quantum algorithms beyond · shor’s algorithm (cont.) repeat steps 1 through 5 until there are...

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QuantumQuantumComputing Computing

??

Samuel J. Lomonaco, Jr.Samuel J. Lomonaco, Jr.Dept. of Comp. Sci. & Electrical EngineeringDept. of Comp. Sci. & Electrical Engineering

University of Maryland Baltimore CountyUniversity of Maryland Baltimore CountyBaltimore, MD 21250Baltimore, MD 21250

Email: Email: [email protected]@UMBC.EDUWebPage: WebPage: http://www.csee.umbc.edu/~lomonacohttp://www.csee.umbc.edu/~lomonaco

Quantum AlgorithmsQuantum Algorithms&&

BeyondBeyond

Quantum StatesQuantum States

SuperpositionSuperposition

UnitaryUnitaryEvolutionEvolutionMeasurementMeasurement

EntanglementEntanglement

Quantum AlgorithmsQuantum Algorithms

Simon’sSimon’s

Shor’sShor’s

Q. HiddenQ. HiddenSubgroup Algs.Subgroup Algs.

AmplitudeAmplitudeAmplif.Amplif.Algs.Algs.

TraceTraceEstimationEstimation

QuantumQuantumSimulationSimulation

of Q. of Q. SystemsSystems

DeutschDeutsch--JozsaJozsaGrover’sGrover’s JonesJones

Poly. Alg.Poly. Alg.

Q. RandomQ. RandomWalksWalks AdiabaticAdiabatic

AlgsAlgs

Exponential Exponential SpeedupSpeedup

Quadratic Quadratic SpeedupSpeedup

NonNon--ClassicalClassicalBehaviorBehavior ??????

OtherOtherAlgs.Algs.

NewNew

RelatedRelated??????

Quantum Algorithms ZooQuantum Algorithms Zoo

Shor’sShor’sAlgorithm ?Algorithm ?

Shor’sShor’sFactoringFactoring

AlgorithmAlgorithm

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•• 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, Shor's Quantum Factoring Shor's Quantum Factoring Algorithm,Algorithm, AMS PSAPM, 58, (2002), 161AMS PSAPM, 58, (2002), 161--179. 179. http://arxiv.org/abs/quanthttp://arxiv.org/abs/quant--ph/0010034ph/0010034

Based on the Following Two PapersBased on the Following Two Papers A Crytanalyst’s Dream A Crytanalyst’s Dream

I’m going to crack the I’m going to crack the RSA Cryto System by RSA Cryto System by finding a Superfast finding a Superfast

Factoring Algorithm !!Factoring Algorithm !!I’ll be rich & famous !!!I’ll be rich & famous !!!

ProblemProblem.. Given an integer Given an integer NN which is which is the product of two unknown primes the product of two unknown primes pp& & qq, i.e., , i.e., N=pqN=pq , find , find pp and and qq, , i.e., factor i.e., factor NN. .

CodeCodeBreakerBreaker

SimplifiedSimplified Shor’s Algorithm Shor’s Algorithm

Step 1Step 1.. Choose an integer Choose an integer s.t. s.t. a ( )gcd , 1a N =

Step 3Step 3.. If is not even, then goto If is not even, then goto Step 1Step 1..P

If is even, then If is even, then Step 4Step 4.. P( ) ( )/2 /21 1 1 modP P Pa a a N− = − +

( ) ( )/ 2 / 2angcd 1, gd cd 1,P Pa N a N− +

So use the Euclidean algorithm to computeSo use the Euclidean algorithm to compute

Step 2Step 2.. Use a Use a Quantum ComputerQuantum Computer to determine the to determine the period of the function:period of the function:P

( ) modxNf x a N=

Step 5Step 5..

SimplifiedSimplified Shor’s Algorithm (Cont.)Shor’s Algorithm (Cont.)

If the above If the above gcdgcd’s are & or’s are & or& , then we have factored . & , then we have factored .

If not, goto If not, goto

pp

qq N

Step 1Step 1..

SimplifiedSimplified Shor’s Algorithm Shor’s Algorithm

Step 1Step 1.. Choose an integer Choose an integer s.t. s.t. a ( )gcd , 1a N =

Step 3Step 3.. If is not even, then goto If is not even, then goto Step 1Step 1..PIf is even, then If is even, then Step 4Step 4.. P

( ) ( )/2 /21 1 1 modP P Pa a a N− = − +

( ) ( )/ 2 / 2angcd 1, gd cd 1,P Pa N a N− +

So use the Euclidean algorithm to computeSo use the Euclidean algorithm to compute

Step 2Step 2.. Use a Use a Quantum ComputerQuantum Computer to determine the to determine the period of the function:period of the function:P

( ) modxNf x a N=

Quantum PartQuantum Partof Algorithmof Algorithm

0, 1, 2,= ± ± … The IntegersThe Integers

:f →P f

ProblemProblem.. Given a periodic function Given a periodic function

Find the period of .Find the period of .,,

ChooseChoose a sufficiently large positive integer , a sufficiently large positive integer , and restrict to the setand restrict to the set

Qf 0,1,2, , 1QS Q= −…

and focus on the restricted functionand focus on the restricted function: Qf S →

SimplificationSimplification.. To avoid minor technicalities, To avoid minor technicalities, we assume that is a multiple of , i.e.,we assume that is a multiple of , i.e.,PQ |P Q

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Shor’s Algorithm (Cont.) Shor’s Algorithm (Cont.)

•• Choose an integer Choose an integer nn s.t. s.t. Q<2Q<2nn & & Max(f)<2Max(f)<2nn

•• Construct two Construct two nn--qubit registers, i.e., qubit registers, i.e., Reg1Reg1and and Reg2Reg2 ..

1 2 0 1 2 0Reg1 Reg2 n n n na a a b b b− − − −=

ArgumentsArgumentsof of ff

ValuesValuesof of ff

ConventionConvention.. 2 11 2 0 0

2n

jn n jja a a a−− − =

= ∑For example, For example, 10111 23=

The QThe Q--Point Fourier TransformPoint Fourier Transform

ωω = Primitive = Primitive QQ--th root of unity, e.g., th root of unity, e.g., ee22ππi/Qi/Q

0, 1, 2, , 1QS Q= −

The Fourier Transform is:The Fourier Transform is:

where where : :Q Qf S f S → →

F

( ) ( ) xyx Q

f y f x ω∈

=∑

RemarkRemark.. We will implement as a unitaryWe will implement as a unitarytransformation.transformation.

F


Step 2.0Step 2.0.. InitializeInitialize 0 0 00 0 00 0= … …

Step 2.1Step 2.1.. Apply to Apply to Reg1Reg1F

1 100 0

0 0 0 0I Q Qj

j jj jω

⊗ − −

= == =∑ ∑

Fi

Step 2.2Step 2.2.. Let be a unitary transformation Let be a unitary transformation that takes to . that takes to . Apply .Apply .

fU0j ( )j f j

fU

1 1

0 00 ( )

fUQ Q

j jj x f x− −

= =∑ ∑

Hence, Hence, Reg1Reg1 now holds all the integersnow holds all the integers0,1,2, … , Q0,1,2, … , Q--11

in superpositionin superposition


0 1

0 1

0 1

1

0

11

1 0 1 00 0

11

1 0 00 0

11

1 0 00 0

( )

( )

(

Reg1 Re

)

2

(

g

)

Q

j

QP P

j j

QP P

j j

QP P

j j

j f j

Pj j f Pj j

Pj j f j

Pj j f j

−

=

−−

= =

−−

= =

−−

= =

=

= + +

= +

= +

∑

∑ ∑

∑ ∑

∑ ∑

0 1

11

1 0 00 0

Reg1 Re 2 ( )g

QP P

j jPj j f j

−−

= =

= +

∑ ∑


Step 2.3Step 2.3.. Measure Measure Reg2Reg2

1

1

1 0 00

( )Reg1 Reg2

QP

jPj r f r

−

=

= + ∑

0 0,1,2, , 1r P∈ −…


Step 2.4Step 2.4.. Apply to Apply to Reg1Reg1F

( )

( )

( )

1 0

1 1

1 0

1

10

1

1 1 1

1 0 0 00 0 0

11

00 0

11

00 0

( ) ( )

( )

( )

Q QQIP P

Pj r k

j j k

QQ P

Pj r k

k j

QQ P jr k Pk

k j

Pj r f r k f r

k f r

k f r

ω

ω

ω ω

− − −⊗+

= = =

−−+

= =

−−

= =

+

=

=

∑ ∑ ∑

∑ ∑

∑ ∑

F

But !But !

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But !But ! ( ) 1

1

1

0

/ 0 mod /if0

QP jPk

j

Q P y Q Potherwise

ω−

=

==

∑

Hence, Hence,

0

0

1

0

: 0,1, , 1

1

00

Reg1 Re (

(

g2 )

)

Pr k

Qk PP

QP rP

k f r

Q f rP

λ λ

λ

λ

ω

ω λ

−

∈ = −

−

=

=

=

∑

∑

…


01

00

Reg1 ( )Reg2QP rP Q f r

Pλ

λω λ

−

=

= ∑

Step 2.4Step 2.4.. Measure Measure Reg1Reg1

0 0Reg1 Reg2 ( )Q f rP

λ =

Hence, we have obtained Hence, we have obtained

for somefor some0QP

λ 0 0,1, , 1Pλ ∈ −…


Repeat steps 1 through 5 until there are Repeat steps 1 through 5 until there are enough multiples of to recover , enough multiples of to recover , and hence . and hence .

/Q P /Q PP

RemarkRemark.. We should remind the We should remind the everyone that the above description everyone that the above description is a simplification of Shor’s factoring is a simplification of Shor’s factoring algorithm which contains most of the algorithm which contains most of the key features of the actual algorithm. key features of the actual algorithm.

The Unsimplified Shor’s AlgorithmThe Unsimplified Shor’s Algorithm

which is to a rational of which is to a rational of the form the form m/P mod1m/P mod1..

Shor’s algorithm selects a random rational of the Shor’s algorithm selects a random rational of the form form k/Q mod 1k/Q mod 1

It then uses the It then uses the continued continued fraction algorithmfraction algorithm to to findfind m/P mod 1m/P mod 1. .

““closestclosest””

FoundFound

FoundFound

FoundFound

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), 139139--202.202. http://xxx.lanl.gov/abs/quanthttp://xxx.lanl.gov/abs/quant--ph/0201095ph/0201095

Lomonaco,Lomonaco, Shor’s Quantum Factoring Shor’s Quantum Factoring AlgorithmAlgorithm,, AMS, PSAPM/58, (2002), 161AMS, PSAPM/58, (2002), 161--179.179. http://arxiv.org/abs/quanthttp://arxiv.org/abs/quant--ph/0010034ph/0010034

More thorough and detailed descriptions of Shor’s More thorough and detailed descriptions of Shor’s algorithm can be found in the following two papers:algorithm can be found in the following two papers:

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Grover’sGrover’sAlgorithm ?Algorithm ?

Grover’s AlgorithmGrover’s Algorithm

Searching for a Needle in a HaystackSearching for a Needle in a Haystack

Skip to refSkip to ref

The ProblemThe Problem

We wish to search an unordered listWe wish to search an unordered list

0, 1, 2, 3, , 2 1 1n N− = −…

for the label for the label 0j

6, 2, 23, 9,L = …

of labels from the set of labels from the set N = 2N = 2nn elementselements

The ProblemThe Problem

We represent the list We represent the list LL as a quantum as a quantum superpositionsuperposition

( )0 01 0 1 2 1j NN

ψ = + + + + + + −… …

and proceed to amplify the amplitudeand proceed to amplify the amplitudeof the label by applying a sequence of of the label by applying a sequence of unitary transformations, unitary transformations,

01 / j

0j

2β

1ψ

It’s All About this PictureIt’s All About this Picture

β

0j

0j⊥

0ψ2β

2ψkψ

MeasureMeasure

MeasureMeasure kψ

Observer

0j

( ) 01 1 1 1k j Nψ ε ε ε ε= + + + − + + −… …

AmplitudeAmplitudelarge as large as possiblepossible

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•• Lomonaco,Lomonaco, Grover's quantum search algorithm,Grover's quantum search algorithm,AMS PSAPM, vol. 58, (2002), 181 AMS PSAPM, vol. 58, (2002), 181 -- 192 192 http://arxiv.org/abs/quanthttp://arxiv.org/abs/quant--ph/0010040ph/0010040

More details can be found More details can be found in on the following Paperin on the following Paper

Skip to QHSASkip to QHSA

Grover’s Algorithm Grover’s Algorithm Finding a Needle in a HaystackFinding a Needle in a Haystack

Consider a large unstructured database consisting of Consider a large unstructured database consisting of records labeled in random order by the records labeled in random order by the

integers integers 0,1, 2, , 1N −…2nN =

E.g., the database could be stored as a linked listE.g., the database could be stored as a linked list

RecordRecord%#*!#*%#*!#*

RecordRecord*!#%*#*!#%*#

953953HeadHead 1025610256 2345723457 ......RecordRecord#%*!*##%*!*#

On average, we have to retrieve labels before On average, we have to retrieve labels before finding the label . finding the label .

/ 2N0x

Hence, the average computational work isHence, the average computational work iscomputational stepscomputational steps

( )O N

This is a practical problem thatThis is a practical problem thatappears in many guisesappears in many guises

Searching Searching a Phone Booka Phone Book

For example, consider a city phone book For example, consider a city phone book containing phone numbers. Find the name containing phone numbers. Find the name associated with the phone numberassociated with the phone number

N

0 (123) 456 7890x = −The best classical algorithm for finding the The best classical algorithm for finding the associated name, say associated name, say Jane DoeJane Doe, would , would search through phone numbers on search through phone numbers on average before finding the name average before finding the name Jane DoeJane Doe. . In other words, it would take on average In other words, it would take on average

computational steps.computational steps.

/ 2N

( )O N

called an called an OracleOracle, such that , such that

we are given a function we are given a function : 0,1,2, , 1 0,1f N − →…

( ) 010if x x

f xOtherwise

==

(“YES”)(“YES”)(“NO”)(“NO”)

By calling it an By calling it an OracleOracle, we mean that we do , we mean that we do not have immediate access to all argumentnot have immediate access to all argument--function pairs . The function pairs . The Oracle Oracle is simply a is simply a blackboxblackbox, which we can query as , which we can query as many times as we like by inputting a number many times as we like by inputting a number

, and then observing the resulting output , and then observing the resulting output . . ButBut each such query comes with an each such query comes with an

associated computational associated computational $$cost$$$$cost$$..

( )( ),x f x ( )f x

x( )f x

More formally,More formally,

The The SearchSearch ProblemProblemforfor anan

UnstructuredUnstructured DatabaseDatabaseis:is:

To find the record labeled as To find the record labeled as with the minimum amount of with the minimum amount of

computational computational $$work$$$$work$$ , i.e., , i.e., with the minimum number of with the minimum number of queries of the oracle .queries of the oracle .f

0x

Another ExampleAnother Example

Consider a Consider a plaintext/ciphertext attack by brute force plaintext/ciphertext attack by brute force key searchkey search on a message encrypted with the on a message encrypted with the Data Data Encryption StandardEncryption Standard ((DESDES), where the key ), where the key KK is a 56 is a 56 bit numberbit number

Given the Given the plaintext/ciphertextplaintext/ciphertext pairpairPlaintextPlaintext TheStolenGoldIsHiddenAtTheStolenGoldIsHiddenAtCiphertextCiphertext xjepPWvZideRkqldievMsFkxjepPWvZideRkqldievMsFk

crack the entire cipher by encrypting the crack the entire cipher by encrypting the plaintextplaintextTheStolenGoldIsHiddenAtTheStolenGoldIsHiddenAt

with each of the keys with each of the keys 0,1,2, … . 20,1,2, … . 25656--11 , in turn, , in turn, until the key until the key KK00 is found that produces the is found that produces the ciphertextciphertext xjepPWvZideRkqldievMsFkxjepPWvZideRkqldievMsFk

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if if ( ),P C

denotes the available denotes the available plaintext/ciphertextplaintext/ciphertextpairpair

then the then the OracleOracle isis

( ) 010if K K

f KOtherwise

==

0Kdenotes the denotes the keykey such thatsuch that

( )0,DES P K C=

, and if, and if

In other words,In other words, Mission Impossible AssignmentMission Impossible Assignment

Your “Your “Mission ImpossibleMission Impossible” assignment, should ” assignment, should you choose to accept it, is to devise an you choose to accept it, is to devise an algorithm which finds the label in algorithm which finds the label in

( )O N

0x

steps. steps.

As always, should you fail, your lecturer will As always, should you fail, your lecturer will disavow any association with your activities.disavow any association with your activities.

Mission Impossible AssignmentMission Impossible Assignment

Lov GroverLov Grover has accepted this “has accepted this “Mission Mission ImpossibleImpossible” challenge, and has successfully ” challenge, and has successfully created an algorithm which finds the label created an algorithm which finds the label in in

0x

( )O N

steps, with a total computational work ofsteps, with a total computational work of

( )logO N N

The Quantum Mechanical PerspectiveThe Quantum Mechanical Perspective

22--D Hilbert space D Hilbert space with orthonormal basiswith orthonormal basis

2H 0 , 1

22nn--D Hilbert space D Hilbert space with orthonormal basiswith orthonormal basis

1

20

n−= ⊗H H

0 , 1 , 2 , , 1N −…

Oracle is given as a blackbox unitary Oracle is given as a blackbox unitary transformation transformation

ffU

2 2

( )x y x f x y⊗ → ⊗⊗ ⊗ −

H H H HfU

The Quantum Mechanical PerspectiveThe Quantum Mechanical Perspective

2 2

( )x y x f x y⊗ → ⊗⊗ ⊗ −

H H H HfUFrom the Oracle From the Oracle

( ) ( )0

( ) 0 01 f xx

x if x xI x x

x otherwise− =

= − =

as follows as follows

x0 1

2− 0 1

2−

( )( 1) f x x−fU

we construct the unitary transformation we construct the unitary transformation 0x

I

is an Inversionis an Inversion0x

I

Note thatNote that0 0 02xI I x x= − , since, since

( )0 0 02 2 |I x x x x x x− = − 0 0x if x xx otherwise− =

=

Also please note that for any unit length ketAlso please note that for any unit length ket ψ2I Iψ ψ ψ= −

is an is an inversioninversion about the hyperplane to ,about the hyperplane to ,⊥ ψ

“A Mirror Image Transformation”“A Mirror Image Transformation”

i.e.,i.e.,

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“A Mirror Reflection”“A Mirror Reflection”

ψLeftLeft

RightRightRightRightLeftLeft

Inversion by Inversion by I ψ

HyperplaneHyperplaneMirrorMirror

HilbertHilbertSpaceSpace

Properties of Properties of I ψDefinitionDefinition.. Let , be unit length kets in Let , be unit length kets in

s.t. . s.t. . χψ

H |χ ψ ∈ ThenThen

( ) , | ,Span a b a bψ χ ψ χ= = + ∈S

is a vector space over lying in with is a vector space over lying in with a real inner product induced by the bracketa real inner product induced by the bracket

in . Hence, if , are in . Hence, if , are linearly independent, then is a 2linearly independent, then is a 2--D D Euclidean plane lying in . Euclidean plane lying in .

ψ χ

H

H|− −

HS

Properties of Properties of I ψ

PropositionProposition.. The plane is invariant under The plane is invariant under the transformations , the transformations ,

SIψ I χ

( )( )

I

Iψ

χ

S = S

S = S

( ) , | ,Span a b a bψ χ ψ χ= = + ∈S

, i.e.,, i.e.,

H& leave the plane invariant& leave the plane invariantIψ I χ S

IψI χ

S

ψ

χ

Let be a ket in perpendicular to ,Let be a ket in perpendicular to ,and let denote the line in passing and let denote the line in passing through the origin and to . through the origin and to .

ψ ⊥S

ψLψ ⊥ S

⊥

ψ

Line Line Lψ⊥

ψ ⊥ψ Iψ

Reflection in lineReflection in lineLψ ⊥

:Iψ →S SS

ThenThen

if is a unit length vector inif is a unit length vector inwhich is to , which is to ,

ψ ⊥

S ⊥ ψ

I Iψ ψ ⊥− =S S

thenthenAnd moreover, And moreover,

And finally, if is a unit length ket in , And finally, if is a unit length ket in , and if is a unitary and if is a unitary transformation, transformation,

ψ H:U →H H

†UUI U Iψ ψ=

thenthen

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SummarySummary

there is a 2there is a 2--D plane (i.e., 2D plane (i.e., 2--D D inner product space over ) “living” in spanned inner product space over ) “living” in spanned by and s.t.by and s.t.ψ χ

Given two unit kets and “living” in with Given two unit kets and “living” in with real, real,

ψ χ|χ ψ

H

HS

•• andand( )Iψ =S S ( )I χ =S S

•• is the lineis the lineI ψS

a reflection ina reflection inoror

an inversion aboutan inversion aboutLψ ⊥

•• I Iψ ψ ⊥− =S S

•• for all unitary transfs for all unitary transfs †

UUI U Iψ ψ= U

Overview of Grover’s AlgorithmOverview of Grover’s Algorithm

Step 0Step 0.. 1

0

10N

jH j

Nψ

−

=

← = ∑0k←

((InitializationInitialization))

( )( )1/ 4sin 1/4

k N Nππ − = ≈ LoopLoop until until Step 1.Step 1.

00 xQ HI HIψ ψ ψ← = −1k k← +

Step 2.Step 2. Measure with respect to the standard Measure with respect to the standard basis basis to obtain the marked unknown state to obtain the marked unknown state with probability with probability

ψ0 , 1 , , 1N −…

0x( )1 1/ N≥ −

What’s Going On ?What’s Going On ?

Let Let

be the Hadamard transform given bybe the Hadamard transform given by

wherewhere

:H →H H

1(2)

0

n

H H−

=⊗

(2) 1 111 12

H = −

The Method in Lov’s The Method in Lov’s MadnessMadness

It’s All About this PictureIt’s All About this Picture

α α

β

β

2β

0x

0x⊥

0ψ

0ψ⊥

0xL ⊥

0Lψ

Thus, we now have “living” in the 2Thus, we now have “living” in the 2--D planeD planeH

S H

0ψ0x

( )0 0,S Span xψ=

(Initialization)(Initialization)Step 0Step 0..

This step creates a superposition of all states, This step creates a superposition of all states, i.e., i.e., 1

00

10 0N

jH j

Nψ

−

=

= =∑

The Iteration LoopThe Iteration LoopStep 1Step 1..

Each iteration rotates (in ) closer to .Each iteration rotates (in ) closer to .S 0x0ψ

( )0 0,Angle xβ ψ⊥=

Let and be unit length kets in Let and be unit length kets in which are to and , respectively. which are to and , respectively. 0ψ

0ψ⊥

0x0x⊥

⊥S

α α

β

β

2β

0x⊥

0ψ

0ψ⊥

0xL ⊥

0Lψ

0x

LetLet

•••••• S

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(Cont.)(Cont.)Step 1Step 1..

α α

β

β

2 β

0x⊥

0ψ

0ψ ⊥

0xL ⊥

0L ψ

0x

With each iterationWith each iteration1k kQψ ψ+ =

where where ( )0 0

0 0 0 00

† †0 0

0

x x

H x x x

Q HI H I HI H I

I I I I I Iψ ψ ⊥

= − = −

= − = − =

is the is the product ofproduct of two inversionstwo inversions..

S α α

β

β

2β

0x⊥

0ψ

0ψ ⊥

0xL ⊥

0L ψ

0x

S

TheoremTheorem

( )2 1

21 2

1 2 2

1 2

&

,L L

L L Lines inL L point O Ref Ref Rot

Angle L Lβ

β

= ⇒ ==

∩

But, But, 0 0 0cos sinx xψ β β⊥= +

So, So, ( )[ ] ( )[ ]0 0 0cos 2 1 sin 2 1kQ k x k xψ β β⊥= + + +

( )[ ] ( )[ ]0 0 0cos 2 1 sin 2 1kQ k x k xψ β β⊥= + + +

But what is ?But what is ?β

We seek to iterate until is as large as We seek to iterate until is as large as possible.possible.

( )[ ]sin 2 1k β+••In other words, we seek the smallest positive integerIn other words, we seek the smallest positive integer

such that is as close as possible tosuch that is as close as possible to..

( )2 1k β+k K=/ 2π

••

( )/ 2k K π β= = This turns out to be This turns out to be ••

But what is ? But what is ? β

α α

β

β

2β

0x⊥

0ψ

0ψ ⊥

0xL ⊥

0L ψ

0x

S

Recall thatRecall that ( )0 0,Angle xβ ψ⊥=

( )0 0,Angle xα ψ=We find by noting that the angleWe find by noting that the angleβ

/ 2α β π+ =is is complementarycomplementary to , i.e., to , i.e., β

SinceSince ( )0 01 0 1 1x NN

ψ π= + + + + + −… …

we havewe have 0 01 | cos cos sin

2x

Nπψ α β β = = = − =

Hence, Hence, ( )1sin 1/ 1/N Nβ −= ≈

andand ( )1/ 4 sin 1 /4

k K N Nππ − = = ≈

Hence, the number of iterations in isHence, the number of iterations in isStep 1Step 1..

( )O N

But each iteration uses the Hadamard transformBut each iteration uses the Hadamard transform1

(2)

0

n

H H−

=⊗at the computational cost of at the computational cost of

( )lgO N

Since is the computationaly dominant part of Since is the computationaly dominant part of Grover’s algorithm, it follows that the computational Grover’s algorithm, it follows that the computational time complexity of this algorithm istime complexity of this algorithm is

( )lgO N N

Step 1Step 1..

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( )[ ]2 2 1Prob sin 2 1 cos 1Success KN

β β= + ≥ = −

The probability that the measurement The probability that the measurement performed in of Grover’s algorithm performed in of Grover’s algorithm will successfully retrieve the unknown label will successfully retrieve the unknown label

is given by is given by 0x

Step 2.Step 2.

The Probability of SuccessThe Probability of Success Grover’s AlgorithmGrover’s Algorithm

Step 0Step 0.. 1

0

10N

jH j

Nψ

−

=

← = ∑0k←

((InitializationInitialization))

( )( )1/ 4sin 1/4

k N Nππ − = ≈ LoopLoop until until Step 1.Step 1.

00 xQ HI HIψ ψ ψ← = −1k k← +

Step 2.Step 2. Measure with respect to the standard Measure with respect to the standard basis basis to obtain the marked unknown state to obtain the marked unknown state with probability with probability

ψ0 , 1 , , 1N −…

0x( )1 1/ N≥ −

Hidden Hidden SubgroupSubgroupAlgorithms ?Algorithms ? QuantumQuantum

HiddenHidden SubgroupSubgroupAlgorithmsAlgorithms

A Grand UnificationA Grand Unification

is commutative.is commutative.

HiddenHidden SubgroupSubgroupStructureStructure

Def.Def. A Map is said to haveA Map is said to haveif there exist if there exist

•• A subgroup of , andA subgroup of , andKϕ A•• An injectionAn injection : /A K Sϕ ϕι →

/

A S

A Kϕ

ϕ

ν ι→

↑

ϕ

AmbientAmbientGroupGroup

TargetTargetSetSet

HiddenHiddenSubgroupSubgroup

Set of RightSet of RightCosetsCosets

Hidden NaturalHidden NaturalSurjectionSurjection

s. t. the diagrams. t. the diagram

hiddenhiddensubgroupsubgroup structurestructure

: A Sϕ →

HiddenHidden SubgroupSubgroup StructureStructure (Cont.)(Cont.)

/

A S

A Kϕ

ϕ

ν ι→

↑

ϕ

, then , then

/H A Kϕ ϕ=and is an and is an

epimorphismepimorphism: /A A Kϕν →

Hidden QuotientHidden QuotientGroupGroup

HiddenHiddenEpimorphismEpimorphism

Kϕ AIf is an If is an invariantinvariant subgroupsubgroup ofof

is a group, is a group,

•11/16/2008

•12

Kitaev observed that finding the period Kitaev 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

modmodn

Nn a N

ϕ→Z Z

P

ϕ

Shor’s Quantum factoring algorithm Shor’s 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

Origins of QHS AlgorithmsOrigins of QHS Algorithms

QuantumQuantum AlgorithmAlgorithm AmbientAmbient GpGp HiddenHidden SubgpSubgp

DeutschDeutsch--JozsaJozsa

SimonSimon

Shor FactoringShor Factoring

A

2

0K ϕ

=

2 2 2⊕ ⊕ ⊕

2

K Pϕ =

2Kϕ ≅

Quantum Hidden Subgroup AlgorithmsQuantum Hidden Subgroup Algorithms

Kϕ

A Grand UnificationA Grand Unification

An algorithm solving this An algorithm solving this problem is called a problem is called a hiddenhidden subgroupsubgroupalgorithmalgorithm ((HSAHSA) )

The The HiddenHidden SubgroupSubgroup ProblemProblem ((HSPHSP))

Given a mapGiven a map

: A Sϕ →

determine determine the the hiddenhidden subgroup of the ambient subgroup of the ambient group .group .

KϕA

with with hidden hidden subgroup structure, subgroup structure,

Some Existing QHSA’sSome Existing QHSA’s

•• Hidden subgroup algorithmsHidden subgroup algorithms

DeutschDeutsch--JozsaJozsa

SimonSimon

ShorShor

Legendre symbol Legendre symbol

Hallgen’s algorithm for solving Pell’s eq. Hallgen’s algorithm for solving Pell’s eq.

Various NonVarious Non--abel. Algorithmsabel. Algorithms

OthersOthers

We will now discuss the following Six HSA’sWe will now discuss the following Six HSA’s

Continuous Shor on Continuous Shor on

Wandering ShorWandering Shor

Lift of Shor toLift of Shor to

HSA on CircleHSA on Circle

Dual Shor HSADual Shor HSA

HSA for Functional IntegralsHSA for Functional Integrals

•• Lomonaco & Kauffman,Lomonaco & Kauffman, Continuous Quantum Hidden Subgroup Continuous Quantum Hidden Subgroup Algorithms, Algorithms, SPIE, 2004,SPIE, 2004, http://xxx.lanl.gov/abs/quanthttp://xxx.lanl.gov/abs/quant--ph/0304084ph/0304084

•• Lomonaco & Kauffman,Lomonaco & Kauffman, Quantum Hidden Subgroup Algorithms:Quantum Hidden Subgroup Algorithms:A Mathematical Perspective,A Mathematical Perspective, AMS, CONM/305, (2002), 139AMS, CONM/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 Shor AlgorithmA Continuous Variable Shor Algorithm,,AMS CONM/381, 2005, 97AMS CONM/381, 2005, 97--108. 108. http://arxiv.org/abs/quanthttp://arxiv.org/abs/quant--ph/0210141ph/0210141

These Six Algorithms Can Be Found These Six Algorithms Can Be Found in the Following Papersin the Following Papers

•• Lomonaco & Kauffman,Lomonaco & Kauffman, Quantum Hidden Subgroup Quantum Hidden Subgroup Algorithms: An Algorithmic Toolkit, Algorithms: An Algorithmic Toolkit, 2006,2006,http://xxx.lanl.gov/abs/quanthttp://xxx.lanl.gov/abs/quant--ph/0607047/ph/0607047/

•11/16/2008

•13

•• 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

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

ϕηLLifted GpLifted Gp

νH ϕ ϕ ι=Approx GpApprox Gp

SAmb. GpAmb. Gp ϕTarget SetTarget SetA

Lifting and PushingLifting and PushingA 3rd Way to Create New Quantum AlgorithmsA 3rd Way to Create New Quantum Algorithms

DualityDuality

A S→ϕAmb. GpAmb. Gp

A S ′→ΦDual GpDual Gp DualDual

QHS AlgQHS Alg

QHS AlgQHS Alg

DualDual

SummarySummary3 Ways to create New Quantum Algorithms3 Ways to create New Quantum Algorithms

•• LiftingLifting

•• PushingPushing

•• DualityDuality

Some Past AlgorithmsSome Past AlgorithmsHidden Subgroup AlgorithmsHidden Subgroup Algorithms

•• Lomonaco & Kauffman,Lomonaco & Kauffman, A Continuous Variable A Continuous Variable Shor AlgorithmShor Algorithm, , AMS CONM/381, 2005, 97AMS CONM/381, 2005, 97--108. 108. 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 ShorWandering Shor

•• Continuous ShorContinuous Shor

•11/16/2008

•14

QZ

ν ι

Approx.Approx.TransversalTransversal

NPZ Z Zν

⊂ →

Recall Shor’s algorithm:Recall Shor’s algorithm:

AmbientAmbientGroupGroupHiddenHidden

SubgroupSubgroup

TargetTargetSetSet

×

Find Approx. Find Approx. PeriodicPeriodic

Obj.: Find PeriodObj.: Find Period Wandering ShorWandering Shor

Q

Sι ν

→↑↓

ϕ ϕ ι=

ϕ

Free AbelFree AbelFinite RkFinite RkAmb GPAmb GP

Approx GpApprox Gp

ShorShorTransvTransv

ApproxApproxMapMap

TargetTargetSetSet

PushPushA⊕ ⊕⊕ ⊕⊕ ⊕⊕ ⊕⊕ ⊕⊕ ⊕

Continuous ShorContinuous Shor

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. Gp of RealsAdd. Gp of Reals

• 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 Shor’s algorithmA quantum algorithm to Shor’s algorithm

CircleCircle

dualdual

functional integralsfunctional integrals

Jeremy BecnelJeremy Becnel has found ahas found afirm mathematical foundationfirm mathematical foundationfor this algorithm.for this algorithm.

Road MapRoad MapShor’s AlgShor’s Alg

QHS Alg forQHS Alg forFunctional Functional IntegralsIntegrals

PushingPushing

Dual of Shor’s AlgDual of Shor’s Alg

QHS Alg on QHS Alg on /

DualityDuality

QHS Alg on QHS Alg on

LiftingLifting Sϕ

Q

/

Q

ϕ S

ϕ~

ϕ~

Lift of ShorLift of ShorAlgorithmAlgorithm

ShorShorAlgorithmAlgorithm

Dual LiftedDual LiftedAlgorithmAlgorithm

Dual ShorDual ShorAlgorithmAlgorithm

DualDual

LiftingLifting & & DualityDuality

•11/16/2008

•15

A Lifting of Shor’s A Lifting of Shor’s Quantum 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 reals.reals.

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 abelian group of an abelian 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

≅ ∈

≅ ∈ =

/⇔DiscretDiscret

ee ContinuousContinuous

•11/16/2008

•16

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

nf x e f nπ

∈

=∑ ( )1

0

1 P

Pn

nx xP P

δ δ−

=

= − ∑

•• Dirac Delta function on Dirac 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 orthonormal basis on with orthonormal 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 orthonormal basis with orthonormal basis

H

:n nn

a n a n∞

=−∞

∈ ∀ ∈ ∑

:n n∈

We now lift Shor’s Quantum We now lift Shor’s Quantum Factoring Algorithm to the Factoring Algorithm to the integers integers

Sϕ

Qϕ~



LiftingLiftingLet be periodic function with hidden minimum period .

Objective:

Find

:ϕ →P

P

Periodic Functions onPeriodic Functions on

•11/16/2008

•17

2 01 0 0in

n ne n nπψ

∈ ∈= = ∈ ⊗∑ ∑ H Hi

0 /0 0ψ = ∈ ⊗ HH

• Step 1.Step 1. Apply 1-1 ⊗F

2 ( )n

n nψ ϕ∈

=∑

• Step 0.Step 0. Initialize

: ( )U n u n u nϕ ϕ+• Step 2.Step 2. Apply

• 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

n e nP P

n nP P

π

π

ππ

π

π

ψ ϕ

ϕ

ϕ

δ ϕ

ϕ

−

∈

−− +

∈ =

−−−

∈ =

−−

=

− −−

= =

−

=

= ∈ ⊗

= +

=

=

=

= Ω

∑∫

∑ ∑∫

∑ ∑∫

∑∫

∑ ∑

∑

H H

• Step 4.Step 4. Measure 1

30

P

n

n nP P

ψ−

=

= Ω ∑

Qydy y yQ

= ∫O

and then proceed to find the corresponding using the continued fraction recursion. /n P

(We assume )22Q P≥

with respect to the observable

/m Qto produce a random eigenvalue

TheTheActualActual


UnUn--LiftedLifted

SkipSkip

The Actual (UnThe Actual (Un--Lifted) Shor AlgorithmLifted) Shor Algorithm

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→ ≈ →

Run the algorithm inRun the algorithm in

Q S⊗H H

1

0

Q

r

r r rQ Q Q

−

=

=∑O

and measure the observableand measure the observable

We have reconstructed theWe have reconstructed theoriginal Shor Algorithm !!!original Shor Algorithm !!!

•11/16/2008

•18

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 ShorLift of ShorAlgorithmAlgorithm



DualDual


•• 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 orthonormal basis on with orthonormal basis

, i.e.,, i.e.,

/H

/H

: /x x∈ ( )x y x yδ= −

( )dx f x x∫

/

Rigged Hilbert SpaceRigged Hilbert Space

Finally, let denote the space of formal Finally, let denote the space of formal sums sums

with orthonormal basiswith orthonormal basis

H

:n nn

a n a n∞

=−∞

∈ ∀ ∈

∑

:n n∈

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/aff

Periodic Admissible Functions onPeriodic Admissible Functions on /

1 2/a aα = ( )1 2gcd , 1a a =

•11/16/2008

•19

• Step 0.Step 0. Initialize

0 0 0ψ = ∈ ⊗H H

2 01 /0 0ixdxe x dx xπψ = = ∈ ⊗∫ ∫i H H

2 ( )dx x xψ ϕ= ∫

1-1 ⊗F••Step 1.Step 1. Apply

: ( )U x u x u xϕ ϕ+••Step 2.Step 2. Apply

• Step 3.Step 3. Apply 1⊗F

( )

( )

23

2

inx

n

inx

n

dxe n x

n dxe x

π

π

ψ ϕ

ϕ

−

∈

−

∈

=

= ∈ ⊗

∑∫

∑ ∫ H H

Letting , we have mmx xa

= −

( ) ( )

( )

11

2 2

0

11 2

0 01

212

0 0

m

maa

inx inx

m ma

maa in xa

m mm

inm aainxa

m

dx e x dx e x

mdx e xa

e dx e x

π π

π

ππ

ϕ ϕ

ϕ

ϕ

+−

− −

=

− − +

=

− − −

=

=

= +

=

∑∫ ∫

∑ ∫

∑ ∫

But But

Thus,Thus,

21

0mod0

0mof di0

inmaa

n am

aotherw s

e ai

neaπ

δ− −

==

== =

∑

( )

( )

( )

( )

23

1/2

0mod0

1/2

0

inx

na

inxn a

n

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

( )3 a aψ∈

= Ω∑

nn n n

∈

=∑O

a

with respect to the observable

to produce a random eigenvalue

TheThe

correspondingcorresponding

algorithmalgorithmdiscretediscrete

•11/16/2008

•20

The Algorithmic Dual The Algorithmic Dual of of

Shor’s Quantum Shor’s Quantum Factoring AlgorithmFactoring Algorithm

Sϕ

Q

/

Q

Φ S

ϕ~

Φ~




Dual ShorDual ShorAlgorithmAlgorithm

DualDual


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

Q S⊗H H

1

0

Q

kk k k

−

=

=∑O

and measure the observableand measure the observable

Quantum Algorithms based on Quantum Algorithms based on Feynman Functional integrals Feynman Functional integrals

SkipSkip

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

Recently, Recently, Jeremy BecnelJeremy Becnel has succeeded in has succeeded in creating a rigorous mathematical foundation creating a rigorous mathematical foundation for this algorithm !for this algorithm !

•11/16/2008

•21

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 sx y s x s y sλ 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 orthonormal basis , orthonormal 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

•11/16/2008

•22

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

( )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 W

where is the where is the Wilson loopWilson loop

( ) ( )( )expK KA tr P A= ∫W

( )K AW

Is Grover’s Algorithm Is Grover’s Algorithm a Quantum Hiddena Quantum Hidden

Subgroup AlgorithmSubgroup Algorithm

??????

•• Lomonaco & Kauffman,Lomonaco & Kauffman, Is Grover’s Algorithm a Is Grover’s Algorithm a Quantum Hidden Subgroup Algorithm ? Quantum Hidden Subgroup Algorithm ? ,, (2006), (2006), http://xxx.lanl.gov/abs/quanthttp://xxx.lanl.gov/abs/quant--ph/0603140ph/0603140

This material can be found This material can be found in the following paperin the following paper

•11/16/2008

•23

Grover’s Alg.Grover’s Alg.:: Oh where is the Hidden Symmetry ???Oh where is the Hidden Symmetry ???

The problem is to find the unknown integer The problem is to find the unknown integer jj00 in in

given the oraclegiven the oracle

01( )

0ifotherwise

j jf j

==

00,1,2,3, , , , 1 2 1nj N − = −… …

is is inversion in the inversion in the hyperplanehyperplane orthogonal to orthogonal to j


Let Let HH be the Hilbert space with orthonormal be the Hilbert space with orthonormal basisbasis 0 , 1 , 2 , , 1N −…

Our oracle is essentially given as the unitary Our oracle is essentially given as the unitary transformationtransformation

( )0

( )

:

1

j

f j

I H H

j j

→

−

wherewhere 2jI I j j= −

Let Let WW denote the denote the Hadamard transformHadamard transform on on HH


StepStep 0.0. (Initialization)(Initialization)1

0

10

0

N

jW j

Nk

ψ −

=← =

←

∑

StepStep 2.2. Meas. w.r.t. standard basisMeas. w.r.t. standard basis

to obtain unknown state withto obtain unknown state with

ψ

0j

0 , 1 , 2 , , 1N −…

Prob 1 1/ N≥ −

StepStep 1.1. Loop untilLoop until / 4k Nπ≈

00

1jQ WI WI j

k k

ψ ψ← = −

← +

Let Let SSNN be the be the symmetric groupsymmetric group on the on the symbolssymbols

00,1,2,3, , , , 1j N −… …

Then Grover’s algorithm is invariant under Then Grover’s algorithm is invariant under thethe

( ) 0 0 0:j N NFix g S g j j S= ∈ = ⊂

If we know the If we know the hidden subgrouphidden subgroup FixFixjj0 0 ,,

then we know then we know jj00, and vice versa., and vice versa.

hiddenhidden subgroupsubgroup

Let denote the transposition , Let denote the transposition , i.e., the permutation that interchanges i.e., the permutation that interchanges iiand and jj , and leaves everything else fixed., and leaves everything else fixed.

( ) Nij S∈ i j↔

Then the set of cosets of the Then the set of cosets of the 0/N jS Fix

hidden subgroup ishidden subgroup is0j

Fix

( ) ( ) ( ) ( )0 0 0 00 0 0 0/ 0 , 1 , 2 , , ( 1)

oN j j j j jS Fix j Fix j Fix j Fix N j Fix = −

…

So Grover’s algorithm solves the following So Grover’s algorithm solves the following hidden subgroup problem:hidden subgroup problem:

0

0 0/

j

j N N jFix S S Fixν

⊂ →



TargetTargetSetSet

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QZ

ν ι

Approx.Approx.TransversalTransversal

Shor’s OracleShor’s Oraclepops up herepops up here

NPZ Z Zν

⊂ →

Recall Shor’s algorithm:Recall Shor’s algorithm:

AmbientAmbientGroupGroupHiddenHidden

SubgroupSubgroup

TargetTargetSetSet

×

( ) ( )0 0

0

: /0 0N NS Fix Sj Fix j

ι →

So Grover’s algorithm solves the following So Grover’s algorithm solves the following hidden subgroup problem:hidden subgroup problem:

0

0 0/

j

j jN NFix S FS ixν

⊂ →


0/NS Fix

0ν


TargetTargetSetSet

0ι

Approx.Approx.

Voila ! Voila ! Grover’s OracleGrover’s Oracle

pops uppops upTransversalTransversal

( ) ( )0

0

0

0

: / /

0 0N N j

j

f S Fix S Fix

j Fix j Fix

→

is Grover’s oracle. is Grover’s oracle. ( ) ( )

0

0

0

0

: / /

0 0N N j

j

f S Fix S Fix

j Fix j Fix

→

( ) ( )0 0

00

0: 0

if

otherwisejj Fix j j

f j FixFix

=

0ifotherwi

:0 se1 j j

f j=

ClaimClaim:: The following The following

For this simplifies to:For this simplifies to:

which is informationally the same as:which is informationally the same as:

Hence, we can conclude that Hence, we can conclude that Grover’s algorithmGrover’s algorithmis a is a Quantum algorithmQuantum algorithm which which solves a hidden solves a hidden subgroup problem.subgroup problem.

Is Grover’s Algorithm a QHS Algorithm Is Grover’s Algorithm a QHS Algorithm ??????

But !!!But !!!

But …But …

The The standardstandard nonnon--abelian QHS algorithm abelian QHS algorithm on on SSNN find the find the hidden subgrouphidden subgroupFixFixjj00

for the following two reasons:for the following two reasons:

•• The subgroups The subgroups FixFixjj are not normal are not normal subgroups of of subgroups of of SSNN. More importantly, . More importantly, the largest normal subgroup of the largest normal subgroup of SSNN lying lying in in FixFixjj is the trivial subgroup. is the trivial subgroup.

•• The subgroups The subgroups FixFix00, … , Fix, … , FixNN--11 are are mutually conjugate subgroups of mutually conjugate subgroups of SSNN

cannotcannot

Moreover, …Moreover, …

Moreover, it is not possible for the standard Moreover, it is not possible for the standard nonnon--abelian QHS algorithm on Sabelian QHS algorithm on SNN to find jto find j00. .

For if it did, it would find For if it did, it would find jj00 exponentially exponentially faster than Grover’s algorithm. But that is faster than Grover’s algorithm. But that is impossible since Zalka has shown that Grover’s impossible since Zalka has shown that Grover’s algorithm is optimal.algorithm is optimal.

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The Shor and Grover algorithms are very The Shor and Grover algorithms are very close relatives.close relatives.

ConjectureConjecture: : There is an extension of the There is an extension of the standard nonstandard non--abelian hidden subgroup abelian hidden subgroup algorithm onalgorithm on SSNN which is Grover’s algorithm.which is Grover’s algorithm.

Conclusion Conclusion ??????

Simplifying theSimplifying theNonNon--Abelian Abelian

Quantum Hidden Quantum Hidden ProblemProblem

This material can be found This material can be found in the following papersin the following papers

•• Lomonaco & Kauffman,Lomonaco & Kauffman, Quantum Hidden Subgroup Quantum Hidden Subgroup Algorithms: An Algorithmic Toolkit, Algorithms: An Algorithmic Toolkit, 2006,2006,http://xxx.lanl.gov/abs/quanthttp://xxx.lanl.gov/abs/quant--ph/0607047/ph/0607047/

•• Lomonaco & Kauffman,Lomonaco & Kauffman, Quantum Hidden Subgroup Quantum Hidden Subgroup Algorithms on Free groupsAlgorithms on Free groups, in preparation, in preparation

A Simplification ofA Simplification ofNonNon--Abelian Quantum Hidden Subgroup Abelian Quantum Hidden Subgroup

Probems & AlgorithmsProbems & Algorithms

Simplification:Simplification:

We have shown that we need only study We have shown that we need only study nonnon--abelian hidden subgroup problems abelian hidden subgroup problems and algorithms on and algorithms on FreeFree GroupsGroups !!!!!!

Free GroupsFree Groups

( )0 1 1, , , :nF x x x −= …

A group A group FF is is freefree if the only relations if the only relations among its generators are those required among its generators are those required for for FF to be a group to be a group

1 1i ix x − =•• Allowed:Allowed:

,i j j ix x x x i j= ≠•• Not Allowed:Not Allowed:

3 1ix =•• Not Allowed:Not Allowed:

NonNon--Abelian Hidden Subgroup Problem (HSP)Abelian Hidden Subgroup Problem (HSP)

K G Sϕ→ →

We assume that the group We assume that the group G/KG/K is finite.is finite.

HiddenHiddenNormalNormalSubgpSubgp

f.g. Nonf.g. Non--Abel.Abel.Ambient GpAmbient Gp

Map with Map with Hidden SymmetryHidden Symmetry TargetTarget

SetSet

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NonNon--abelian Hidden Subgroup Problem (HSP)abelian Hidden Subgroup Problem (HSP)

ResultResult 1.1. Every nonEvery non--abelian HSP on a group abelian HSP on a group GG can be can be lifted in polytime to a HSP on a f.g. free group lifted in polytime to a HSP on a f.g. free group FF..

( )1 K F

K G Sϕ

ηη

− →↓ Φ

→ →

ResultResult 2.2. A solution to the above HSP on A solution to the above HSP on FF can can then be used to create a solution to the HSP on then be used to create a solution to the HSP on GGby applying the by applying the ReidemeisterReidemeister--Schreier theoremSchreier theorem. . This is done in polytime.This is done in polytime.

ConclusionConclusion.. We need only consider the We need only consider the nonnon--abelian HSP on f.g. free groups.abelian HSP on f.g. free groups.

We have also shown that the best way to We have also shown that the best way to lift an hidden subgroup algorithm is to lift an hidden subgroup algorithm is to use a use a 22--sided Schreier Transversalsided Schreier Transversal..

NonNon--abelian Hidden Subgroup Problem (HSP)abelian Hidden Subgroup Problem (HSP)

Hence, we consider the HSG problems of the formHence, we consider the HSG problems of the form

( )0 1, , nF x x S

Qι ν

−= →↑↓…

ϕ

ϕ ϕ ι=

Approx.Approx.GroupGroup

22--sidedsidedSchreier Schreier

TransversalTransversal

AmbientAmbientFree GroupFree Group

ApproxApproxMapMap

TargetTargetSetSet

What is a Schreier Transversal?What is a Schreier Transversal?

•• Free GroupFree Group( )0 1, , :nF x x −= =…

•• Examples of words inExamples of words in F1 1 1 1 1 1 1 1

2 1 1 1 5 5 5 5 2 1 5 5x x x x x x x x x x x x− − − − − − − −=WordWord Reduced Reduced

WordWord

•• Typical wordTypical word

where where 1 2 kw a a a=

1jj ka x j±= ∀

1Ki ''w K'w K

TransversalsTransversals Given a group and a normal subgroup ,Given a group and a normal subgroup ,we can construct a new group , called the quotient of we can construct a new group , called the quotient of

F K/F K

( )0 1 1, , , nF F x x x−= …K 'w K ''w K

5 7/ , ,G K K g K g K=

ιTransversalTransversal

11ww’’

w’w’’’

What is a Schreier Transversal?What is a Schreier Transversal?

A A 22--sided Schreier transversalsided Schreier transversal for infor inis a transversal is a transversal

such that such that

K F: /F K Fι →

( )wK wK wι =

1)1) is a reduced word is a reduced word ( ) 1 2 kw K w a a aι = =

2) The word formed by removing2) The word formed by removingthe the rightmostrightmost symbol is a coset representative.symbol is a coset representative.

1 2 1R tw a a a −=ta

3) The word formed by removing3) The word formed by removingthe the leftmostleftmost symbol is a coset representative.symbol is a coset representative.

2 1L t tw a a a−=1a

•• Free GroupFree Group

•• subgroup of subgroup of ( )0 1, , nF F x x −= …

FK

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•27

The EndThe End

quantum algorithms beyond · shor’s algorithm (cont.) repeat steps 1 through 5 until there are...

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