adiabatic evolution algorithm
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Adiabatic Evolution Algorithm. Chukman So (19195004) Steven Lee (18951053) December 3, 2009 for CS C191 - PowerPoint PPT PresentationTRANSCRIPT
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
Quantum Computation by Adiabatic Evolution, E. Farhi, J Goldstone et al, Los Alamos arXiv 0001106
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
Quantum Computation by Adiabatic Evolution, E. Farhi, J Goldstone et al, Los Alamos arXiv 0001106
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
Quantum Computation by Adiabatic Evolution, E. Farhi, J Goldstone et al, Los Alamos arXiv 0001106
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
Quantum Computation by Adiabatic Evolution, E. Farhi, J Goldstone et al, Los Alamos arXiv 0001106
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