Grover’s Search Algorithm and Quantum Lower Bounds
Lecture Notes for CS 294-6, Quantum Computing
Lecturer: Umesh Vazirani
Scribe: Jie-Hong Jiang
October 6, Fall 2000
Searching an item in an unsorted database with size N costs a classical computer O(N)
running time. Can a quantum computer search a needle in a haystack much more
efficient than its classical counterpart? Grover, in 1996, affirmatively answered this
question by proposing a search algorithm, which consults the database only O( N )
times. In contrast to algorithms based on the quantum Fourier transform, with
exponential speedups, the search algorithm only provides a square-root speedup.
However, the algorithm is quite important because it has broad applications and the
same technique can be used to improve the solution of NP-complete problems.
One might think of having better improvements over the search algorithm.
Nevertheless, it turns out that Grover’s search algorithm is optimal. At least ( N )
queries are needed to solve the problem. This note details the quantum search algorithm
and its lower bound in Section 1 and Section 2 respectively.
1. Grover’s Search Algorithm
1.1 The quantum oracle
Let f: {0, 1}n  {0, 1}, be a Boolean function. We are given a quantum black box Uf for
computing f:
Uf : x
n
y  x
n
y  f ( x) .
Set y as 0 , we have
Uf : x
n
0  x
n
f ( x) .
As we have seen before, if we initialize y to ( 0  1 )/ 2 , the oracle acts as
1
Uf : x
n
 0 1


2


  (1) f ( x ) x


n
 0 1


2


.


Now suppose that there is a single value k such that f(k) = 1. If f is specified by a
black box, what is the fewest queries we must make to f to determine such k?
1.2 The iterative procedure
On searching an N-item database, Grover’s search algorithm uses the operator D
defined as
2
 2
( N  1)
N
 2
2
(  1)
D N
N


 
2
2

N
 N
2 
N 
2 


N .

 
2
 (  1)
N


D has two properties:
(1) It is unitary and can be efficiently realized.
(2) It can be seen as an “inversion about mean.”
Proof:
(1) For N = 2n, operator D can be decomposed and rewritten as:
2
0
HN 


0
D

1 0
0  1
HN 
 

0 0
0
0

0




0
0
HN  I


0
 0
 0 
HN
 

  1




HN 



  2 0  0 


0 0  0 


HN
I H
       N

 
 0 0  0 

2
2
N
0

0
N
0

0




2 
N 
0 I
 

0 

2
N

2

 N




2
N
 I
2

N 
Observe that D be expressed as the product of three unitary matrices (two Hadamard
matrices separated by a conditional phase shift matrix). Therefore, D is also unitary.
Regarding the implementation, both Hadamard and the conditional phase shift
transforms can be efficiently realized within O(n) gates.
(2) When D operates on a vector  and generates another vector  , that is,
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 1 
 1 
  
  
 
 
D  i     i  ,
 
 
  
  
 N 
  N 
the ith amplitude  i 
2
 j  i  2   i    (   i ) can be considered as an
N j
“inversion about mean” with respect to  i , where  is the average amplitude.

As shown in Figure 1, the operation of D increases (decreases) amplitudes that are
originally below (above) the mean value  .
 i
(a)


i
 i
(b)


i
Figure 1: Effects of D operation. (a) States before operation. (b) States after operation.
The quantum search algorithm iteratively improves the probability of measuring a
solution. In each iteration, this algorithm performs two operations: first consult the
oracle Uf and then apply the “inversion about mean” operator D. The quantum state
evolves as i1 = DUf i along with iteration i to iteration (i+1).
Suppose we are finding one out of N items. In the first step, as shown in Figure 2
(a), we prepare the initial state as a uniform superposition over these N items. In each
iteration, Uf marks the only solution k, f(k)=1, with a phase shift as indicated in (b).
Then D operation amplifies k, the amplitude of the marked item, and suppresses those
of all other items as in (c). Repeating the process before measurement increases the
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probability of measuring k. For example, after the first iteration, k  3
second iteration, k  5
N ; after the
N . More formally, at iteration t, k and l (l = 0, 1, , …, N –1;
l  k) are
 (kt )  (1 
 l(t )  (
Initially,  (k0)   l( 0)  1
2 (t 1)
2
) k  (2  ) l(t 1)
N
N
2 (t 1)
2
) k  (1  ) l(t 1) .
N
N
N . After O( N ) steps, k becomes constant. Therefore, in
the measurement, the probability of observing k becomes constant. Notice that
repeating iterations does not always increase the chance of measuring the right answer.
The amplitude of the marked solution goes up and down as a cycle. If we do not stop at
the right time, we might not have a good chance to measure the correct item.
 i
(a)



i
k
 i
(b)

k


i
 i
(c)



k
i
Figure 2: Finding 1 out of N items. (a) Uniform superposition is prepared initially. Every item has equal
amplitude (1 /
N ). (b) Oracle Uf recognizes and marks the solution item k. (c) Operator D amplifies
the amplitude of the marked item and suppresses amplitudes of other items.
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1
M
k 
x
x f 1 (1)
DU f  0
0
2


u 
1
N M
x
x f 1 ( 0 )
U f 0
Figure 3: Geometric interpretation of the iterative procedure.
1.3 The geometric interpretation
Suppose to find M solutions from a sample space with N entries. We can cluster these
1
items into two orthogonal bases, say k 
 x (collection of the M solutions)
M x f 1 (1)
and u 
1
N M
x
(collection of the remaining items). Hence, Figure 3 can
x f 1 ( 0 )
visualize iterative steps in a single plane spanned by these two vectors. For original
state  0 
0 
1
N
M
N
N 1
x
, it can be rewritten as
x 0
 1

 M


N M
x
1   N
x  f (1) 

1

 N M


M
N M
x
1   N k  N u .
x f ( 0) 
In the oracle consultation, Uf shifts the phase in the k component and therefore
reflects the acted vector about u . Meanwhile, since D is a reflection about 000 in
the Hadamard basis (refer the proof of property (1) of D in page 2), it reflects the acted
vector about 0 . The product of these two operators, DUf, performs an equivalent
2 -rotation operation, where   sin 1
M
N M
. After i such iterations,
 cos 1
N
N
5
the state becomes
DU 
i
f
 0  sin 2 i  1  k  cos 2 i  1  u .
In the special case of finding 1 out of N items (N » 1),   sin   1
N , to
maximize the probability of the correct measurement, the needed number of iterations 
 2  2   4N . Consequently, Grover’s search algorithm makes O(
N ) queries.
Through this visualization, it can be seen that if the number of iterations is not
chosen properly, the final vector might not be rotated to a right angle; a small
magnitude is projected onto the k direction, that is, we can only measure the right
answer with a small probability.
2. Quantum Lower Bounds
In light of previously developed quantum algorithms, one might ask if a quantum
computer can solve NP-complete problems in polynomial time. Consider the
satisfiability (SAT) problem, the first proven NP-complete problem. It can be
formulated as a search problem. That is, given a Boolean formula f(x1, x2, …, xn), search
an assignment under which the value of the expression is 1. Ask whether we can devise
a quantum algorithm to search within poly(n), or log N (N = 2n), steps. In the
following discussion, quantum lower bounds show that such a quantum speedup is
unlikely.
2.1 The hybrid argument
Consider any quantum algorithm A for solving the search problem. First do a test run of
A on function f  0. Define the query magnitude of x to be

2
x ,t
, where  x,t is the
t
amplitude with which A queries x at time t. The expectation value of the query
magnitudes

E   
x
t
2
x ,t
2
 T

 = N . Thus, min    x ,t   T N . Fix such an x, by
x

 t

Cauchy-Schwarz inequality,

x,t
 T
N
t
.
Now we modify the query on another function g: g(x) = 1, g(y) = 0 y  x. Let
 0 , 1 , …, T be the states of Af. Suppose the final state of Ag is  T . We will
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show that
T   T
must be small.
Claim:  T = T + E0 + E1 + … + ET 1 , where Ei is the error due to Step
i.
2.2 The quantum adversary method
References
[1]
[2]
[4]
[5]
A. Ambainis. Quantum lower bounds by quantum arguments. quant-ph/0002066.
C. H. Bennett, E. Bernstein, G. Brassard and U. Vazirani. Strengths and
weaknesses of quantum computing. SIAM Journal on Computing, Vol. 26, No.5,
pp 1510-1523, Oct. 1997.
L. K. Grover. A fast quantum mechanical algorithm for database search.
Proceedings of the 28th ACM Symposium on Theory of Computing, pp. 212-219,
1996.
U. Vazirani. On the power of quantum computation. Philosophical Transactions
of the Royal Society of London, Series A: Mathematical and Physical Sciences,
356: 1759-1768, August 1998.
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