VersionQuantum Versionofof
Henry FordHenry Ford’’ssAssembly LineAssembly Line
Three Methods for Three Methods for Creating New Quantum Creating New Quantum
AlgorithmsAlgorithms
Two Ways to Create New Quantum AlgorithmsTwo Ways to Create New Quantum Algorithms
GivenGiven : A Sϕ →
PushPush
LiftLift
ι
ϕη
LLifted Lifted GpGp
νH ϕ ϕ ι=Approx Approx GpGp
SAmbAmb. . GpGp ϕTarget SetTarget SetA
Lifting and PushingLifting and Pushing
A 3rd Way to Create New Quantum AlgorithmsA 3rd Way to Create New Quantum AlgorithmsDualityDuality
A S→ϕAmbAmb. . GpGp
A S ′→ΦDual Dual GpGp DualDual
QHS QHS AlgAlg
QHS QHS AlgAlg
DualDual
SummarySummary3 Ways to create New Quantum Algorithms3 Ways to create New Quantum Algorithms
•• LiftingLifting
•• PushingPushing
•• DualityDuality
5
Some Past AlgorithmsSome Past AlgorithmsHidden Subgroup AlgorithmsHidden Subgroup Algorithms
•• Lomonaco & Kauffman,Lomonaco & Kauffman, A Continuous A Continuous Variable Variable ShorShor AlgorithmAlgorithm, , http://xxx.lanl.gov/abs/quanthttp://xxx.lanl.gov/abs/quant--ph/0210141ph/0210141
•• Lomonaco & Kauffman,Lomonaco & Kauffman, Quantum Hidden Quantum Hidden Subgroup Algorithms:Subgroup Algorithms: A Mathematical A Mathematical Perspective,Perspective, AMS, CONM/305, (2002).AMS, CONM/305, (2002).http://xxx.lanl.gov/abs/quanthttp://xxx.lanl.gov/abs/quant--ph/0201095ph/0201095
•• Wandering Wandering ShorShor
•• Continuous Continuous ShorShor
Wandering Wandering ShorShor
Q
S
ι ν→
↑↓ϕ ϕ ι=
ϕ
Free AbelFree AbelFinite Finite RkRkAmbAmb GPGP
Approx Approx GpGp
ShorShorTransvTransv
ApproxApproxMapMap
TargetTargetSetSet
PushPushA⊕ ⊕⊕ ⊕⊕ ⊕⊕ ⊕⊕ ⊕⊕ ⊕
Continuous Continuous ShorShor
A S→ϕAmbient GroupAmbient Group
Key Idea: Key Idea: of discrete algorithms to of discrete algorithms to a continuous groupsa continuous groups
S→
LiftingLifting
Add. Add. GpGp of of RealsReals
? Quantum algorithm for ? Quantum algorithm for the Jones polynomialthe Jones polynomial
• A highly speculative quantum algorithm for A highly speculative quantum algorithm for
Three Recent QHS AlgorithmsThree Recent QHS Algorithms
• A quantum algorithm on the A quantum algorithm on the
⇒
• A quantum algorithm to A quantum algorithm to ShorShor’’ss algorithmalgorithm
CircleCircle
dualdual
functional integralsfunctional integrals
Road MapRoad MapShorShor’’ss AlgAlg
QHS QHS AlgsAlgs forforFunctional Functional IntegralsIntegrals
PushingPushing
Dual of Dual of ShorShor’’ss AlgAlg
QHS QHS AlgAlg on on /
DualityDuality
QHS QHS AlgAlg on on
LiftingLifting Sϕ
Q
/
Q
ϕS
ϕ~
ϕ~
Lift of Lift of ShorShorAlgorithmAlgorithm
ShorShorAlgorithmAlgorithm
Dual LiftedDual LiftedAlgorithmAlgorithm
Dual Dual ShorShorAlgorithmAlgorithm
DualDual
LiftingLifting & & DualityDuality
6
A Lifting of A Lifting of ShorShor’’ssQuantum Factoring Quantum Factoring
Algorithm toAlgorithm toIntegers Integers
Fourier AnalysisFourier Analysison theon the
CircleCircle
A Momentary DigressionA Momentary Digression
The Circle as a GroupThe Circle as a Group
TheThe circlecircle groupgroup can be viewed ascan be viewed as
•• AA multiplicativemultiplicative groupgroup, i.e., as the unit , i.e., as the unit circle in the complex planecircle in the complex plane
2 :ixe xπ ∈( )22 2 i x yix iye e e ππ π +=i
where denotes the additive group of where denotes the additive group of realsreals..
The Circle as a GroupThe Circle as a Group
TheThe circle groupcircle group cancan alsoalso be viewed asbe viewed as•• AnAn additiveadditive groupgroup, i.e., as, i.e., as
where denotes the additive group of where denotes the additive group of integers.integers.
/ mod1reals=
mod 1x y+
The Character GroupThe Character Group
TheThe character groupcharacter group of an of an abelianabelian group group is defined asis defined as
( ),A Hom A Circle= : :A Circle a morphismχ χ= →
with group operation (in multiplicative notation),with group operation (in multiplicative notation),
( )( ) ( ) ( )1 2 1 2a a aχ χ χ χ=i i
or (in additive notation) asor (in additive notation) as
( )( ) ( ) ( )1 2 1 2a a aχ χ χ χ+ = +
A A
The Character Groups of The Character Groups of andand
• TheThe character groupcharacter group of isof is
•• TheThe character groupcharacter group of isof is
/
2: : /inxx n e xπχ= ∈ =
/
2/ : :
: mod 1:
inxn
n
x e n
x nx n
πχ
χ
≅ ∈
≅ ∈ =
/⇔DiscreteDiscrete
ContinuousContinuous
7
Fourier Analysis on the CircleFourier Analysis on the Circle /
TheThe Fourier transformFourier transform of of is defined as the map is defined as the map
given by given by
TheThe inverse Fourier transforminverse Fourier transform is defined asis defined as
: /f →
:f →
2( ) ( )inxf n dxe f xπ−= ∫
2( ) ( )inx
n
f x e f nπ
∈
=∑ ( )1
0
1 P
Pn
nx x
P Pδ δ
−
=
= − ∑
•• DiracDirac Delta function on Delta function on ( )xδ /
•• For a nonFor a non--zero integer, we will zero integer, we will also need on the generalized also need on the generalized functionfunction
P/
Needed Mathematical MachineryNeeded Mathematical Machinery
•• The elements of are formal integrals The elements of are formal integrals of the formof the form
•• denotes the rigged Hilbert spacedenotes the rigged Hilbert spaceon with on with orthonormalorthonormal basis basis
, i.e.,, i.e.,
/H
( )dx f x x∫
/H
: /x x ∈ ( )x y x yδ= −/
Rigged Hilbert SpaceRigged Hilbert SpaceFinally, let denote the space of formal Finally, let denote the space of formal sums sums
with with orthonormalorthonormal basis basis
H
:n nn
a n a n∞
=−∞
∈ ∀ ∈
∑
:n n ∈
A Lifting of A Lifting of ShorShor’’ssQuantum Factoring Quantum Factoring
Algorithm toAlgorithm toIntegers Integers
Sϕ
Q
ϕ~Lift of Lift of ShorShorAlgorithmAlgorithm
ShorShorAlgorithmAlgorithm
LiftingLifting & & DualityDuality
8
Let be periodic function with hidden minimum period .
Objective:
Find
:ϕ →P
P
Periodic Functions onPeriodic Functions on • Step 0.Step 0. Initialize
• Step 1.Step 1. Apply
• Step 2.Step 2. Apply
0 /0 0ψ = ∈ ⊗ HH
2 01 0 0in
n n
e n nπψ∈ ∈
= = ∈ ⊗∑ ∑ H Hi
1-1 ⊗F
: ( )U n u n u nϕ ϕ+
2 ( )n
n nψ ϕ∈
=∑
• Step 3.Step 3. Apply 1⊗F( )
( ) ( )
( )
( ) ( )
( )
1 0
1 0
01
1 0
0
0
0
0
23 /
12
1 00
122
00
12
00
1 12
00 0
1
0
1
inx
n
Pi n P n x
n n
Pin xin Px
n n
Pin x
Pn
P Pin x
n n
P
n
dx x e n
dx x e n P n
dx x e e n
dx x x e n
ne n
P P
n nP P
π
π
ππ
π
π
ψ ϕ
ϕ
ϕ
δ ϕ
ϕ
−
∈
−− +
∈ =
−−−
∈ =
−−
=
− −−
= =
−
=
= ∈ ⊗
= +
=
=
=
= Ω
∑∫
∑ ∑∫
∑ ∑∫
∑∫
∑ ∑
∑
H H
• Step 4.Step 4. Measure
with respect to the observable
to produce a random eigenvalueand then proceed to find the corresponding
using the continued fraction recursion. (We assume )
1
30
P
n
n nP P
ψ−
=
= Ω ∑
Qydy y y
Q = ∫O
/m Q
/n P22Q P≥
TheTheActualActual
ShorShorAlgorithmAlgorithm
UnUn--LiftedLifted
The Actual (UnThe Actual (Un--Lifted) Lifted) ShorShor AlgorithmAlgorithm
Make the following approximations by selecting Make the following approximations by selecting a sufficiently large integer :a sufficiently large integer :Q
is only approximately periodic !is only approximately periodic !ϕ
: 0Q k k Q≈ = ∈ ≤ <
/ mod 1: 0,1, , 1Q
rr Q
Q ≈ = = −
…
: : Qϕ ϕ→ ≈ →
9
Run the algorithm inRun the algorithm in
and measure the observableand measure the observable
Q S⊗H H
1
0
Q
r
r r rQ Q Q
−
==∑O
A Quantum Hidden A Quantum Hidden Subgroup Algorithm Subgroup Algorithm
on the on the
CircleCircle
The Dual AlgorithmThe Dual Algorithmon theon the
CircleCircle
Sϕ
Q
/ϕ
S
ϕ~Lift of Lift of ShorShorAlgorithmAlgorithm
ShorShorAlgorithmAlgorithm
Dual LiftedDual LiftedAlgorithmAlgorithm
DualDual
LiftingLifting & & DualityDuality
•• The elements of are formal The elements of are formal integrals of the form integrals of the form
•• denotes the rigged Hilbert spacedenotes the rigged Hilbert spaceon with on with orthonormalorthonormal basis basis
, i.e.,, i.e.,
/H
/H
: /x x ∈ ( )x y x yδ= −
( )dx f x x∫
/
Rigged Hilbert SpaceRigged Hilbert SpaceFinally, let denote the space of formal Finally, let denote the space of formal sums sums
with with orthonormalorthonormal basisbasis
H
:n nn
a n a n∞
=−∞
∈ ∀ ∈
∑
:n n ∈
10
Let be an admissible periodic function of minimum rational period
Proposition:Let (with ) be a period of . Then is also a period of .
Remark: Hence, the minimum rational period is always the reciprocal of an integer modulo 1 .
: /f →
/α ∈
21/ af
f
Periodic Admissible Functions onPeriodic Admissible Functions on /
1 2/a aα = ( )1 2gcd , 1a a =
• Step 0.Step 0. Initialize
• Step 1.Step 1. Apply
• Step 2.Step 2. Apply
0 0 0ψ = ∈ ⊗H H
1-1 ⊗F
2 01 /0 0ixdxe x dx xπψ = = ∈ ⊗∫ ∫i H H
: ( )U x u x u xϕ ϕ+
2 ( )dx x xψ ϕ= ∫
• Step 3.Step 3. Apply 1⊗F
( )
( )
23
2
inx
n
inx
n
dx e n x
n dx e x
π
π
ψ ϕ
ϕ
−
∈
−
∈
=
= ∈ ⊗
∑∫
∑ ∫ H H
Letting , we have m
mx x
a= −
( ) ( )
( )
1
12 2
0
1
1 2
0 0
121
2
0 0
m
maa
inx inx
m ma
maa in xa
m mm
inm aainxa
m
dx e x dx e x
mdx e x
a
e dx e x
π π
π
ππ
ϕ ϕ
ϕ
ϕ
+−
− −
=
− − +
=
− − −
=
=
= +
=
∑∫ ∫
∑ ∫
∑ ∫
But But
Thus,Thus,
21
0mod0
0mof di
0
inmaa
n am
a
otherw se a
i
n
e
aπ
δ− −
==
== =
∑
( )
( )
( )
( )
23
1/2
0mod0
1/2
0
inx
n
ainx
n an
ai ax
n dx e x
n dx e x
a dx e x
a a
π
π
π
ψ ϕ
δ ϕ
ϕ
−
∈
−=
∈
−
∈
∈
=
=
=
= Ω
∑ ∫
∑ ∫
∑ ∫∑
• Step 4.Step 4. Measure
with respect to the observable
to produce a random eigenvalue
( )3 a aψ∈
= Ω∑
n
n n n∈
=∑O
a
11
TheThe
correspondingcorresponding
algorithmalgorithm
discretediscrete
The Algorithmic Dual The Algorithmic Dual of of
ShorShor’’ss Quantum Quantum Factoring AlgorithmFactoring Algorithm
Sϕ
Q
/
Q
ϕS
ϕ~
ϕ~
Lift of Lift of ShorShorAlgorithmAlgorithm
ShorShorAlgorithmAlgorithm
Dual LiftedDual LiftedAlgorithmAlgorithm
Dual Dual ShorShorAlgorithmAlgorithm
DualDual
LiftingLifting & & DualityDuality
is only approximately periodic !is only approximately periodic !
We now create a corresponding We now create a corresponding discrete algorithmdiscrete algorithm
The approximations are:The approximations are:
: : Qϕ ϕ→ ≈ →
/ mod 1: 0,1, , 1Q
rr Q
Q ≈ = = −
…
: 0Q k k P≈ = ∈ ≤ <
ϕ
Run the algorithm inRun the algorithm in
and measure the observableand measure the observable
Q S⊗H H
1
0
Q
k
k k k−
=
=∑O
Quantum Algorithms based on Quantum Algorithms based on Feynman Functional integrals Feynman Functional integrals
The following algorithm is The following algorithm is highly speculativehighly speculative. . In the spirit of Feynman, the following In the spirit of Feynman, the following quantum algorithm is quantum algorithm is based on functional based on functional integrals whose existence is difficult to integrals whose existence is difficult to determinedetermine, let alone approximate., let alone approximate.
CaveatCaveat EmptorEmptor
12
The SpaceThe Space PathsPaths
PathsPaths = all continuous paths= all continuous pathswhich are with respect to the inner which are with respect to the inner productproduct
PathsPaths is a vector space over with is a vector space over with respect torespect to
[ ]: 0,1 nx →2L
1
0( ) ( )x y ds x s y s= ∫i i
( )( )
( ) ( )
( ) ( ) ( )
x s x s
x y s x s y s
λ λ= + = +
The Problem to be SolvedThe Problem to be Solved
Let be a functional with a Let be a functional with a hiddenhidden subspacesubspace of such thatof such that
: Pathsϕ →V Paths
( ) ( )x v x v Vϕ ϕ+ = ∀ ∈
Objective. Create a quantum algorithm Create a quantum algorithm that finds the hidden subspace .that finds the hidden subspace .V
The Ambient Rigged Hilbert SpaceThe Ambient Rigged Hilbert Space
Let be the rigged Hilbert space with Let be the rigged Hilbert space with orthonormalorthonormal basis , basis ,
and with bracket product and with bracket product
PathsH
:x x Paths∈
( )|x y x yδ= −
Parenthetical Remark
Please note that can be written as the Please note that can be written as the following disjoint union: following disjoint union:
( )v V
Paths v V ⊥
∈
= +∪
Paths
•• Step 0.Step 0. InitializeInitialize
•• Step 1.Step 1. Apply Apply
•• Step 2.Step 2. Apply Apply
0 0 0 Pathsψ = ∈ ⊗H H
1-1 ⊗F
2 01 0 0ix
Paths Paths
x e x x xπψ = =∫ ∫iD D
: ( )U x u x u xϕ ϕ+
2 ( )Paths
x x xψ ϕ= ∫ D
• Step 3. Apply 1⊗F
( )
( )
23
2
ix y
Paths Paths
ix y
Paths Paths
y x e y x
y y x e x
π
π
ψ ϕ
ϕ
−
−
=
=
∫ ∫
∫ ∫
i
i
D D
D D
13
ButBut
( ) ( )
( ) ( )
( )
2 2
2
2 2
ix y ix y
Paths V v V
i v x y
V V
iv y ix y
V V
xe x v xe x
v xe v x
ve xe x
π π
π
π π
ϕ ϕ
ϕ
ϕ
⊥
⊥
⊥
− −
+
− +
− −
=
= +
=
∫ ∫ ∫
∫ ∫
∫ ∫
i i
i
i i
D D D
D D
D D
However,However,
So, So,
( )2 iv y
V V
ve u y uπ δ⊥
− = −∫ ∫iD D
( )
( ) ( )
( )
( )
2 23
2
2
n
n
iv y ix y
Paths V V
ix y
Paths V V
ix u
V V
V
y y v e x e x
y y u y u x e x
u u x e x
u u u
π π
π
π
ψ ϕ
δ ϕ
ϕ
⊥
⊥ ⊥
⊥ ⊥
⊥
− −
−
−
=
= −
=
= Ω
∫ ∫ ∫
∫ ∫ ∫
∫ ∫
∫
i i
i
i
D D D
D D D
D D
D
••Step 4.Step 4. Measure Measure
with respect to the observable with respect to the observable
to produce a random element ofto produce a random element of
( )3
V
u u uψ⊥
= Ω∫ D
Paths
A w w w w= ∫ D
V ⊥
Can the above path integral quantum algorithm Can the above path integral quantum algorithm be modified in such a way as to create a be modified in such a way as to create a quantum algorithm for the Jones polynomial ?quantum algorithm for the Jones polynomial ?
I.e., can it be modified by replacing I.e., can it be modified by replacing by the by the space of gauge connectionsspace of gauge connections, and by , and by making suitable modifications?making suitable modifications?
QuestionQuestion
Paths
( ) ( ) ( )KK A A Aψ ψ= ∫D W
where is the where is the Wilson loopWilson loop
( ) ( )( )expK KA tr P A= ∫W
( )K AW
The EndThe End
Quantum Computation:Quantum Computation: A Grand Mathematical Challenge A Grand Mathematical Challenge for the Twentyfor the Twenty--First Century and the Millennium,First Century and the Millennium,Samuel J. Lomonaco, Jr.Samuel J. Lomonaco, Jr. (editor),(editor), AMS PSAPM/58, AMS PSAPM/58, (2002). (2002).
14
Quantum Computation and InformationQuantum Computation and Information,, Samuel J. Samuel J. Lomonaco, Jr. and Howard E. BrandtLomonaco, Jr. and Howard E. Brandt (editors),(editors), AMS AMS CONM/305, (2002). CONM/305, (2002).