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

Adiabatic Quantum Computation Adiabatic 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 toour optimization problem.

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• Language of Hamiltonians.Language of Hamiltonians.

• New approach to designing quantum New approach to designing quantum algorithmsalgorithms

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

• Natural fault-tolerance propertiesNatural fault-tolerance properties

• Laid back approach! Laid back approach!

Properties of Adiabatic ComputationProperties of Adiabatic Computation

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The Conventional Model of The Conventional Model of Quantum ComputersQuantum Computers

InputInput

UU11

……..

UU55

UU44

UU33

UU22

)0(|)(| 11 UUUL LL

01...011|)0(| Output: Output: measuremeasure

Quantum Computing of Quantum Computing of ““Classical” functionsClassical” functions““Quantum states”Quantum states”

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)(|)()(| ttHi dt

td

Schrodinger’s Schrodinger’s Equation:Equation:

Ground StatesGround States

Ground state:Ground state: Eigenvector with lowest Eigenvector with lowest eigenvalueeigenvalue

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 Optimization in 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=x N, f(x) for x=x11,…..x,…..xnn, , ObjectiveObjective: find x: find xminmin which minimizes f which minimizes f

are the eigenvectorsare the eigenvectorsf(x) are the eigenvaluesf(x) are the eigenvaluesThe answer = state with minimal eigenvalue The answer = state with minimal eigenvalue

x|

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Special Quantum StatesSpecial Quantum States [AharonovTa-Shma’02][AharonovTa-Shma’02]

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

0

v2

v1

vv

nS

nG

)(|

!1

As well as Factoring, Discrete Log… [A’TaShma’02]

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Apply a Hamiltonian with the desired Apply a Hamiltonian with the desired ground state ground state

AND…. AND….

??Adiabatic Computation

A method to help the system reach a 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)

)(|)()(| ttHi dt

td

Adiabatic Evolution Adiabatic Evolution

)(| T

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

2)}({min1

tsT

)()()( 01 tEtEt

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Adiabatic Systems as Computation Devices Adiabatic 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)II• Slowly transform HSlowly transform H00 into H into HTT

Efficient: T< nEfficient: T< nc c i.e.i.e. cns 1)(

TsHHssH 0)1()(

HH00

HHTT

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

)(poly1)( ns

Physics:Physics: Periodic Hamiltonians, nPeriodic Hamiltonians, n∞∞ γγ > const > const or or γγ00

Adiabatic computation:Adiabatic computation: Tailored Hamiltonians , nTailored Hamiltonians , n∞∞ The interesting line is The 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 ChainsHamiltonian Hamiltonian Transition rate matrix Transition rate matrixGroundstate Groundstate Limiting Distribution Limiting DistributionSpectral gap Spectral gap Spectral gap for rapid Spectral gap for rapid mixingmixing

)(|)()(| ttHi dt

td

Quantum Simulated AnnealingQuantum Simulated Annealing

)0(| )(| THH00

HHTT

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Remark 3:Remark 3: Adiabatic Optimization Adiabatic Optimization [FGGS’00][FGGS’00] Adiabatic Computation Adiabatic Computation [ADKLLR’03][ADKLLR’03]

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 is Final state is the groundstate the groundstate

of a local Hamiltonianof a local Hamiltonian

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A Natural Model of ComputationA Natural Model of Computation

Adiabatic Computation Adiabatic Computation The set of computations that can be The set of computations that can be

performed by performed by Quantum systems, evolving adiabatically Quantum systems, evolving adiabatically

under the under the action local Hamiltonians with non negligible action local Hamiltonians with non negligible

spectral gaps.spectral gaps. What is theWhat is the

computational power computational power of Adiabatic of Adiabatic ComputersComputers

??

What are the What are the possible dynamics ofpossible dynamics of

Adiabatic systemsAdiabatic systems

??

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Overview Overview

1 Adiabatic Computation

2 Previous Results Adiabatic Optimization

3 Main Result: Adiabatic Computers Can perform any Quantum Computation

4 Adding Geometry: True even if the adiabatic computation is on 2 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 Optimization Adiabatic Algorithms for Optimization

||)(}1,0{

xxxfHnx

T

GivenGiven: f: {0,1}: f: {0,1}nn N, f(x) for x=x N, f(x) for x=x11,…..x,…..xnn, , ObjectiveObjective: find x: find xminmin which minimizes f which minimizes f

min|)(| xT

[FarhiGoldstoneGutmanSipser’[FarhiGoldstoneGutmanSipser’00].00].

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 (Cont’d) Adiabatic Algorithms for Optimization (Cont’d)

||)(}1,0{

xxxfHnx

T

2

1|0|

2

1|0|

2

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

min|)(| xT

[FarhiGoldstoneGutmanSipser’[FarhiGoldstoneGutmanSipser’00].00].

TsHHssH 0)1()(HH00

HHTT

? )( )(poly1

ns

))((1

2

|1|0

2

1|0|0

n

jjH

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

[vanDamVazirani’01, Reichhardt’03]. • Quadratic speed up: Adiabatic algorithm to solve NP in √2n.

Classical NP algorithm: 2n [RolandCerf’01,vanDamMoscaVazirani’01]

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

[FGGRV’03][FGGRV’03]

))((1

2

|1|0

2

1|0|0

n

jjH

|11|1

n

jjTH

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(x)w(x)00 nn

Adiabatic optimization isAdiabatic optimization isExponentially faster than Exponentially 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

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Result Result [A’TaShma’02,A’02,A’vanDamKempeLandauLloydRegev’03][A’TaShma’02,A’02,A’vanDamKempeLandauLloydRegev’03]

All of Quantum Computation can be done All of Quantum Computation can be done adiabatically!adiabatically!

Unitary gates

Unitary gatesSpectral gaps,

Spectral gaps,

Eigenstates

Eigenstates Condensed matter &Condensed matter &Mathematical PhysicsMathematical Physics

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

1.1. Designing new Designing new algorithmsalgorithms: change of langauge, new tools.: change of langauge, new tools. 2.2. Adiabatic Computation is resilient to certain types of errors Adiabatic Computation is resilient to certain types of errors [ChildsFarhiPreskill’01] [ChildsFarhiPreskill’01] Possible applications for Possible applications for fault tolerance. fault tolerance. (2-dim architecture) (2-dim architecture)

Implications for PhysicsImplications for Physics: : Understanding ground states, Adiabatic Dynamics from Understanding ground states, Adiabatic Dynamics from an information perspective. an information perspective.

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Want to construct adiabatic computation with γ(t)>1/Lc from which 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 a Hamiltonian we need to know !

What’s the Problem?What’s 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 be Correct History can be checked locally. checked locally.

Classical computation:Classical computation:

Kitaev’99, based on Feynman:Kitaev’99, 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||

kkhistory

L

kL

|)(||0

11

Correct History can be Correct History can be checked locally. checked locally.

Classical computation:Classical computation:

Kitaev’99, based on Feynman:Kitaev’99, 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|

kkT

L

kL

|)(|)(|0

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● ● Test that input is 0Test that input is 0

● ● Test correctTest correct propagation: propagation: Energy Energy penalty penalty

HHT:T:

HH0:0:

|1| |1|

|11| || 0

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kkUkkU

kkIkkIH

HH

kk

k

L

kkT

Local interaction:Local interaction:

L

kk

n

jjH

110 |11| |11|

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4.Adding Geometry:

Adiabatic Computation on a

Two-D Lattice

4.Adding Geometry:

Adiabatic Computation on a

Two-D Lattice

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Particles on a 2-d LatticeParticles on a 2-d Lattice WantedWanted: : Evolution of the form Evolution of the form Problem:Problem:

Not enough interaction between clock and Not enough interaction between clock and computercomputer

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 encode States will no longer be tensor products but will encode time in their time in their geometric shape.geometric shape.

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

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

|1| |1|

|11| ||

kkUkkU

kkIkkIH

kk

k

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The 2-Dim Lattice ConstructionThe 2-Dim Lattice Construction Six 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 Check correct propagation by checking

each move; Each move involves only two particles. each 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 Space of legal states is no longer invariant. 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 Fortunately, can be checked by checking nearest neighbors:neighbors:

00 0000 00

00 00 000000 0000

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To SummarizeTo Summarize

Ground states:Ground states:All states are ground states of local All states are ground states of local Hamiltonians, Hamiltonians, Adiabatic dynamics are general.Adiabatic dynamics are general.

Algorithm DesignAlgorithm Design: : New New languagelanguage: : Ground states, spectral gaps.Ground states, spectral gaps.

What states can we reach? What states can we reach? What statesWhat states are ground states of local are ground states of local Hamiltonians?Hamiltonians?

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

Methods from Mathematical Physics? Methods from Mathematical Physics?

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

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SlowSlow down,down, youyou move move too too fast……fast……