From Schr\"odinger's Equation to the Quantum Search Algorithm

The quantum search algorithm is a technique for searching N possibilities in only sqrt(N) steps. Although the algorithm itself is widely known, not so well known is the series of steps that first led to it, these are quite different from any of the generally known forms of the algorithm. This paper describes these steps, which start by discretizing Schr\"odinger's equation. This paper also provides a self-contained introduction to the quantum search algorithm from a new perspective.Comment: Postscript file, 16 pages. This is a pedagogical article describing the invention of the quantum search algorithm. It appeared in the July, 2001 issue of American Journal of Physics (AJP

Simple Algorithm for Partial Quantum Search

Quite often in database search, we only need to extract portion of the information about the satisfying item. Recently Radhakrishnan & Grover [RG] considered this problem in the following form: the database of $N$ items was divided into $K$ equally sized blocks. The algorithm has just to find the block containing the item of interest. The queries are exactly the same as in the standard database search problem. [RG] invented a quantum algorithm for this problem of partial search that took about $0.33\sqrt{N/K}$ fewer iterations than the quantum search algorithm. They also proved that the best any quantum algorithm could do would be to save $0.78 \sqrt(N/K)$ iterations. The main limitation of the algorithm was that it involved complicated analysis as a result of which it has been inaccessible to most of the community. This paper gives a simple analysis of the algorithm. This analysis is based on three elementary observations about quantum search, does not require a single equation and takes less than 2 pages.Comment: 3 pages, 3 figure

Quantum computers can search rapidly by using almost any transformation

A quantum computer has a clear advantage over a classical computer for exhaustive search. The quantum mechanical algorithm for exhaustive search was originally derived by using subtle properties of a particular quantum mechanical operation called the Walsh-Hadamard (W-H) transform. This paper shows that this algorithm can be implemented by replacing the W-H transform by almost any quantum mechanical operation. This leads to several new applications where it improves the number of steps by a square-root. It also broadens the scope for implementation since it demonstrates quantum mechanical algorithms that can readily adapt to available technology.Comment: This paper is an adapted version of quant-ph/9711043. It has been modified to make it more readable for physicists. 9 pages, postscrip

Quantum computers can search arbitrarily large databases by a single query

This paper shows that a quantum mechanical algorithm that can query information relating to multiple items of the database, can search a database in a single query (a query is defined as any question to the database to which the database has to return a (YES/NO) answer). A classical algorithm will be limited to the information theoretic bound of at least O(log N) queries (which it would achieve by using a binary search).Comment: Several enhancements to the original pape

Hamiltonian and measuring time for analog quantum search

We derive in this study a Hamiltonian to solve with certainty the analog quantum search problem analogue to the Grover algorithm. The general form of the initial state is considered. Since the evaluation of the measuring time for finding the marked state by probability of unity is crucially important in the problem, especially when the Bohr frequency is high, we then give the exact formula as a function of all given parameters for the measuring time.Comment: 5 page

Realization of generalized quantum searching using nuclear magnetic resonance

According to the theoretical results, the quantum searching algorithm can be generalized by replacing the Walsh-Hadamard(W-H) transform by almost any quantum mechanical operation. We have implemented the generalized algorithm using nuclear magnetic resonance techniques with a solution of chloroform molecules. Experimental results show the good agreement between theory and experiment.Comment: 11 pages,3 figure. Accepted by Phys. Rev. A. Scheduled Issue: 01 Mar 200

Observation of tunable exchange bias in Sr2_2YbRuO6_6

The double perovskite compound, Sr$_{2}$YbRuO$_{6}$, displays reversal in the orientation of magnetic moments along with negative magnetization due to an underlying magnetic compensation phenomenon. The exchange bias (EB) field below the compensation temperature could be the usual negative or the positive depending on the initial cooling field. This EB attribute has the potential of getting tuned in a preselected manner, as the positive EB field is seen to crossover from positive to negative value above $T_{\mathrm{comp}}$.Comment: 4 Pages, 4 Figure

Energy and Efficiency of Adiabatic Quantum Search Algorithms

We present the results of a detailed analysis of a general, unstructured adiabatic quantum search of a data base of $N$ items. In particular we examine the effects on the computation time of adding energy to the system. We find that by increasing the lowest eigenvalue of the time dependent Hamiltonian {\it temporarily} to a maximum of $\propto \sqrt{N}$, it is possible to do the calculation in constant time. This leads us to derive the general theorem which provides the adiabatic analogue of the $\sqrt{N}$ bound of conventional quantum searches. The result suggests that the action associated with the oracle term in the time dependent Hamiltonian is a direct measure of the resources required by the adiabatic quantum search.Comment: 6 pages, Revtex, 1 figure. Theorem modified, references and comments added, sections introduced, typos corrected. Version to appear in J. Phys.

Nested quantum search and NP-complete problems

A quantum algorithm is known that solves an unstructured search problem in a number of iterations of order $\sqrt{d}$, where $d$ is the dimension of the search space, whereas any classical algorithm necessarily scales as $O(d)$. It is shown here that an improved quantum search algorithm can be devised that exploits the structure of a tree search problem by nesting this standard search algorithm. The number of iterations required to find the solution of an average instance of a constraint satisfaction problem scales as $\sqrt{d^\alpha}$, with a constant $\alpha<1$ depending on the nesting depth and the problem considered. When applying a single nesting level to a problem with constraints of size 2 such as the graph coloring problem, this constant $\alpha$ is estimated to be around 0.62 for average instances of maximum difficulty. This corresponds to a square-root speedup over a classical nested search algorithm, of which our presented algorithm is the quantum counterpart.Comment: 18 pages RevTeX, 3 Postscript figure

Measurement of an integral of a classical field with a single quantum particle

A method for measuring an integral of a classical field via local interaction of a single quantum particle in a superposition of 2^N states is presented. The method is as efficient as a quantum method with N qubits passing through the field one at a time and it is exponentially better than any known classical method that uses N bits passing through the field one at a time. A related method for searching a string with a quantum particle is proposed.Comment: 3 page