Continuous Quantum Hidden Subgroup Algorithmslomonaco/conf/uva2003/lecture...xnx n χ π χ ≅∈...

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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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Transcript of Continuous Quantum Hidden Subgroup Algorithmslomonaco/conf/uva2003/lecture...xnx n χ π χ ≅∈...

  • 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 algorithmsDeutschDeutsch--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

    modnN

    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

    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 FordAn 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 ϕ Target 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 ior (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( ) ( )inxn

    f x e f nπ

    =∑ ( )1

    0

    1 PP

    n

    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

    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 0

    in

    n n

    e n nπψ∈ ∈

    = = ∈ ⊗∑ ∑ H Hi1-1 ⊗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, , 1Qr

    r QQ

    ≈ = = −

    : : 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

    Q

    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 0

    ixdxe 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 mm

    x xa

    = −

    ( ) ( )

    ( )

    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

    aπδ

    − −

    ==

    == =

    ( )

    ( )

    ( )

    ( )

    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

    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, , 1Qr

    r QQ ≈ = = −

    { }: 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 0

    ix

    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 yV 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

    ( )3V

    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 Wwhere 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).