Quantum algorithms: entanglement-enhanced information processing

We discuss the fundamental role of entanglement as the essential non-classical feature providing the computational speed-up in the known quantum algorithms. We review the construction of the Fourier transform on an Abelian group and the principles underlying the fast Fourier transform (FFT) algorithm. We describe the implementation of the FFT algorithm for the group of integers module 2(n) in the quantum context, showing how the group-theoretic formalism leads to the standard quantum network, and identify the property of entanglement that gives rise to the exponential speed-up (compared to the classical FFT). Finally, we outline the use of the Fourier transform in extracting periodicities, which underlies its utility in the known quantum algorithms.