Showing papers on "Quantum sort published in 1996"

A fast quantum mechanical algorithm for database search

Abstract: were proposed in the early 1980’s [Benioff80] and shown to be at least as powerful as classical computers an important but not surprising result, since classical computers, at the deepest level, ultimately follow the laws of quantum mechanics. The description of quantum mechanical computers was formalized in the late 80’s and early 90’s [Deutsch85][BB92] [BV93] [Yao93] and they were shown to be more powerful than classical computers on various specialized problems. In early 1994, [Shor94] demonstrated that a quantum mechanical computer could efficiently solve a well-known problem for which there was no known efficient algorithm using classical computers. This is the problem of integer factorization, i.e. testing whether or not a given integer, N, is prime, in a time which is a finite power of o (logN) . ----------------------------------------------

Tight bounds on quantum searching

Abstract: We provide a tight analysis of Grover''s recent algorithm for quantum database searching. We give a simple closed-form formula for the probability of success after any given number of iterations of the algorithm. This allows us to determine the number of iterations necessary to achieve almost certainty of finding the answer. Furthermore, we analyse the behaviour of the algorithm when the element to be found appears more than once in the table and we provide a new algorithm to find such an element even when the number of solutions is not known ahead of time. Using techniques from Shor''s quantum factoring algorithm in addition to Grover''s approach, we introduce a new technique for approximate quantum counting, which allows to estimate the number of solutions. Finally we provide a lower bound on the efficiency of any possible quantum database searching algorithm and we show that Grover''s algorithm nearly comes within a factor 2 of being optimal in terms of the number of probes required in the table.

Quantum Computation and Shor's Factoring Algorithm

Abstract: Current technology is beginning to allow us to manipulate rather than just observe individual quantum phenomena. This opens up the possibility of exploiting quantum effects to perform computations beyond the scope of any classical computer. Recently Peter Shor discovered an efficient algorithm for factoring whole numbers, which uses characteristically quantum effects. The algorithm illustrates the potential power of quantum computation, as there is no known efficient classical method for solving this problem. The authors give an exposition of Shor's algorithm together with an introduction to quantum computation and complexity theory. They discuss experiments that may contribute to its practical implementation. [S0034-6861(96)00303-0]

Quantum networks for elementary arithmetic operations.

Abstract: Quantum computers require quantum arithmetic We provide an explicit construction of quantum networks effecting basic arithmetic operations: from addition to modular exponentiation Quantum modular exponentiation seems to be the most difficult (time and space consuming) part of Shor's quantum factorizing algorithm We show that the auxiliary memory required to perform this operation in a reversible way grows linearly with the size of the number to be factorized \textcopyright{} 1996 The American Physical Society

A quantum algorithm for finding the minimum

Abstract: Let T[0..N−1] be an unsorted table of N items, eachholding a value from an ordered set. For simplicity,assume that all values are distinct. The minimumsearchingproblem is to find the index ysuch that T[y]is minimum. This clearly requires a linear number ofprobes on a classical probabilistic Turing machine.Here, we give a simple quantum algorithm whichsolves the problem using O(√N) probes. The mainsubroutine is the quantum exponential searching al-gorithm of [2], which itself is a generalization ofGrover’s recent quantum searching algorithm [3].Due to a general lower bound of [1], this is withina constant factor of the optimum.

Threshold Accuracy for Quantum Computation

Abstract: We have previously (quant-ph/9608012) shown that for quantum memories and quantum communication, a state can be transmitted over arbitrary distances with error $\epsilon$ provided each gate has error at most $c\epsilon$. We discuss a similar concatenation technique which can be used with fault tolerant networks to achieve any desired accuracy when computing with classical initial states, provided a minimum gate accuracy can be achieved. The technique works under realistic assumptions on operational errors. These assumptions are more general than the stochastic error heuristic used in other work. Methods are proposed to account for leakage errors, a problem not previously recognized.

Quantum Randomness and Nondeterminism

Abstract: Does the notion of a quantum randomized or nondeterministic algorithm make sense, and if so, does quantum randomness or nondeterminism add power? Although reasonable quantum random sources do not add computational power, the discussion of quantum randomness naturally leads to several definitions of the complexity of quantum states. Unlike classical string complexity, both deterministic and nondeterministic quantum state complexities are interesting. A notion of \emph{total quantum nondeterminism} is introduced for decision problems. This notion may be a proper extension of classical nondeterminism.

Quantum computing and phase transitions in combinatorial search

Abstract: We introduce an algorithm for combinatorial search on quantum computers that is capable of significantly concentrating amplitude into solutions for some NP search problems, on average. This is done by exploiting the same aspects of problem structure as used by classical backtrack methods to avoid unproductive search choices. This quantum algorithm is much more likely to find solutions than the simple direct use of quantum parallelism. Furthermore, empirical evaluation on small problems shows this quantum algorithm displays the same phase transition behavior, and at the same location, as seen in many previously studied classical search methods. Specifically, difficult problem instances are concentrated near the abrupt change from underconstrained to overconstrained problems.

On The Power of Exact Quantum Polynomial Time

Abstract: We investigate the power of quantum computers when they are required to return an answer that is guaranteed correct after a time that is upper-bounded by a polynomial in the worst case. In an oracle setting, it is shown that such machines can solve problems that would take exponential time on any classical bounded-error probabilistic computer.

Error-Correcting Code Keeps Quantum Computers on Track

Abstract: Quantum MechanicsQuantum computers, which could exploit systems such as the energy states of a row of atoms to perform vast numbers of computations at the same time, have remained in the realm of fantasy. A major barrier has been the fragility of quantum states, which makes quantum systems vulnerable to errors. Classical error-correction schemes won't work in a quantum computer, because they rely on duplicate bits, and in quantum systems information vanishes when you read and copy it. A new mathematical scheme, however, makes it possible—at least in theory—to nudge a quantum system back into line without directly looking at it. Computers/Math

A Framework for Quantum Search Heuristics

Abstract: A quantum algorithm for combinatorial search is presented that provides a simple framework for utilizing search heuristics. The algorithm is evaluated in a new case that is an unstructured version of the graph coloring problem. It performs significantly better than the direct use of quantum parallelism, on average, in cases corresponding to previously identified phase transitions in search difficulty. The conditions underlying this improvement are described. Much of the algorithm is independent of particular problem instances, making it suitable for implementation as a special purpose device.