Approximate quantum Fourier transform and decoherence.

AbstractWe discuss the advantages of using the approximate quantum Fourier trans-form (AQFT) in algorithms which involve periodicity estimations. We analysequantum networks performing AQFT in the presence of decoherence and showthat extensive approximations can be made before the accuracy of AQFT (ascompared with regular quantum Fourier transform) is compromised. We showthat for some computations an approximation may imply a better performance.PACS: 89.70.+c, 03.65.-w, 42.50.Lc Submitted to Phys. Rev. A. (Jan. 96) 1 Introduction In the course of history many ingenious mechanical, acoustic and optical devices havebeen invented for performing Fourier transforms [1] (including nature’s own such asthe human ear). Most of them are now of merely historical interest since the arrival ofthe computer–based algorithm known as the fast Fourier transform(FFT) [2, 3] whichefficiently computes the discrete Fourier transform. The FFT algorithm can also bephrased in terms of quantum dynamics, i.e., in terms of unitary operations performedby a quantumcomputeron quantumregisters. Indeed, all known quantumalgorithmsemploy the quantum version of Fourier transforms, either explicitly or indirectly. Itis used for the periodicity estimation in the Shor algorithm [4] and its approximateversion (the Hadamard transform) is commonly used to prepare quantum registersin coherent superpositions of different values.In this paper we analyse the performance of the quantum Fourier transform(QFT) in the presence of decoherence. In particular we show that as far as the peri-1