L. Grover, "How fast can a quantum computer search?," quantph /9809029v3, Bell Labs, April 1999.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....this section we shortly review the optimal quantum database search algorithm later on a proper preparation of the database will be given. Consider a large unsorted database, which contains entries of to find the desired value with any classical algorithms would need at least steps. Grover [12], 13] 14] has derived a quantum search algorithm that took precisely ( iterations to carry out the search, which is the optimal solution, as it was proved in [15] Boyer et al. [16] introduced upper bounds for the required number of evaluations for several number of identical solutions in ....

....sent no decision is possible or 1. The qregisters have to be built up just one time at all. It is obvious to choose a suitable database searching algorithm, to see which qregister contains the received bit, if any at all. We apply the optimal quantum search algorithm proposed by Grover [12] and depicted in Figure 1 for our purpose. We feed the received signal (1) to the oracle , where the function = G=4 Z is evaluated such that SJ 1 ifSN (21) Z ( SU K L 8 Z , lk . Assuming, there is again solutions for the search in ....

L. Grover, "How fast can a quantum computer search?," quantph /9809029v3, Bell Labs, April 1999.

....by polynomials. Our quantum adversary method can be used to give more uni ed proofs for many (but not all) results that were previously shown using di erent variants of hybrid and or polynomials method. There is also a new proof of the p N) lower bound on unordered search by Grover[14]. This proof is based on considering the sum of distances between superpositions on di erent inputs. While the motivation for Grover s proof (the sum of distances) is fairly di erent from ours (quantum adversary) these two methods are, in fact, closely related. We discuss this relation in section ....

....j is the variable for which y j = 1 and x is the only permutation that di ers from y only in these two places. Therefore, l x;i = n=2, l y;i = 1 and l max = n=2. By Theorem 6.1, this implies that any quantum algorithm needs q n 2 n ) p n) queries. 7. RELATION TO GROVER S PROOF Grover[14] presents a proof of the p n) lower bound on the search problem based on considering the sum of distances (t) X i;j2f1; ng;i6=j k t i t 0 k 2 where t i is the state of the algorithm after t queries on the input x 1 = x i 1 = 0, x i = 1, x i 1 = xn = 1 and ....

[Article contains additional citation context not shown here]

L. Grover. How fast can a quantum computer search? quant-ph/9809029.

....polynomials. Our quantum adversary method can be used to give more unified proofs for many (but not all) results that were previously shown using different variants of hybrid and or polynomials method. There is also a new proof of the Omega Gamma p N ) lower bound on unordered search by Grover[10]. This proof is based on considering the sum of distances between superpositions on different inputs. While the motivation for Grover s proof (sum of distances) is fairly different from ours (quantum adversary) these two methods are, in fact, closely related. We discuss this relation in section ....

....which y j = 1 and x is the only permutation that differs from y only in these two places. Therefore, l x;i = n=2, l y;i = 1 and l max = n=2. By Theorem 6, this implies that any quantum algorithm needs Omega Gamma q n 2 n ) Omega Gamma p n) queries. 2 7 Relation to Grover s proof Grover[10] presents a proof of the Omega Gamma p n) lower bound on the search problem based on considering the sum of distances Delta(t) X i;j2f1; ng;i6=j kOE t i Gamma OE t 0 k 2 where OE t i is the state of the algorithm after t queries on the input x 1 = x i Gamma1 = 0, x i = 1, ....

[Article contains additional citation context not shown here]

L. Grover. How fast can a quantum computer search? quant-ph/9809029.

....N has to query at least N=2 items of the list in order to have success probability 2=3. In contrast, a quantum computer can make queries in superposition and can search such a list using only O( p N) queries [Gro96] It is known that the O( p N) is optimal [BBBV97, BBHT98, Zal97, BBC 98, Gro98] If we do not want to allow a small error probability then even a quantum computer needs N queries [BBC 98] Until recently, not much attention had been paid to the quantum complexity of searching a list which is ordered according to some key field of the items. Classically, we can search ....

L. K. Grover. How fast can a quantum computer search? quant-ph/9809029, 10 Sep 1998.

....finds the desired item with high probability using only O( p N) queries. The following is known about the error probability in quantum search: ffl can be made an arbitrarily small constant using O( p N) queries [Gro96] but not using o( p N) queries [BBBV97, BBHT98, Zal97, BBC 98, Gro98a] ffl can be made 1=2 N ff using O(N 0:5 ff ) queries [BCW98, Theorem 1.16] ffl If we want no error at all ( 0) then we need N queries [BBC 98, Corollary 6.2] Many applications of quantum computing will need to apply quantum search several times as a subroutine. We should ....

....item in an unordered list of N items with high probability, using only O( p N) queries (a.k.a. database look ups) whereas a classical algorithm needs Theta(N ) queries for this. There exist several lower bound proofs that show that the O( p N) is optimal [BBBV97, BBHT98, Zal97, BBC 98, Gro98a] What about search in a list of N items which is ordered according to some key value of each item A classical deterministic algorithm can search such a list using log N queries by means of binary search (each query can effectively halve the relevant part of the list: looking at the key of the ....

L. K. Grover. How fast can a quantum computer search? quant-ph/9809029, 10 Sep 1998.

.... claims of the proof of ultimate security of quantum cryptography, i.e. security against all possible attacks [85] 87] 88] 89] Finally, although tangentially related to this paper, it should be mentioned that a new quantum algorithm for searching databases has been developed [71] [72], 73] 12 Appendix A. The no cloning theorem In this appendix, we prove that there can be no device that produces exact replicas or copies of a quantum system. If such a quantum copier existed, then Eve could eavesdrop without detection. This proof is taken from [99] It is an amazingly ....

Grover, Lov K., How fast can a quantum computer search?, quant-ph/9809029.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC