Adiabatic Evolution Algorithm

Adiabatic Evolution AlgorithmChukman So (19195004) Steven Lee (18951053)

December 3, 2009 for CS C191

In this presentation we will talk about the quantum mechanical principle of a new quantum computation technique, based on

the adiabatic approximation. Two examples of its application is introduced, and its equivalence to tradition unitary-based

quantum computer is demonstrated.

Adiabatic Evolution Algorithm• Formulation

– For a Hamiltonian H– Characterised by a some parameter λ (think box size in particle-in-box problem)– Solve the eigensystem

• Adiabatic approximation– If the parameter is t, does not give the right evolution– e.g. Start from certain , at a later time, may not be – But if “slow enough”, this is approximately true

• If is a ground state of the initialtime evolution will yield the ground state of

( ) ( ) ( ) ( )n n nH E

time evolution( 0) ( )n nt t

( 0)n t ( 0)H t ( )H t

Quantum Computation by Adiabatic Evolution, E. Farhi, J Goldstone et al, Los Alamos arXiv 0001106

( )n t

3( 0)t 3( )t

• A tool to obtain the fulfilling assignment to a clause– E.g. Solve A OR B

– Start with some initial ground state

– We need a final with the fulfilling assignment as lowest energy state

(this may seem useless, but clauses like this can be added = AND’ed)

– By slowly varying H(t) from t = 0 to T, the initial ground state can be evolved into a fulfilling final state

– But how slow?

1 00 00 0 01 01 0 10 10 0 11 11 00 00H

Adiabatic Evolution Algorithm

Violating assignmentEnergy = 1

Fulfilling assignmentEnergy = 0

A B

( 0) iH t H

( ) fH t T H

0 0( 0) where is the ground state of it H

Adiabatic Approximation• For a time-dependent Hamiltonian H(t)

– Time-dependent Schrodinger equation

– For any given time, instantaneous eigenstates can be found

– We can always expend instantaneously using these kets, treating t just as a parameter

( ) ( ) ( )H t t i tt

( ) ( ) ( ) ( )n n nH t t E t t

( )t

0( ') '

( ) ( ) ( )t

ni E t dt

n nn

t A t t e

Introduced without lose of generality

Introduction to Quantum Mechanics, Bransden & Ostlie 2006

Adiabatic Approximation– The exact functional form of is governed by TDSE; to make use of it we need

– Putting into TDSE, two terms cancel, leaving

( )nA t

0( ') '

( ) ( ) ( ) ( ) ( )t

ni E t dt

n n nn

H t t e A t E t t

0( ') '

( ) ' ( ) ( ) ( ) ( ) ( ) ( ) ( )t

ni E t dt

n n n n n n nn

it e A t t A t t A t t E tt t

0 0

0 0

0

( ') ' ( ') '

( ') ' ( ') '

( ( ') ( ')) '

( ) ' ( ) ( ) ( ) ( ) ( )

' ( ) ( ) ( ) ( )

' ( ) ( ) ( ) ( )

t tn n

t tm n

tn m

i iE t dt E t dt

m n n m n nn n

i iE t dt E t dt

m m n nn

i E t E t dt

m n m nn m

t e A t t t e A t tt

A t e t e A t tt

A t A t e t tt

Need to find this: TISE

Adiabatic Approximation– To find for m ≠ n, we differentiate TISE on both sides by t

– Putting this back, we have

– So far everything is exact – no approximations

( ) ( )m nt tt

( ) ( ) ( ) ( )

for

1 1

n n n

nm n n m n n n

m n m m n n m n

m n m n m nn m mn

H t t E t tt t

EH H Et t t t

H E E m nt t t

H Ht E E t t

0'1' ( ) ( )

tmni dt

m n m nn m mn

HA t A t et

Adiabatic Approximation

– Adiabatic Approximation• Assuming the initial wavefunction is a pure eigenstate i

only one , all other zero• Assuming (a priori) at later time, other amplitudes stay small

i.e. for all time, all other(justified later by looking at the evolution)

– Then we can simplify:

– Integrating with time:

0'1' ( ) ( )

tmni dt

m n m nn m mn

HA t A t et

( 0) 1iA t

( ) 1iA t 0

0'1' ( )

tfii dt

f i f ifi

HA t et

'

0( '') ''

0

1 ( ')( ) '( ') '

tfit i t dt

f i f ifi

H tA t dt et t

Adiabatic Approximation

– Now we can try to justify our a priori assumption– a crude way to approximate the order of this integral:

ignore time dependence

'

0( '') ''

0

1 ( ')( ) '( ') '

tfit i t dt

f i f ifi

H tA t dt et t

( ) '

0

( )

222 ( )

2 4

2

2 4

2

2 4

1 ( )( ) '( )

1 ( ) 1( ) ( )

1 ( )( ) ( ) 1( )

2 ( ) 1 cos( ( ) )( )

4 ( )( )

fi

fi

fi

t i t tf i f i

fi

i t t

f ifi fi

i t tf f i f i

fi

f i fifi

f ifi

H tA t dt et t

H t et t i t

H tP t A t et t

H t t tt t

H tt t

Adiabatic Approximation– i.e. For our a priori assumption to work, we require

– For adiabatic approximation to work, T must be large enough / ramp slow enough

2

2 4

2

2 4

4 ( )( ) 1 ,( )

4 ( ) 1 Putting back ( )

f f ifi

f i fi f ifi

H tP t f tt t

H t E Et t

T is the total ramp time from i to f state2

2

( )

1 where ( )( )

( )

( )

f i

f i

f i

f i

Hd tt

E E dt T

HT

E E

Adiabatic Approximation– This measure is important

• Determines how fast the computation can be performed• Since

the smaller the gap is, the more likely a transition is• The 1st excited state dominates• T chosen wrt. smallest gap during evolution

• If states cross & matrix element non-zero → computation fail

which makes choosing the initial important

2

1( )f i

TE E

iH

( )f i

H

What is SAT?• Boolean satisfiability problem (SAT)

– Clause: A disjunction of literals

– Literal: a variable or negation of variables• Basically a huge Boolean expression, which we try to find a valid set of values

for the variables to make the given problem TRUE overall• Adiabatic approximation setup:

– N-bit problem maps to n variables; use time evolution to solve for problem• SAT is NP-complete

1 2 MC C C

1 2kC x x

NP-complete• Nondeterministic polynomial time (NP)

– Verifiable in polynomial time by deterministic Turing machine– Solvable in polynomial time by nondeterministic Turing machine

• NP-complete is a class of problems having two properties:– Being NP– Problem (in class) can be solved quickly (polynomial time) → all NP problems can

be solved quickly as well• Showing that a NPC problem reducing to a given NP problem is sufficient to

show the problem is NPC• P != NP? So far most believe that is not the case, thus NP-complete problems

are at best deterministically solvable in exponential time

SAT quantum algorithm• Create a time-dependent Hamiltonian which is a linear ramp between the

initial/starting Hamiltonian and final/problem Hamiltonian– Idea is to, given enough time T, to slowly evolve the initial ground state (easy to

find) to final ground state (hard to find)

• Note n-bit SAT problems mean that the Hamiltonian we are working with exist in a Hilbert space spanned by N = 2n basis vectors

• Thus finding ground state of problem Hamiltonian in general requires exponential time

• Adiabatic approximation efficiency all depends on T, which is related to gmin

1

1

i f

i f

t tH t H HT T

H s s H sH


Initial Hamiltonian Hi• Set up an initial Hamiltonian whose ground state is easy to find

• Noticing that 3-SAT is equivalent to SAT:

12

0 11

1 0

1 11 10 01 12 2

k k ki x x

ki k k

k k

H with

H x x x x x

x and x

,

,

c c ci j ki C B B B

i i CC

H H H H

H H

Initial Hamiltonian Hi• Ground state for HB is xk = 0 for all kth bit

• Reason why we use a ground state in the x-axis instead of the z-axis is to prevent gmin from becoming zero, else adiabatic approximation fails

1 2

1 2 1 2/2

1

10 0 02

n

n nnz z z

nk

i k ik

x x x z z z

H d H

Problem Hamiltonian Hf

• Energy function of clause C: 0 if the bits satifsy the clause, else 1• Total energy can be defined as sum of individual HC’s• Hf can be defined as follows:

• Ground state is solution to SAT problem• If no solution exists, will minimize number of violated clauses (lowest energy)

, ,C C CC i j kh z z z

, 1 2 1 2

,

, ,C C Cf C n C i j k n

f f CC

H z z z h z z z z z z

H H

1-bit problem• Consider a problem with one 1-bit clause satisfied with 1 bit

• Setup time-dependent ramped Hamiltonian

• Eigenvalues:

1

1 12 2

1 12 2

i i

i

H H

H

0 0

1 00 0

f

f

H

H

1

1 111 12

i fH s s H sH

s sH s

s s

2 1 (1 ) 02

1 1 2 (1 )2

E E s s

s sE

1-bit problem

0

1 1 2 (1 )2

s sE

2

1min 2

1 2 (1 )

2 2 112

E s s

E s s

g when s

11 1 2 (1 )

2s s

E


Grover Problem• A quantum search problem

– Locate a specific entry in unstructured database– Using the following notation for states

– Given a quantum oracle Hamiltonian

– To find , we start with an initial Hamiltonian for which ground state is known

0

0

0 0

1 if0 if

1

f

f

H

H

1 2 3

...

...z z z znm m m m

0

A total of n bits, each a spin measure in z

Lowest energy state

0

0 11 if

20 if

1

n

i

i

hH

h

H h h

i.e. Lowest state = Hadamard state

How fast is Adiabetic Quantum Computation?, W. van Dam, M. Mosca et al, Los Alamos arXiv 0206003

Grover Problem– Linear ramp between the two Hamiltonian:

– Using Adiabatic Approx.• Solve instantaneous eigenvalues• Find out two lowest states separation• Get the bound on ramping rate

0 0

0 0

( ) 1 1

(1 )(1 ) (1 )

1 (1 )

i f i ft tH t H H s H s HT Ts h h s

s h h s

where tsT

2 2

( )1

( )

f i

f i

H

TE E E

Grover Problem• Solve instantaneous eigenvalues

– dot both sides with and

0 0(1 )

E H

E s h h s

h 0

0 0

0 0 0

0 0

0 0

0 0

2

0

( 1) (1 )

( 1) (1 )

( )( 1 ) (1 )

1( )1

( )( 1 ) (1 )

E h s h s hE s h h s

E s h s hE s s h h

sE s h s h h hE s

E s E s s s h

need to solve this

or 0h corresponding to E=1 roots

Grover Problem– Where the dot product is well defined:

– Therefore the eigenvalues are given by as

0 1 2

1 2/2

/2

0 1 0 1 0 1... ...

2 2 21 0 1 0 1 ... 0 1

2

12

z z zn

z z znn

n

h m m m

m m m

2

0 h

mz1 must be either 0 or 1

2

2

(1 ) (1 )2

(1 )(1 2 ) 0

1 1 4 (1 )(1 2 ) or 1

2

n

n

n

E E s s s s

E E s s

s sE

2

0( )( 1 ) (1 )E s E s s s h

Grover Problem• Eigenvalue spectrum, from n=2 to 20• Against s (or time)

1 1 4 (1 )(1 2 )2

ns sE

1 1 4 (1 )(1 2 )2

ns sE

1E

ground state

1st excited state

n-2 degenerate states

Energy

s

1 4 (1 )(1 2 )n

E

s s

Grover Problem– i.e.

where 1 4 (1 )(1 2 )nE s s

2 2

( )1

( )

f i

f i

H

tE E E

E

s

1 4 (1 )(1 2 )nE s s

Grover Problem– Allowing the ramping rate to adjust to the gap

– Compared with conventional search,

i.e. quantum quadratic speed up

/22nquantumT

total bit combinations 2nconventionalT

1

20

1

0

1 1/2

0 1/22 2

1/2 1/2

2 20 02

1 /2

1

11 4 (1 )(1 2 )

1 11 1 11 14 4 4

2 1 2 11 1

4

2 1tan 22 8

s

ns

s u

u u

n

T t dsE

dss s

ds dus s u

du duuu

214

4 1 2 4n

u s

2 1 1 1 1 2

4 4 2 1 44 1 2

n

nn

Grover Problem• Choice of initial Hamiltonian Hi is important

– Bad choice changes gap dependence → longer ramp time– e.g if we choose

– Eigenvalues calculation in quant-ph/0001106

( )

1

1 12

ni

i xi

H

Energy

s

1st excited state

ground state

2nquantumT

No quantum speed up


Approximating adiabatic with unitaries• Discretize the interval 0 to T into M intervals

– Unitary written as product of M factors

• Note that we want to make the intervals small enough so that the Hamiltonian is near-constant in each discrete interval

0 0, ,

,0 0

,0 , , 2 ,0

di U t t H t U t tdt

T U T

U T U T T U T T U

1 2 1 21 , , 1

1 , i H l

if H t H t t t l lM

then U l l e

/where T M


Approximating adiabatic with unitaries• Then we substitute the Hamiltonian with the ramped Hamiltonian between the

initial and final Hamiltonians

• M is T times polynomial in n• Trotter formula for self-adjoint matrices:

• Thus n in the above equation needs to be large enough to be used as sufficient approximation

1f iH H

( ) / /limnA B A n B n

ne e e

Approximating adiabatic with unitaries

• Then by using a large K, we can approximate using Trotter formula:

2

//

1

fi

i f

Ki vH Ki H l i uH K

if K M H H

then e e e

1 ,i fH l uH vH where u l T v l T

Approximating adiabatic with unitaries• Thus the whole equation can be written in 2K terms, half being each of these

terms:

• Hi is sum of n commuting 1-bit operators, so related unitary can be written as product of n 1-qubit unitary operators

• Hf is sum of commuting operators (each for each clause), so related unitary can be written as product of unitary operators, each acting only on qubits related to clause

• Thus total number of factors is T2 times polynomial in n– If T is polynomial as well, then number of factors is also polynomial

fi i vH Ki uH Ke or e

Conclusion• We have talked about:

– Physical principle of quantum adiabatic evolution algorithm– Its equivalence to traditional unitary quantum computation– Its application in two examples: a one-bit SAT problem, and Glover problem

• Much like tradition QC– Adiabatic evolution leads to quantum speed up in specialised problems– “Smartness” is needed

• picking unitary vs picking initial Hamiltonian– No general rule

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Adiabatic Evolution Algorithm