Chen and Diao’s quantum search algorithm is not exponentially fast C. C. Tu1,2 and G. L. Long1,2,3,4∗ 1. 2 3
Department of Physics, Tsinghua University, Beijing, 100084, China
Key Laboratory For Quantum Information and Measurement, Beijing, 100084, China
Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100080, P. R. China 4
Center for Atomic, Molecular and NanoSciences,
Tsinghua University, Beijing 100084, P. R. China. (October 17, 2001) In this paper, we analysed the Chen and Diao quantum search algorithm where exponential speedup was claimed over classical algorithm. We calculated the total number of queries in this algorithm and showed that Chen and Diao’s “Quantum quart-section method” need totally
3n −1 2
or 3n − 1
queries in finding the target from a large database with 22n items. Although it is faster than Classical algorithm(O(4n ) steps), it is much slower compared with Grover’ algorithm(O(2n ) steps).
∗ Corresponding
author:
[email protected]
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I. INTRODUCTION
L. K. Grover’s algorithm can search for a unique item(the marked state) from an un√ sorted database of N items with O( N ) steps, and each step includes one query of the oracle [1]. Zalka has proved that Grover’s algorithm is optimal [2]. The probability of finding the marked state is not always exactly 1. In general, zero failure probability can be achieved by replacing the phase inversions with phase rotations at specific angles [3,4]. The search for an exponentially fast quantum search algorithm is very appealing, since once it is found then in quantum computer it means that NP C = P . Recently Chen and Diao reported their discovery of an “Exponentially” fast quantum search algorithm, the “quantum quart-section method”, which is supposed to have the ability of finding a single target in a large database with 22n items in n or 2n steps [5]. In this paper, it will be shown that Chen and Diao’s “quantum quart-section method” need totally
3n −1 2
or 3n − 1 queries of the black box. It is faster than classical algorithm,
but is slower than Grover’s algorithm. It is not useful in speeding up the searching process. However, there are some advantages in synthesizing a superposition of some marked basis states. II. AN EXAMINATION OF CHEN AND DIAO ALGORITHM
Suppose there is only one marked state(|τ i) in a database with N = 22n items. One artificially marks 22(n−1) − 1 items, for instance, |ii, i = 1, · · · , 22(n−1) − 1, i.e. states with first and second qubits labeled as |0i except |0i = |000 · · · 000i. There are
N 4
= 22(n−1)
marked items(the target state plus the artificially marked states). The superposition |S1 i =
1
2n−1
N −1 4
P
i=1
|ii + |τ i can be synthesized from the initial state |S0 i =
1 2n
N P
i=1
|ii by one
step of Grover’s searching program. In other words, the overall Hilbert space H is divided into four quarter subspaces, the one containing the target is “marked”. One step Grover’s searching program move the state from H to the “marked quarter subspace”. Continuing this quarter division for n times , the outcome is exactly the target |τ i. If it happens that the target state is just among those artificially marked states. The 1
superposition of these
N 4
− 1 marked items
|S10 i
= √N
N −1 4
P
1
4
−1
i=1
|ii can not be synthesized
from the initial state |S0 i. The serach process will go wrong. In this case, one will have to repeat the algorithm with some other artificially marked states, say states with the first 2 qubits as 11. Then the target state is not included in these artificially marked states. The target will be reached after n steps of this quarter divisions. Now, let us discuss this algorithm more explicitly. There are N = 22n states which are represented as 2n qubit strings |ii, i = 0, · · · , N − 1, where i is the binary representation of the i. A unique state, say |τ i, satisfies the condition f (|τ i) = 1, whereas for all other states |ii, f (|ii) = 0. The problem is to find the state |τ i. Define a auxiliary condition, 1 0 < i < 22(n−j) , fj (|ii) = j = 1, 2, · · · , n 0 otherwise.
There are 22(n−j) − 1 states satisfy the condition fj = 1, and for j = 1, there are
(1) N 4
−1
states satisfy the condition f1 = 1. Further define auxiliary oracle function Fj = f ∨ fj , where “∨” is the “OR” logic operation, and Fn = f . Now suppose f1 (|τ i) = 0, then fk (|τ i) = 0, for k = 1, · · · , n. 22(n−j) states satisfy the condition Fj = 1. Initialize the system from |0i to the superposition −1 1 NX |S0 i = √ |ii N i=0
(2)
by Walsh-Hadmamard transformation. Define |Sk+1 i ≡ Qk |Sk i = −ISk Ik |Sk i,
k = 0, · · · , n − 1,
(3)
where Qk ≡ −ISk Ik , Ik ≡ I − 2
P
Fk+1 (|ii)=1
ISk ≡ I − 2|Sk ihSk |. 2
|iihi|,
(4)
I is the identity operator. There are
N 4
= 22(n−1) states satisfying F1 = 1. As we discussed in section I, after
one step of Grover searching operation Q0 , the initial superposition of all N states |S0 i is converted into |S1 i, the superposition of those
N 4
= 22(n−1) states satisfying F1 = 1. And
the states satisfying Fk+1 = 1 are just a quarter of the states satisfying Fk = 1. So under recursion relation, the Grover searching operation Qk convert |Sk i, the superposition of 22(n−k) states satisfying Fk = 1, to |Sk+1 i, the superposition of 22(n−k−1) states satisfying Fk+1 = 1. And |Sn i is the target |τ i. Explicitly, |τ i = |Sn i =
n−1 Y k=0
(−ISk Ik )|S0 i.
(5)
It seems that the target can be obtained in n steps. If f1 (|τ i) = 1, which means that the first two qubits of the target state are 00, the program will go wrong during the quater dviding process. This problem can be checked by evaluating the f function of the result. Then we can redefine the artificial marked states as states satisfying 1 3 N < i < N, 4 fj (|ii) = 0 otherwise.
(6)
That is states that begin with the first two qubits 11. Then repeat the quarter division procedure, we will have fk (|τ i) = 0. |Sn i must be the target |τ i at the end of the procedure. In the next section, this situation is ignored for simplicity. III. COMPUTATIONAL COMPLEXITY OF THE CHEN DIAO ALGORITHM
The computational complexity of a unsorted database search is measured by the number of queries of the oracle blackbox. There is just one query of the oracle in each operation of Ik . We examine ISk now. First, IS0 = I − 2|S0 ihS0 | = I − 2W |0ih0|W = W I0 W, 3
(7)
where I0 = I − 2|0ih0|.
(8)
This is the inversion about the average operation. There is no query of the oracle in this step. So there is only one query in first iteration step Q0 . Then we examine IS1 : IS1 = I − 2|S1 ihS1 | = I − 2IS0 I0 |S0 ihS0 |I0 IS0 = IS0 I0 IS0 I0 IS0 .
(9)
This is the inversion about |S1 i operation. Since we have no knowledge of the target state except the query oracle, we can only complete this inversion by the repeated use of the oracle. It is crucial to the computational complexity of the Chen and Diao algorithm. During the 2nd iteration of the Chen and Diao algorithm, Q1 = −IS1 I1 = −IS0 I0 IS0 I0 IS0 I1 .
(10)
There are 3 queries in the second iteration Q1 . Generally, ISk+1 = I − 2|Sk+1ihSk+1 | = I − 2ISk Ik |Sk ihSk |Ik ISk = ISk Ik ISk Ik ISk , Qk+1 = −ISk+1 Ik+1 = −ISk Ik ISk Ik ISk Ik+1 .
(11)
It is easy to calculate the total number of queries by a recursion relation. Suppose there are ak queries in Qk operation, then 3ak queries are needed in Qk+1 . So there are 3k queries in Qk+1 . Totally, we need n−1 X i=0
3i =
3n − 1 2
(12)
queries in finding the target. Here we have ignored the situation f1 (|τ i) = 1. If we are not so lucky, and the target state has its first two qubits as 00, then we have to run the program twice using a different artificial marking set, and this makes the total number of all queries to 3n − 1. In table I, it is shown the total number of queries in classical searching algorithm(N/2), Grover’s quantum algorithm and the Chen and Diao algorithm in finding a target with N = 22n items in a database. 4
n
N 2
= 22n−1
Grover’s
3n −1 2
1
2
1
1
2
8
3
4
3
32
6
13
4 .. .
128 .. .
13 .. .
40 .. .
10
524288
804
29524
TABLE I. Comparisons of number of queries in classical, Grover and Chen and Diao algorithms
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IV. SUMMARY
From the equation (11), we can rewrite the whole operation · · · {(W I0 W I0 W I0 W )I1 (W I0 W I0 W I0 W ) I2 }(W I0 W I0 W I0 W I1 )(W I0W I0 )|0i. |
{z
I S2
}
|
{z
IS1
}
| {z }
(13)
I S0
It just like Grover’s algorithm except that some I0 operation is substitute with Ik . It certainly falls into the kind of quantum search algorithm of Zalka’s optimal theorem. Thus Zalka’s theorem is applicable. In summary, we have pointed out that the Chen Diao algorithm is not an exponentially fast quantum algorithm. It requires more number of queries than the Grover’s quantum search algorithm.
[1] L. K. Grover, Quantum Mechanics Helps in Searching for a Needle in a Haystack, Phys. Rev. Lett. 79, 325(1997); [2] C. Zalka, Grover’s Quantum Searching Algorithm is Optimal, Phys. Rev. A. 60, 2746(1999); [3] G. L. Long, Grover Algorithm with zero theoretical failure rate, Accepted for publication in Physical Review A, e-print quant-ph/0106071; [4] G. L. Long, C. C. Tu, Y. S. Li, W. L. Zhang and H. Y. Yan, An SO(3) Picture for Quantum Searching, J. Phys. A. 34, 861-866(2001); [5] G. Chen and Z. Diao, Exponentially fast quantum search algorithm, e-print quant-ph/0011109; [6] L. K. Grover, Synthesis of Quantum Superpositions by Quantum Computation, Phys. Rev. Lett. 85, 1334-1337(2000).
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