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arXiv:quant-ph/9903057 · Physical Review A

Quantum search without entanglement

Seth Lloyd

Published Dec 8, 1999• 142 citations
arXivHTMLPDF

Abstract

Entanglement of quantum variables is usually thought to be a prerequisite for obtaining quantum speedups of information processing tasks such as searching databases. This paper presents methods for quantum search that give a speedup over classical methods, but that do not require entanglement. These methods rely instead on interference to provide a speedup. Search without entanglement comes at a cost: although they outperform analogous classical devices, the quantum devices that perform the search are not universal quantum computers and require exponentially greater overhead than a quantum computer that operates using entanglement. Quantum search without entanglement is compared to classical search using waves.

DOI
10.1103/physreva.61.010301

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Abstract. Quantum computation, in particular Grover’s algorithm, has aroused a great deal of interest since it allows for a quadratic speedup to be obtained in search procedures. Classical search procedures for an N element database require at most O(N) time complexity. Grover’s algorithm is able to find a solution with high prob-ability in O( N) time through an amplitude amplification scheme. In this work we draw elements from both classical and quantum computation to develop an alternative search proposal based on quantum entanglement detection schemes. In 2002, Horodecki and Ekert proposed an efficient method for direct detection of quantum entanglement. Our proposition to quantum search combines quantum entanglement detection along-side entanglement inducing operators. Grover’s quantum search relies on measuring a quantum superposition after having applied a unitary evolution. We deviate from the standard method by focusing on fine-tuning a unitary operator in order to infer the solution with certainty. Our proposal sacrifices space for speed and depends on the mathematical properties of linear positive maps Λ which have not been operationally characterized. Whether such a Λ can be easily determined remains an open question.
Grover's quantum search algorithm has recently been implemented without entanglement, by replacing multiple particles with a single particle having exponentially many states. We recall that all physical resources must be accounted for to quantify algorithm complexity, and that this scheme typically incurs exponential costs in some other resource(s). In particular, we demonstrate that a recent experimental realization requires exponentially increasing precision. There is, however, a quantum algorithm which searches a "sophisticated" database (not unlike a Web search engine) with a single query, but which we show does not require entanglement even for multiparticle implementations.
Quantum computing introduces an efficient way for unfolding complex problems on computing systems. It uses concept of superposition and entanglement, & hence supports intrinsic parallelism for obtaining the speedup in context of time over classical computing. Quantum algorithms can be efficiently implemented over quantum computing system. Most popular quantum algorithms like Grover's and Shor's provide a great foundation to specifically design the algorithms. These algorithms work on improving the lower runtime bounds of classical algorithms. Grover's algorithm exhibits a quadratic polynomial speedup while searching an element among `N' elements unstructured database. The algorithmic strategy can be applied over various computer science related applications and some important are being considered in this article i.e. pattern matching, data mining and machine learning. This article explores quantum fundamentals; history based evolution followed by Grover's search algorithm in detail with cascading advantages, limitations & applications and also finally concludes with the significance of quantum computing & quantum algorithmic domains.