1 Samuel J. Lomonaco, Jr. Samuel J. Lomonaco, Jr. Dept. of Comp. Dept. of Comp. Sci Sci. & Electrical Engineering . & Electrical Engineering University of Maryland Baltimore County University of Maryland Baltimore County Baltimore, MD 21250 Baltimore, MD 21250 Email: Email: [email protected][email protected]WebPage WebPage: : http:// http://www.csee.umbc.edu/~lomonaco www.csee.umbc.edu/~lomonaco Quantum Computing Quantum Computing Overview Overview Four Talks Four Talks • A Rosetta Stone for Quantum Computation A Rosetta Stone for Quantum Computation • Three Quantum Algorithms Three Quantum Algorithms • Quantum Hidden Subgroup Algorithms Quantum Hidden Subgroup Algorithms • An Entangled Tale of Quantum Entanglement An Entangled Tale of Quantum Entanglement Elementary Elementary Advanced Advanced Samuel J. Lomonaco, Jr. Dept. of Comp. Sci. & Electrical Engineering University of Maryland Baltimore County Baltimore, MD 21250 Email: [email protected]WebPage: http://www.csee.umbc.edu/~lomonaco Continuous Quantum Continuous Quantum Hidden Subgroup Hidden Subgroup Algorithms Algorithms Defense Advanced Research Projects Agency (DARPA) & Defense Advanced Research Projects Agency (DARPA) & Air Force Research Laboratory, Air Force Materiel Command, USAF Air Force Research Laboratory, Air Force Materiel Command, USAF Agreement Number F30602 Agreement Number F30602-01 01-2-0522 0522 Lecture 3 Lecture 3 This work is in collaboration with This work is in collaboration with Louis H. Kauffman Louis H. Kauffman • The Defense Advance Research Projects Agency (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. L L - - O O - - O O - - P P This work is supported by: This work is supported by:
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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
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:
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
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))
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
•• 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 /
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
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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).
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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).