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AIP conference proceedings

Quantum Noise and Information in Quantum Search Algorithm

Demosthenes Ellinas

Published Jan 1, 2004• 1 citation
Publisher

Abstract

External influences in the form of quantum noise on Grover’s search algorithm are investigated. The study shows that the algorithm can be robust under such external dissipation. The effect of noise is described by a completely positive trace preserving map, acting on an unsorted N‐dimensional database made of projective density matrices. To evaluate the efficiency of the algorithm two figures of merit are used, the radial fidelity giving the projection between the Bloch vectors of target and final density operators, and the cosine of angular fidelity between the same vectors. Explicitly we prove that the resulting search positive map depends on x, the strength of the coupling to the environment, and that there are infinitely many x values, named good x’s and bad x’s, for which search is successful after O(N) queries, or fails, respectively . Next we study the information theoretic aspects of the dissipative quantum search algorithm. To this end we view the initial state as a pure qubit signal entering through a cascade of quantum channels, each channel described by a completely positive map, and corresponds to the iterations of the algorithm. This map is identical to the dissipative search map of the algorithm for which the parameter x now characterizes the influence of the noisy channel exercised upon the incoming quantum signal. The process of searching is understood as transmission of the initial signal/item through the composite channel made by O(N) elementary channels. The quality of this transmission corresponds to the fidelity of targeting the wanted item in the quantum database, and is studied in terms of the majorization/quantum entropy of the input‐output density matrix, and by the entropy exchanged between the qubit signal/item and an external environment.

DOI
10.1063/1.1834410

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External influences in the form of quantum noise on Grover’s search algorithm are investigated. The study shows that the algorithm can be robust under such external dissipation. The effect of noise is described by a completely positive trace preserving map, acting on an unsorted N‐dimensional database made of projective density matrices. To evaluate the efficiency of the algorithm two figures of merit are used, the radial fidelity giving the projection between the Bloch vectors of target and final density operators, and the cosine of angular fidelity between the same vectors. Explicitly we prove that the resulting search positive map depends on x, the strength of the coupling to the environment, and that there are infinitely many x values, named good x’s and bad x’s, for which search is successful after O(N) queries, or fails, respectively . Next we study the information theoretic aspects of the dissipative quantum search algorithm. To this end we view the initial state as a pure qubit signal entering through a cascade of quantum channels, each channel described by a completely positive map, and corresponds to the iterations of the algorithm. This map is identical to the dissipative search map of the algorithm for which the parameter x now characterizes the influence of the noisy channel exercised upon the incoming quantum signal. The process of searching is understood as transmission of the initial signal/item through the composite channel made by O(N) elementary channels. The quality of this transmission corresponds to the fidelity of targeting the wanted item in the quantum database, and is studied in terms of the majorization/quantum entropy of the input‐output density matrix, and by the entropy exchanged between the qubit signal/item and an external environment.
External environment influences on Grover’s search algorithm modeled by quantum noise are investigated. The algorithm is shown to be robust under that external dissipation. Explicitly we prove that the resulting search positive maps acting on unsorted N-dimensional database made of projective density matrices depend on x the strength of the environment, and that there are infinitely many x values for which search is successful O ( N ) after queries. These algorithms are quantum entropy increasing.
External environment influences on Grover's search algorithm modeled by quantum noise are investigated. The algorithm is shown to be robust under that external dissipation. Explicitly we prove that the resulting search positive maps acting on unsorted N-dimensional database made of projective density matrices depend on x the strength of the environment, and that there are infinitely many x values for which search is successful after O(\sqrt{N}) queries. These algorithms are quantum entropy increasing.
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 are 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 $\ensuremath{\epsilon}.$ The noise terms at different calls to U are assumed to be uncorrelated. For a search space of size N=2n{N=2}^{n}N=2n (where n is the number of qubits used to span this space) it is found that as long as $\ensuremath{\epsilon}<{O(n}^{\ensuremath{-}1/2}{N}^{\ensuremath{-}1/4})$ 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.
An algebraic reformulation of the quantum search algorithm associated to a k‐valued oracle function, is introduced in terms of the so called oracle matrix algebra, by means of which a Bloch sphere like description of search is obtained. A parametric family of symmetric completely positive trace preserving (CPTP) maps, that formalize the presence of quantum noise but preserves the complexity of the algorithm, is determined. Dimensional reduction of representations of oracle Lie algebra is introduced in order to determine the reduced density matrix of subsets of qubits in database. The L1 vector‐induced norm of reduced density matrix is employed to define an index function for the quantum entanglement between database qubits, in the presence of non invariant noise CPTP maps. Analytic investigations provide a causal relation between entanglement and fidelity of the algorithm, which is controlled by quantum noise parameter.
Abstract Grover’s quantum search algorithm promises a quadratic speedup for unstructured search over its classical counterpart. But this advantage is affected by noise acting on the search space. Here, we show that a quantum switch can act as a resource to mitigate the effects of noise. In this scenario, the noise is modeled by a depolarizing channel, which coherently acts on the entire quantum register. We show that a quantum switch can significantly reduce the error in Grover’s search algorithm. We consider the success probability of finding the marked item as the sole quantifier of diminishing the effect of noise in the search space in the presence of quantum switch. We propose two frameworks for the application of quantum switches. In the first framework, we apply the superposition of channel’s orders in the form of a switch and do a post-selection at every iteration of the applications of the Grover operator. In the second framework, we delay this measurement and post-selection until the very end. The number of post selections is minimal in the second scenario, and hence the noise reduction can be attributed more to the presence of quantum switch. We illustrate with an example of significant advantage in the success probability of Grover’s algorithm using quantum switch.
The quantum random walk is a possible approach to construct new quantum search algorithms. It has been shown by Shenvi et al. [Phys. Rev. A 67 (2003) 52307 that a kind of algorithm can perform an oracle search on a database of N items with O ( N 1/2 ) calling to the oracle, yielding a speedup similar to other quantum search algorithms. We study the effect of white or Gaussian noise on this algorithm. The algorithm loses efficiency when noise is added. We also show that noise on the target state plays a more important role than that on other states. Finally we compare the effects of similar types of noise in the quantum random walk search algorithm and Grover's search algorithm.