Taiwan Adiabatic

8/8/2019 Taiwan Adiabatic

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Dorit AharonovDorit AharonovHebrew Univ. & UC BerkeleyHebrew Univ. & UC Berkeley

Adiabatic Quantum ComputationAdiabatic Quantum Computation


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Ground State Solutions

Which spin distributionminimizes the number of

red edges with similar spins

and green edges withopposite spins?

(1 violation.)

1) A combinatorial minimization problem.2) A lowest energy question for magnetic materials.

The ground state of the magnet is the solution to

our optimization problem.


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Language of Hamiltonians.Language of Hamiltonians.

New approach to designing quantumNew approach to designing quantum

algorithmsalgorithms

Equivalent in power to quantum ckts.Equivalent in power to quantum ckts.

Natural fault-tolerance propertiesNatural fault-tolerance properties

Properties of Adiabatic ComputationProperties of Adiabatic Computation


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The Conventional Model ofThe Conventional Model ofQuantum ComputersQuantum Computers

InputInput

UU11

..

UU55

UU44

UU33

UU22

)0(|)(| 11 UUUL LL =

01...011|)0(| =

Output:Output:measuremeasure

Quantum Computing ofQuantum Computing of Classical functionsClassical functionsQuantum statesQuantum states


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)(|)()(|

ttHidt

td

=

Schrodingers Equation:Schrodingers Equation:

Ground StatesGround States

Ground state:Ground state:Eigenvector with lowest eigenvalueEigenvector with lowest eigenvalue

The Hamiltonian (A Hermitian Matrix)The Hamiltonian (A Hermitian Matrix)

Eigenvectors (eigenstates)Eigenvectors (eigenstates)Eigenvalues (Energies)Eigenvalues (Energies)

j|

jE

=lk

lk tHtH,

, )()(


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Classical OptimizationClassical Optimizationin terms ofin terms of

Quantum statesQuantum states

=

)(

.

.

.

)(

111

000

xf

xf

H

GivenGiven: f: {0,1}: f: {0,1}nn N, f(x) for x=xN, f(x) for x=x11,..x,..xnn,,

ObjectiveObjective: find x: find xminmin which minimizes fwhich minimizes f

are the eigenvectorsare the eigenvectorsf(x) are the eigenvaluesf(x) are the eigenvalues

The answer = state with minimal eigenvalueThe answer = state with minimal eigenvalue

x|


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Special Quantum StatesSpecial Quantum States[AharonovTa-Shma02][AharonovTa-Shma02]

1. Graph Isomorphism1. Graph Isomorphism 2. Closest Lattice Vector2. Closest Lattice Vector

0

v2v1

vv

n

S

nG

)(|!

1

As well as Factoring, Discrete Log [ATaShma02]


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Apply a Hamiltonian with the desiredApply a Hamiltonian with the desiredground stateground stateAND.AND.

??Adiabatic Computation

A method to help the system reacha desired groundstate


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Adiabatic theorem:[BornFock 28, Kato 51][BornFock 28, Kato 51]

Ground state of H(0) ground state of H(T)

)(|)()(|

ttHidt

td

=

Adiabatic EvolutionAdiabatic Evolution

)(| T

H(0)H(0) H(T)H(T))0(|

2)}({m in

1

ts

T

> > )()()( 01 tEtEt =


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Adiabatic Systems as Computation DevicesAdiabatic Systems as Computation Devices

)0(| )(| T

InputInput OutputOutputAlgorithm:Algorithm:

HHTT Hamiltonian with ground state |Hamiltonian with ground state |

(T)(T)ii

HH00 Hamiltonian with known ground state |Hamiltonian with known ground state | (0)(0)IISlowly transform HSlowly transform H00 into Hinto HTT

Efficient: T< nEfficient: T< ncc i.e.i.e. cns1

)( >

TsHHssH += 0)1()(

HH00HHTT


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Remark 1:Remark 1: Non Negligible Spectral GapsNon Negligible Spectral Gaps

)(poly1)(

ns >

Physics:Physics:Periodic Hamiltonians, nPeriodic Hamiltonians, n > const> const oror 00

Adiabatic computation:Adiabatic computation:Tailored Hamiltonians , nTailored Hamiltonians , nThe interesting line isThe interesting line is

Allow it to go to zero if sufficiently slowly.Allow it to go to zero if sufficiently slowly.


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Remark 2:Remark 2: Connection to Simulated AnnealingConnection to Simulated Annealing

Adiabatic Rapidly mixingAdiabatic Rapidly mixingComputation Markov ChainsComputation Markov ChainsHamiltonianHamiltonian Transition rate matrixTransition rate matrixGroundstateGroundstate Limiting DistributionLimiting DistributionSpectral gapSpectral gap Spectral gap for rapid mixingSpectral gap for rapid mixing

)(|)()(|

ttHidt

td

=

Quantum Simulated AnnealingQuantum Simulated Annealing

)0(|)(| T

HH00HHTT

R k 3


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Remark 3:Remark 3:

Adiabatic OptimizationAdiabatic Optimization [FGGS00][FGGS00]

Adiabatic ComputationAdiabatic Computation [ADKLLR03][ADKLLR03]

Without increasing the physical resources:

||)(}1,0{

xxxfHnx

T

= =ji

jiT HH,

,

Diagonal HDiagonal HTT

Final state is a basis stateFinal state is a basis state

General Local HGeneral Local HTT

Final state isFinal state isthe groundstatethe groundstate

of a local Hamiltonianof a local Hamiltonian


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A Natural Model of ComputationA Natural Model of Computation

Adiabatic ComputationAdiabatic ComputationThe set of computations that can be performed byThe set of computations that can be performed byQuantum systems, evolving adiabatically under theQuantum systems, evolving adiabatically under the

action local Hamiltonians with non negligibleaction local Hamiltonians with non negligible

spectral gaps.spectral gaps.

What is theWhat is the

computational power ofcomputational power ofAdiabatic ComputersAdiabatic Computers

??

What are theWhat are the

possible dynamics ofpossible dynamics ofAdiabatic systemsAdiabatic systems

??


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OverviewOverview

1 Adiabatic Computation

2 Previous Results AdiabaticOptimization

3 Main Result:

Adiabatic Computers Can perform anyQuantum Computation

4Adding Geometry: True even if the adiabatic computation is on2 dim grid, nearest neighbor interactions

Implications and Open Questions


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2.

Examples:Adiabatic Optimization

2.

Examples:Adiabatic Optimization


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Adiabatic Algorithms for OptimizationAdiabatic Algorithms for Optimization

||)(}1,0{ xxxfH nx

T

=

GivenGiven: f: {0,1}: f: {0,1}nn N, f(x) for x=xN, f(x) for x=x11,..x,..xnn,,

ObjectiveObjective: find x: find xminmin which minimizes fwhich minimizes f

min|)(| xT =

[FarhiGoldstoneGutmanSipser00].[FarhiGoldstoneGutmanSipser00].

f(x) is number of unsatisfied clausesf(x) is number of unsatisfied clauses

...)()()...( 7423211 = xxxxxxxxF n

Energy Penalty: Project on Unsatisfying values of xEnergy Penalty: Project on Unsatisfying values of x

=c

)(Clauses

cHTH ....7,4,23,2,1 101|000|++=


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Adiabatic Algorithms for Optimization (Contd)Adiabatic Algorithms for Optimization (Contd)

||)(}1,0{

xxxfHnx

T

=

2

1|0|

2

1|0|

2

1|0|.....)0(|

+++=

min|)(| xT =

[FarhiGoldstoneGutmanSipser00].[FarhiGoldstoneGutmanSipser00].

TsHHssH += 0)1()(HH00

HHTT

?)()(poly

1n

s >

))((1 2

|1|0

2

1|0|

0 =

=

n

j

jH

20 bits: promising simulation [Farhi et al.00,01] Mounting evidence that (s) is exponentially small in worst case

[vanDamVazirani01, Reichhardt03].

Quadratic speed up: Adiabatic algorithm to solve NP in 2n. Classical NP

algorithm: 2n [RolandCerf01,vanDamMoscaVazirani01]

lT li


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Tunneling:Tunneling:Simulated Annealing vs Adiabatic OptimizationSimulated Annealing vs Adiabatic Optimization

[FGGRV03][FGGRV03]

))((1

2

|1|0

2

1|0|

0

=

=

n

jjH

|11|1

=

=n

j

jTH

E(x)E(x)

w(x)w(x)00 nn

0|....0|0||)(|min

== xT

2

1|0|

2

1|0|

2

1|0|.....)0(|

+++=

n

x

xwxE

0....00

s1'ofNumber)()(

min =

==

E(x)E(x)

w(xw(x00 nn

Adiabatic optimization isAdiabatic optimization isExponentially faster thanExponentially faster than

simulated annealing!simulated annealing!But finding 0 is easy.But finding 0 is easy.


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3.

How to Implement any Quantum

Algorithm

Adiabatically

3.

How to Implement any Quantum

Algorithm

Adiabatically

R s lts t [AT Sh 02 A02 A D K L d Ll dR 03][AT Sh 02 A02 A D K L d Ll dR 03]


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Resultesu t [ATaShma02,A02,AvanDamKempeLandauLloydRegev03][ATaShma02,A02,AvanDamKempeLandauLloydRegev03]

All of Quantum Computation can be doneAll of Quantum Computation can be done

adiabatically!adiabatically!

Unitarygate

s

Unitarygat

esSpe

ctralgaps,

Spectralgaps,

Eigenstates

Eigenstates Condensed matter &Condensed matter &Mathematical PhysicsMathematical Physics

Implication for QuantumImplication for Quantumcomputationcomputation::Equivalence: New Language, new tools !Equivalence: New Language, new tools !New vantage point to tackle the challenges of quantum computation:New vantage point to tackle the challenges of quantum computation:

1.1. Designing newDesigning new algorithmsalgorithms: change of langauge, new tools.: change of langauge, new tools.

2.2. Adiabatic Computation is resilient to certain types of errorsAdiabatic Computation is resilient to certain types of errors[ChildsFarhiPreskill01][ChildsFarhiPreskill01] Possible applications forPossible applications forfault tolerance.fault tolerance. (2-dim architecture)(2-dim architecture)

Implications for PhysicsImplications for Physics::

Understanding ground states, Adiabatic Dynamics fromUnderstanding ground states, Adiabatic Dynamics froman information ers ective.an information perspective.


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Want to constructadiabatic computationwith (t)>1/Lc fromwhich we can deduce

the answer.

1...0110|)(|

,,

1

1

UUL

UU

L

L

=

H(0)H(0)H(T)H(T)

First try: Make the ground state of H(T).)(| L

)(| L

Problem: To specify such aHamiltonian

we need to know !

Whats the Problem?Whats the Problem?

Local unitary gatesLocal unitary gates

UU11

..UU55UU44 UU33 UU22


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Key IdeaKey Idea

Instead of , use a local Hamiltonian H(T)whose ground state is the History.

)(| L

Correct History can beCorrect History can bechecked locally.checked locally.

Classical computation:Classical computation:

Kitaev99, based on Feynman:Kitaev99, based on Feynman:

TimeTimestepssteps

)(| k

)0(|

)1(|

::


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Key IdeaKey Idea

Instead of , use a local Hamiltonian H(T)whose ground state is the History.

)(| L

=

kLk

k 0..001..11||

=

=+

kkhistoryL

kL

|)(||01

1

Correct History can beCorrect History can bechecked locally.checked locally.

Classical computation:Classical computation:

Kitaev99, based on Feynman:Kitaev99, based on Feynman:TimeTimestepssteps


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The Hamiltonian H(s)The Hamiltonian H(s)

1,,1|100110||1| + kkkkk

== 0..0|0..01|0..0|)0(|)0(|

1|00|

= =+ kkTL

kL |)(|)(|

01

1

Test that input is 0Test that input is 0

Test correctTest correct

propagation:propagation:Energy penaltyEnergy penalty

HHT:T:

HH0:0:

|1||1|

|11|||

021

kkUkkU

kkIkkIH

HH

kk

k

L

k

kT

+ =

=

+

=

Local interaction:Local interaction:

== + =L

k

k

n

jH11

0 |11||11|


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4.

Adding Geometry:

Adiabatic Computationon a

Two-D Lattice

4.

Adding Geometry:

Adiabatic Computationon a

Two-D Lattice


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Particles on a 2-d LatticeParticles on a 2-d LatticeWantedWanted::

Evolution of the formEvolution of the formProblem:Problem:

Not enough interaction between clock and computerNot enough interaction between clock and computer

to have terms like:to have terms like:

Solution:Solution:Relax notion of computation/clock particles.Relax notion of computation/clock particles.

Each particle will have both types of degrees of freedom.Each particle will have both types of degrees of freedom.States will no longer be tensor products but will encodeStates will no longer be tensor products but will encodetime in theirtime in theirgeometric shape.geometric shape.

To do this we use a like evolution.To do this we use a like evolution.

Lkkk ,...,0,|)(| =

|1||1||11|||

kkUkkUkkIkkIH

kk

k

+ =+

Th 2 Di L i C iTh 2 Di L tti C t ti


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**

The 2-Dim Lattice ConstructionThe 2-Dim Lattice ConstructionSix states particles:Six states particles:

Unborn First Phase Second Phase DeadUnborn First Phase Second Phase Dead

11

00

11

00 11 000011

RR

nn

00

00** ** ** ** **

** ************

******

** ****

** ****** ****

** ****** ****


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The HamiltonianThe HamiltonianAs before:As before:Check correct propagation by checking eachCheck correct propagation by checking each

move; Each move involves only two particles.move; Each move involves only two particles.

Except:Except: Moves may seem correct locally but are not.Moves may seem correct locally but are not.Space of legal states is no longer invariant.Space of legal states is no longer invariant.

Solution:Solution:Add penalty for all forbidden shapes:Add penalty for all forbidden shapes:

HHclockclock ==

00 00 00000000

Fortunately, can be checked by checking nearest neighbors:Fortunately, can be checked by checking nearest neighbors:

00 0000 00

00 00 000000 0000

To SummarizeTo Summarize


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To SummarizeTo Summarize

Ground states:Ground states:All states are ground states of local Hamiltonians,All states are ground states of local Hamiltonians,

Adiabatic dynamics are general.Adiabatic dynamics are general.

Algorithm DesignAlgorithm Design::

New languageNew language::round states, spectral gaps.round states, spectral gaps.

What states can we reach?What states can we reach?

What statesWhat statesare ground states of local Hamiltonians?are ground states of local Hamiltonians?

Fault ToleranceFault Tolerance::Adiabatic comp. is naturally robust.Adiabatic comp. is naturally robust.

Adiabatic Fault Tolerance?Adiabatic Fault Tolerance?

Methods from Mathematical Physics?Methods from Mathematical Physics?

Saw how to implement any Q algorithm adiabatically.Saw how to implement any Q algorithm adiabatically.


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SlowSlowdown,down,

youyoumovemovetootoo

fastfast

Taiwan Adiabatic

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