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Title: Implementation of Grover's quantum search algorithm in a scalable system

Abstract

We report the implementation of Grover's quantum search algorithm in the scalable system of trapped atomic ion quantum bits. Any one of four possible states of a two-qubit memory is marked, and following a single query of the search space, the marked element is successfully recovered with an average probability of 60(2)%. This exceeds the performance of any possible classical search algorithm, which can only succeed with a maximum average probability of 50%.

Authors:
; ; ; ; ;  [1]
  1. FOCUS Center and Department of Physics, University of Michigan, Ann Arbor, Michigan 48109 (United States)
Publication Date:
OSTI Identifier:
20786420
Resource Type:
Journal Article
Resource Relation:
Journal Name: Physical Review. A; Journal Volume: 72; Journal Issue: 5; Other Information: DOI: 10.1103/PhysRevA.72.050306; (c) 2005 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
74 ATOMIC AND MOLECULAR PHYSICS; ALGORITHMS; ATOMIC IONS; PROBABILITY; QUANTUM COMPUTERS; QUANTUM MECHANICS; QUBITS; TRAPPING

Citation Formats

Brickman, K.-A., Haljan, P. C., Lee, P. J., Acton, M., Deslauriers, L., and Monroe, C. Implementation of Grover's quantum search algorithm in a scalable system. United States: N. p., 2005. Web. doi:10.1103/PHYSREVA.72.0.
Brickman, K.-A., Haljan, P. C., Lee, P. J., Acton, M., Deslauriers, L., & Monroe, C. Implementation of Grover's quantum search algorithm in a scalable system. United States. doi:10.1103/PHYSREVA.72.0.
Brickman, K.-A., Haljan, P. C., Lee, P. J., Acton, M., Deslauriers, L., and Monroe, C. Tue . "Implementation of Grover's quantum search algorithm in a scalable system". United States. doi:10.1103/PHYSREVA.72.0.
@article{osti_20786420,
title = {Implementation of Grover's quantum search algorithm in a scalable system},
author = {Brickman, K.-A. and Haljan, P. C. and Lee, P. J. and Acton, M. and Deslauriers, L. and Monroe, C.},
abstractNote = {We report the implementation of Grover's quantum search algorithm in the scalable system of trapped atomic ion quantum bits. Any one of four possible states of a two-qubit memory is marked, and following a single query of the search space, the marked element is successfully recovered with an average probability of 60(2)%. This exceeds the performance of any possible classical search algorithm, which can only succeed with a maximum average probability of 50%.},
doi = {10.1103/PHYSREVA.72.0},
journal = {Physical Review. A},
number = 5,
volume = 72,
place = {United States},
year = {Tue Nov 15 00:00:00 EST 2005},
month = {Tue Nov 15 00:00:00 EST 2005}
}
  • We describe an implementation of Grover's fixed-point quantum search algorithm on a nuclear magnetic resonance quantum computer, searching for either one or two matching items in an unsorted database of four items. In this algorithm the target state (an equally weighted superposition of the matching states) is a fixed point of the recursive search operator, so that the algorithm always moves towards the desired state. The effects of systematic errors in the implementation are briefly explored.
  • Grover's quantum search algorithm is analyzed for the case in which the initial state is an arbitrary pure quantum state |{phi}> of n qubits. It is shown that the optimal time to perform the measurement is independent of |{phi}>, namely, it is identical to the optimal time in the original algorithm in which |{phi}>=|0>, with the same number of marked states, r. The probability of success P{sub s} is obtained in terms of the amplitudes of the state |{phi}> and is shown to be independent of r. A class of states, which includes fixed points and cycles of the Grovermore » iteration operator, is identified. The relevance of these results in the context of using the success probability as an entanglement measure is discussed. In particular, the Groverian entanglement measure, previously limited to a single marked state, is generalized to the case of several marked states.« less
  • The Grover quantum search algorithm is generalized to deal with an arbitrary mixed initial state. The probability to measure a marked state as a function of time is calculated, and found to depend strongly on the specific initial state. The form of the function, though, remains as it is in the case of initial pure state. We study the role of the von Neumann entropy of the initial state, and show that the entropy cannot be a measure for the usefulness of the algorithm. We give few examples and show that for some extremely mixed initial states (carrying high entropy),more » the generalized Grover algorithm is considerably faster than any classical algorithm.« less
  • The effect of unitary noise on the performance of Grover's quantum search algorithm is studied. This type of noise may result from tiny fluctuations and drift in the parameters of the (quantum) components performing the computation. The resulting operations are still unitary, but not precisely those assumed in the design of the algorithm. Here we focus on the effect of such noise in the Hadamard gate W, which is an essential component in each iteration of the quantum search process. To this end W is replaced by a noisy Hadamard gate U. The parameters of U at each iteration aremore » taken from an arbitrary probability distribution (e.g., a Gaussian distribution) and are characterized by their statistical moments around the parameters of W. For simplicity, we assume that the noise is unbiased and isotropic, namely, all noise variables in the parametrization we use have zero average and the same standard deviation {epsilon}. The noise terms at different calls to U are assumed to be uncorrelated. For a search space of size N=2{sup n} (where n is the number of qubits used to span this space) it is found that as long as {epsilon}<O(n{sup -1/2}N{sup -1/4}) the algorithm maintains significant efficiency, while above this noise level its operation is hampered completely. It is also found that below this noise threshold, when the search fails, it is likely to provide a state that differs from the marked state by only a few bits. This feature can be used to search for the marked state by a classical postprocessing, even if the quantum search has failed, thus improving the success rate of the search process.« less
  • In a fundamental paper [Phys. Rev. Lett. 78, 325 (1997)] Grover showed how a quantum computer can find a single marked object in a database of size N by using only O({radical}(N)) queries of the oracle that identifies the object. His result was generalized to the case of finding one object in a subset of marked elements. We consider the following computational problem: A subset of marked elements is given whose number of elements is either M or K, our task is to determine which is the case. We show how to solve this problem with a high probability ofmore » success using iterations of Grover's basic step only, and no other algorithm. Let m be the required number of iterations; we prove that under certain restrictions on the sizes of M and K the estimation m<2{radical}(N)/({radical}(K)-{radical}(M)) obtains. This bound reproduces previous results based on more elaborate algorithms, and is known to be optimal up to a constant factor. Our method involves simultaneous Diophantine approximations, so that Grover's algorithm is conceptualized as an orbit of an ergodic automorphism of the torus. We comment on situations where the algorithm may be slow, and note the similarity between these cases and the problem of small divisors in classical mechanics.« less