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Efficient implementations of the quantum Fourier transform

Dieter Suter, Kavita Dorai

Published Jun 1, 2005• 5 citations
Publisher

Abstract

The Quantum Fourier transform (QFT) is a key ingredient in most quantum algorithms. We have compared various spin-based quantum computing schemes to implement the QFT from the point of view of their actual time-costs and the accuracy of the implementation. We focus here on an interesting decomposition of the QFT as a product of the non-selective Hadamard transformation followed by multiqubit gates corresponding to square- and higher-roots of controlled-NOT gates. This decomposition requires only O(n) operations and is thus linear in the number of qubits n. The schemes were implemented on a two-qubit NMR quantum information processor and the resultant density matrices reconstructed using standard quantum state tomography techniques. Their experimental fidelities have been measured and compared.

DOI
10.17877/de290r-2457

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The quantum Fourier transform (QFT) is a key ingredient of several quantum algorithms and a qudit-specific implementation of the QFT is hence an important step toward the realization of qudit-based quantum computers. This work develops a circuit decomposition of the QFT for hybrid qudits based on generalized Hadamard and generalized controlled-phase gates, which can be implemented using selective rotations in NMR. We experimentally implement the hybrid qudit QFT on an NMR quantum emulator, which uses four qubits to emulate a single qutrit coupled to two qubits.
A quantum Fourier transform (QFT) has been implemented on a three qubit nuclear magnetic resonance (NMR) quantum computer to extract the periodicity of an input state. Implementation of a QFT provides a first step towards the realization of Shor's factoring and other quantum algorithms. The experimental implementation of the QFT on a periodic state is presented along with a quantitative measure of its efficiency measured through state tomography. Experimentally realizing the QFT is a clear demonstration of the ability of NMR to control quantum systems.
Quantum computing (QC) provides an efficient platform to solve complex problems such as number factoring and searching. The quantum Fourier transform (QFT) is an integral part of quantum algorithms for integer number factoring, phase estimation, discrete algorithms, interchange of position and momentum states, quantum key distribution protocol, multiparty quantum telecommunication, and quantum arithmetic. The theoretical and experimental implementations of QFT on various platforms have been proposed by researchers. Spin-torque-based qubit(s) manipulation is one of the encouraging solid-state device technologies. However, to date, QFT is not realized by spin-torque-based QC architecture. In this article, the spin-torque-based architecture has been modeled with the help of optimized decomposition of quantum circuits for the QFT. Moreover, an optimal-depth Clifford + T gates set-based quantum circuit is utilized to implement the QFT. The performance analysis in terms of fidelity (>99%), magnitude, and phase difference of respective density matrices for different forms of three-qubit QFT provides a novel way of its physical realization to address the complex problems.
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We found that the actual computational time-cost of the QFT is O(n 2^n) for large n in a quantum computer using nuclear spins. The computational cost of a quantum algorithm has usually been estimated as the sum of the universal gates required in such ideal mathematical models as the Quantum Turing Machine(QTM) and the quantum circuit. This cost is proportional to an actual time-cost in the physical implementation where all quantum operations can be achieved in the same time. However, if the implementation takes a different time for each quantum gate, there is a possibility that the actual time-cost will have a different behavior from the ideal cost. So we estimated the actual time-cost of the QFT in these implementations by considering the gating time. The actual time-cost is drastically different from O(n^2) estimated by complexity analysis.