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Communications in Theoretical Physics

Scheme for Implementing Quantum Search Algorithm in a Cluster State Quantum Computer

Yanhui Wang, Yong Zhang

Published Jun 1, 2008• 2 citations
Publisher

Abstract

Using cluster state and single qubit measurement one can perform the one-way quantum computation. Here we give a detailed scheme for realizing a modified Grover search algorithm using measurements on cluster state. We give the measurement pattern for the cluster-state realization of the algorithm and estimated the number of measurement needed for its implementation. It is found that O(2 3n/2 n 2 ) number of single qubit measurements is required for its realization in a cluster-state quantum computer.

DOI
10.1088/0253-6102/49/6/30

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Using cluster state and single qubit measurement one can perform the one-way quantum computation. Here we give a detailed scheme for realizing a modified Grover search algorithm using measurements on cluster state. We give the measurement pattern for the cluster-state realization of the algorithm and estimated the number of measurement needed for its implementation. It is found that O(2 3n/2 n 2 ) number of single qubit measurements is required for its realization in a cluster-state quantum computer.
We report an experimental realization of one-way quantum computing on a two-photon four-qubit cluster state. This is accomplished by developing a two-photon cluster state source entangled both in polarization and spatial modes. With this special source, we implemented a highly efficient Grover's search algorithm and high-fidelity two-qubit quantum gates. Our experiment demonstrates that such cluster states could serve as an ideal source and a building block for rapid and precise optical quantum computation.
Quantum computers promise to be more efficient and powerful than their classical counterparts. In the one-way quantum computer model [1], a sequence of measurements processes qubits, which are initially prepared in a highly entangled cluster state. We present here an optical implementation of a 4-qubit cluster state and we show how different algorithms can be experimentally realized by performing the corresponding sequence of measurements. The key advantage of this scheme over the standard network approach of quantum computing is that inherent, randomly induced measurement errors can classically be fed-forward and corrected by adapting the basis of subsequent measurement.
The translation of Grover's search algorithm from its standard version, designed for implementation on a single quantum system amenable to projective measurements, into one suitable for an ensemble of quantum computers, whose outputs are expectation values of observables on individual qubits, is described in detail. A filtering scheme, which effectively determines expectation values on a limited portion of the quantum state, is presented and used to locate a single item for searches involving more than one marked item. A truncated version of Grover's algorithm, requiring fewer steps than the translated standard version but locating marked items just as successfully, is proposed. For quantum computational devices that only return expectation values, the truncated version is superior to its standard counterpart. This indicates that it is possible to modify quantum algorithms so as to reduce the required temporal resources by using the ensemble's spatial resources.
We give a detailed account of the one-way quantum computer, a scheme of quantum computation that consists entirely of one-qubit measurements on a particular class of entangled states, the cluster states. We prove its universality, describe why its underlying computational model is different from the network model of quantum computation, and relate quantum algorithms to mathematical graphs. Further we investigate the scaling of required resources and give a number of examples for circuits of practical interest such as the circuit for quantum Fourier transformation and for the quantum adder. Finally, we describe computation with clusters of finite size.
We provide a set of prescriptions for implementing a quantum circuit model algorithm as measurement based quantum computing (MBQC) algorithm1, 2 via a large cluster state. As means of illustration we draw upon our numerical modeling experience to describe a large graph state capable of searching a logical 8 element list (a non-trivial version of Grover's algorithm3 with feedforward). We develop several prescriptions based on analytic evaluation of cluster states and graph state equations which can be generalized into any circuit model operations. Such a resulting cluster state will be able to carry out the desired operation with appropriate measurements and feed forward error correction. We also discuss the physical implementation and the analysis of the principal 3-qubit entangling gate (Toffoli) required for a non-trivial feedforward realization of an 8-element Grover search algorithm.