120 research outputs found
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
Grover's quantum searching algorithm is optimal
I improve the tight bound on quantum searching by Boyer et al.
(quant-ph/9605034) to a matching bound, thus showing that for any probability
of success Grovers quantum searching algorithm is optimal. E.g. for near
certain success we have to query the oracle pi/4 sqrt{N} times, where N is the
size of the search space. I also show that unfortunately quantum searching
cannot be parallelized better than by assigning different parts of the search
space to independent quantum computers. Earlier results left open the
possibility of a more efficient parallelization.Comment: 13 pages, LaTeX, essentially published versio
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 items was
divided into 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 fewer iterations
than the quantum search algorithm. They also proved that the best any quantum
algorithm could do would be to save 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
Eigenvector Approximation Leading to Exponential Speedup of Quantum Eigenvalue Calculation
We present an efficient method for preparing the initial state required by
the eigenvalue approximation quantum algorithm of Abrams and Lloyd. Our method
can be applied when solving continuous Hermitian eigenproblems, e.g., the
Schroedinger equation, on a discrete grid. We start with a classically obtained
eigenvector for a problem discretized on a coarse grid, and we efficiently
construct, quantum mechanically, an approximation of the same eigenvector on a
fine grid. We use this approximation as the initial state for the eigenvalue
estimation algorithm, and show the relationship between its success probability
and the size of the coarse grid.Comment: 4 page
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
Comment on "Probabilistic Quantum Memories"
This is a comment on two wrong Phys. Rev. Letters papers by C.A.
Trugenberger. Trugenberger claimed that quantum registers could be used as
exponentially large "associative" memories. We show that his scheme is no
better than one where the quantum register is replaced with a classical one of
equal size.
We also point out that the Holevo bound and more recent bounds on "quantum
random access codes" pretty much rule out powerful memories (for classical
information) based on quantum states.Comment: REVTeX4, 1 page, published versio
A General SU(2) Formulation for Quantum Searching with Certainty
A general quantum search algorithm with arbitrary unitary transformations and
an arbitrary initial state is considered in this work. To serach a marked state
with certainty, we have derived, using an SU(2) representation: (1) the
matching condition relating the phase rotations in the algorithm, (2) a concise
formula for evaluating the required number of iterations for the search, and
(3) the final state after the search, with a phase angle in its amplitude of
unity modulus. Moreover, the optimal choices and modifications of the phase
angles in the Grover kernel is also studied.Comment: 8 pages, 2 figure
Quantum complexities of ordered searching, sorting, and element distinctness
We consider the quantum complexities of the following three problems:
searching an ordered list, sorting an un-ordered list, and deciding whether the
numbers in a list are all distinct. Letting N be the number of elements in the
input list, we prove a lower bound of \frac{1}{\pi}(\ln(N)-1) accesses to the
list elements for ordered searching, a lower bound of \Omega(N\log{N}) binary
comparisons for sorting, and a lower bound of \Omega(\sqrt{N}\log{N}) binary
comparisons for element distinctness. The previously best known lower bounds
are {1/12}\log_2(N) - O(1) due to Ambainis, \Omega(N), and \Omega(\sqrt{N}),
respectively. Our proofs are based on a weighted all-pairs inner product
argument.
In addition to our lower bound results, we give a quantum algorithm for
ordered searching using roughly 0.631 \log_2(N) oracle accesses. Our algorithm
uses a quantum routine for traversing through a binary search tree faster than
classically, and it is of a nature very different from a faster algorithm due
to Farhi, Goldstone, Gutmann, and Sipser.Comment: This new version contains new results. To appear at ICALP '01. Some
of the results have previously been presented at QIP '01. This paper subsumes
the papers quant-ph/0009091 and quant-ph/000903
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