Adiabatic Evolution Algorithm
Chukman 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
H ( ) n ( )  En ( ) n ( )
• Adiabatic approximation
– If the parameter is t, n (t ) does not give the right evolution
– e.g. Start from certain 3 (t  0) , at a later time, may not be 3 (t )
– But if “slow enough”, this is approximately true
time evolution
n (t  0) 
 n (t )
• If n (t  0) is a ground state of the initial H (t  0)
time evolution will yield the ground state of H (t )
Quantum Computation by Adiabatic Evolution, E. Farhi, J Goldstone et al, Los Alamos arXiv 0001106
Adiabatic Evolution Algorithm
• A tool to obtain the fulfilling assignment to a clause
– E.g. Solve A OR B
– Start with some initial H (t  0)  H i
ground state  (t  0)  0 where 0 is the ground state of Hi
– We need a final H (t  T )  H f with the fulfilling assignment as lowest energy state
AB
H  1 00 00  0 01 01  0 10 10  0 11 11  00 00
Violating assignment
Energy = 1
Fulfilling assignment
Energy = 0
(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?
Adiabatic Approximation
• For a time-dependent Hamiltonian H(t)
– Time-dependent Schrodinger equation

 (t )
t
– For any given time, instantaneous eigenstates can be found
H (t )  (t )  i
H (t ) n (t )  En (t ) n (t )
– We can always expend  (t ) instantaneously using these kets, treating t just as a
parameter
i t
  En ( t ') dt '
 (t )  A (t )  (t ) e 0

n
n
n
Introduced without lose of generality
Introduction to Quantum Mechanics, Bransden & Ostlie 2006
Adiabatic Approximation
– The exact functional form of An (t ) is governed by TDSE; to make use of it we need
i
  En ( t ') dt ' 


 (t )   e 0
A
'
(
t
)

(
t
)

A
(
t
)
n (t )  An (t ) n (t )
n
n
 n
t

t
n

i t

En ( t ') dt '
H (t )  (t )  e 0
A (t ) E (t )  (t )
t

n
n
 i


E
(
t
)
n




n
n
– Putting into TDSE, two terms cancel, leaving
i
t
En ( t ') dt '
m (t )  e 0
  A 'n (t ) n (t )   m (t )

n
i
t
 A 'm (t )e 0

Em ( t ') dt '
0 En (t ') dt '  A (t )   (t ) 
e
n
n
 n

t



i
t
En ( t ') dt ' 



0
 m (t )  e
A
(
t
)

(
t
)
n
 n

t


n

i
t
( En ( t ')  Em ( t ')) dt '


0
A 'm (t )    An (t )e
m (t )
n (t )

t
nm

i
t
Need to find this: TISE
Adiabatic Approximation

n (t ) for m ≠ n, we differentiate TISE on both sides by t
t


H
(
t
)

(
t
)

En (t ) n (t ) 



n
t
t


 E

 H

m 
n  H n   m  n n  En n 
t
t
 t

 t

H


m
n  Em m
n  En m
n
for m  n
t
t
t

1
H
1
H
m
n 
m
n  
m
n
t
En  Em
t
mn
t
– To find m (t )
– Putting this back, we have
A 'm (t )   An (t )
nm
1
mn
i mn dt '
H
m
n e 0
t
– So far everything is exact – no approximations
t
Adiabatic Approximation
A 'm (t )   An (t )
nm
i mn dt '
H
m
n e 0
t
t
1
mn
– Adiabatic Approximation
• Assuming the initial wavefunction is a pure eigenstate i
only one Ai (t  0)  1 , all other zero
• Assuming (a priori) at later time, other amplitudes stay small
i.e. Ai (t )  1 for all time, all other  0
(justified later by looking at the evolution)
– Then we can simplify:
A ' f i (t ) 
1
 fi
i  fi dt '
H
f
i e 0
t
t
– Integrating with time:
i  fi ( t '') dt ''
H (t ')
Af i (t )   dt '
f
i e 0
0
 fi (t ')
t '
t
1
t'
Adiabatic Approximation
i  fi ( t '') dt ''
H (t ')
Af i (t )   dt '
f
i e 0
0
 fi (t ')
t '
t'
1
t
– Now we can try to justify our a priori assumption
– a crude way to approximate the order of this integral:
ignore time dependence
Af i (t )
1
f
 fi (t )
H (t )
i
t

t
0
dt ' e
i fi ( t ) t '
 ei fi (t )t  1 
H (t )

i 
 i (t ) 
 fi (t ) f t
fi


1
Pf (t )  Af i (t )
2
H (t )

i
f
2
4
 fi (t )
t
1
H (t )

i
f
2
4
 fi (t )
t
2
H (t )
 2 4
f
i
 fi (t )
t
2
2
4
1  cos(
2
e
fi
i fi ( t ) t
(t )t ) 
1
2
Adiabatic Approximation
– i.e. For our a priori assumption to work, we require
H (t )
Pf (t )  2 4
f
i
 fi (t )
t
4
H (t )

i
2
 fi 4 (t ) f t
4
H ( )
i

( E f  Ei ) 2
f
1
f , t
2
d
dt
H ( )
i

( E f  Ei ) 2
f
2
1
Putting back  fi  E f  Ei
1
where  (t ) 
t
T
T is the total ramp time
from i to f state
T
– For adiabatic approximation to work, T must be large enough / ramp slow enough
Adiabatic Approximation
– This measure is important
• Determines how fast the computation can be performed
• Since
1
T
( E f  Ei )2
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  f
H ( )
i non-zero → computation fail

which makes choosing the initial H i important
What is SAT?
• Boolean satisfiability problem (SAT)
C1  C2 
 CM
– Clause: A disjunction of literals
Ck  x1   x2
– 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
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)
t

t
H  t   1   H i    H
 T
T 
H  s   1  s  H i  sH f
f
• 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
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
k 
Hi

1
2
1   
k 
k 
with  x
x
0 1


1
0


H i k  xk  x  x xk  x
1  1
xk  0 
  and
2  1
1 1
xk  0 
 
2  1
• Noticing that 3-SAT is equivalent to SAT:
H i ,C  H B c   H B c   H B c 
i
H i   H i ,C
C
j
k
Initial Hamiltonian Hi
• Ground state for HB is xk = 0 for all kth bit
x1  0 x2  0
1
xn  0  n /2
2
  z
1
z1
z2
z2
zn
n
H i   d k H i k 
k 1
• 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
zn
Problem Hamiltonian Hf

hC ziC , z jC , zkC

• 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:
H f ,C  z1 z2
zn
  h z
C
iC

, z jC , zkC z1 z2
zn
H f   H f ,C
C
• Ground state is solution to SAT problem
• If no solution exists, will minimize number of violated clauses (lowest energy)
1-bit problem
• Consider a problem with one 1-bit clause satisfied with 1 bit
H i  H i1
 12
Hi   1
 2
Hf  0 0
 12 
1 
2 
1 0
Hf 

0
0


• Setup time-dependent ramped Hamiltonian
H  s   1  s  H i  sH f
1 1  s s  1
H s  

2  s 1 1 s 
• Eigenvalues:
1
E 2  E  s (1  s )  0
2
1  1  2 s (1  s)
E
2
1-bit problem
E  1  2 s (1  s )
E1 
E  2s 2  2s  1
1  1  2 s(1  s)
2
E0 
g min 
1
2
 when s  12 
1  1  2 s (1  s )
2
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 0
– Using the following notation for states
 


 ... 
 mz1  mz 2  mz 3  ...  mzn
A total of n bits, each a spin measure in z
– Given a quantum oracle Hamiltonian
1 
Hf  
0  
H f  1  0 0
if   0
if   0
Lowest energy state
– To find 0 , we start with an initial Hamiltonian for which ground state is known
n

 0 1 
1  if   
  h
Hi   
2 


i.e. Lowest state = Hadamard state
if   h
0  
Hi  1  h h
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:
t

t
H (t )  1   H i    H f  1  s  H i   s  H f
 T
T 
 (1  s)(1  h h )  s(1  0 0 )
 1  (1  s) h h  s 0 0
– Using Adiabatic Approx.
• Solve instantaneous eigenvalues
• Find out two lowest states separation
• Get the bound on ramping rate
H ( )
i

( E f  Ei ) 2
f
T
1
E 2
where s 
t
T
Grover Problem
• Solve instantaneous eigenvalues
E  H 
E     (1  s) h h   s 0 0 
– dot both sides with h and 0
 ( E  1) h   (1  s ) h   s h 0 0 

( E  1) 0   (1  s ) 0 h h   s 0 

( E  s ) h    s h 0 0 

( E  1  s ) 0   (1  s ) 0 h h 
 ( E  s ) h    s h 0
1  s
0 h h 
E 1 s
 ( E  s )( E  1  s )  s (1  s ) 0 h
2
or h   0
corresponding to E=1 roots
need to solve this
Grover Problem
– Where the dot product 0 h
2
is well defined:
 0 1
0 h   mz1  mz 2  ... mzn  


1
2n /2
 m 0
z1
1
2
  m  0
z2

1
0 1
2
 ... 
  ...  m  0
zn
0 1 

2 
1

mz1 must be either 0 or 1
1
2n /2
– Therefore the eigenvalues are given by ( E  s)( E  1  s)  s(1  s) 0 h

E 2  E  s (1  s )  s (1  s )2 n
E 2  E  s (1  s )(1  2 n )  0
1  1  4 s(1  s)(1  2 n )
E
or 1
2
2
as
Grover Problem
• Eigenvalue spectrum, from n=2 to 20
• Against s (or time)
Energy
E 1
n-2 degenerate states
E 
1  1  4s(1  s)(1  2 n )
E
2
1st excited state
1  4s(1  s)(1  2 n )
1  1  4s(1  s)(1  2 n )
E
2
ground state
s
Grover Problem
– i.e. t
H ( )
i

( E f  Ei )2
f
1
E 2
n
where E  1  4s (1  s )(1  2 )
E
E  1  4s (1  s )(1  2 n )
s
Grover Problem
– Allowing the ramping rate to adjust to the gap
1
s 0 E 2 ds
1
1
s 0 1  4s(1  s)(1  2 n ) ds
1
1/2
1
1

ds  
du
s 0
u

1/2
1
1
1




1    s2  s   
1   u2  
4 4
4


2 1/2
1
2 1/2
1
 
du  
du
 u 0  1 1  2
 u 0  2  u 2
  u
 4 
 1   n /2
2

tan 1 
 2

2

8


T   t
1
2

1

u

s




4


  4 1  2 n  4



1
1 1 1
2 n
 
 

4 4 2n  1
4
4 1  2 n
2


– Compared with conventional search, Tconventional  total bit combinations
Tquantum
2n/2
i.e. quantum quadratic speed up
2n
Grover Problem
• Choice of initial Hamiltonian Hi is important
– Bad choice changes gap dependence → longer ramp time
– e.g if we choose
n
1
H i   1   x(i ) 
i 1 2
– Eigenvalues calculation in quant-ph/0001106
Energy
1st excited state
Tquantum 2n
No quantum speed up
ground state
s
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
d
i U  t , t0   H  t  U  t , t 0 
dt
 T   U T , 0    0 
where   T / M
U  T , 0   U  T , T    U  T  , T  2  U   , 0 
• Note that we want to make the intervals small enough so that the Hamiltonian is
near-constant in each discrete interval
if
H  t1   H  t2 
1
M
t1 , t2  l ,  l  1  
then U   l  1 , l   e iH l  
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
 H f  Hi
1
• M is T times polynomial in n
• Trotter formula for self-adjoint matrices:
 ( A B )
e
 lim  e

 A/ n  B / n n
n
e
• Thus n in the above equation needs to be large enough to be used as sufficient
approximation
Approximating adiabatic with unitaries
H  l    uHi  vH f where u  1  l  T  , v  l  T
• Then by using a large K, we can approximate using Trotter formula:
if
K
then e

M 1   Hi   H f
 iH  l  
e
 iuH i / K  ivH f / K
e


2
K
Approximating adiabatic with unitaries
• Thus the whole equation can be written in 2K terms, half being each of these
terms:
e
 iu H i K
or e
 ivH f K
• 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
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