Database search: the simplest example of quantum
speed-up
Armen Allahverdyan
(Yerevan Physics Institute)
-- Classical unstructured database search
-- Reminder on quantum mechanics
-- Quantum (Grover's) search
Classical unstructured database search
N items, N-1 identical, 1 special item to be found
selector function
f [ ] 1
f[ ]0
Worst case: search takes N-1 steps. Best case: 1 step.
In average: (N-1)/2 steps
probability to find the red ball randomly is 1/N
Databases are normally structured, but the unstructured
assumption is still reasonable
N
Time
 log N
prior information (structure)
Lot of prior information (structure); the red ball is heavier.
The balls are arranged into regular structure and
can be easily divided into equal parts
search time
 2 log N
Quantum mechanics =
waves+ Hamiltonian mechanics + probability theory
|  ( 1 ,..., N )T ,  | ( 1*,..., N *)
N
 |   k k *
k 1
N
N
k 1
k 1
H   Ek | wk wk |, H |    Ek | wk wk |
wk | wl   kl
| wk | |2
probability
Mapping to Hamiltonian of N-level quantum system
| w1 , 0
| w ,E
| wN 1 , 0
N 1
Hamiltonian
H 0  E | w w |   0 | wk wk |
wk |wl   kl
k 1
{| wk }
is unknown
E is a known parameter
Interaction Hamiltonian and initial state
H  E | w w| E | s s |
|s
known interaction Hamiltonian
switched on at time zero
is a known vector
Schroedinger equation
We want:
i  t | (t )  H | (t ) , | (0)  | s
| (T )  | w
Estimating the characteristic time via short-time expansion
it | (t )  H | s  E(| w w | s  | s ), | (0) | s
| (t )  ei E t /  | s
amplitude determines the change
pure phase
random
known, controlled base
|w
| s1 | s , | s2 , ..., | sN
sk |sl   kl
averages
N 1
| w   sk | w | sk , | s | w |  | s2 | w |  ...  | s N | w |  1 / N
k 1
  1/ | w | s | N
characteristic time
Exact solution of the Schroedinger equation
|w
|s
motion sticks to
two-dimensional Hilbert space
 s' | H | s'
ˆ
h  
 s | H | s'
 s'|
i t 
 s |
| s'
 1 | w | s |2
s' | H | s 
  E
 s | w w | s'
s | H | s 

 ˆ  s'|
h 

 s |






s'| w w | s 

1 | w | s |2 
two linear equations
w | (t )
2
 sin (t /  )  w | s


E | w| s |
2
2
cos 2 (t /  )
characteristic time
Tquantum  N
w |  (  / 2)  1
Tclassical  N
quantum motion is determined by amplitudes, not by probabilities
supersposition principle
Alternative interaction Hamiltonians
H  E | w w |  E | s s |  E | w w |  E' (| s w | | w s |)
it | (t )  H | s  .....  E'| w , | (0) | s


E'
does not depend on N
but you cannot apply this in
search problems
The simplest Hamiltonian is optimal at least for one-body physics