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IEEE Transactions on Magnetics

Spin-Torque-Based Quantum Fourier Transform

Anant Aravind Kulkarni, Brajesh Kumar Kaushik

Published Aug 27, 2019• 4 citations
Publisher

Abstract

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.

DOI
10.1109/tmag.2019.2931278

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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.
Optimal Boolean Logic Quantum Circuit Decomposition for Spin-Torque-Based n\boldsymbol{n}n -Qubit Architecture
The semiconductor industry is facing a twofold challenge at sub-nanometer level; first, the high power dissipation in irreversible computing architectures due to information loss and second, inability to handle large data due to sequential information processing. These issues create obstacles in producing the presumed low-power complex computing outcomes. The heat dissipation can be reduced by utilizing the complementary metal-oxide-semiconductor-based reversible computing architectures. However, these architectures fail to enhance the performance owing to inability to process large data. Quantum computing (QC) can circumvent these problems due to its fundamental ineffaceable characteristics of quantum mechanics-based reversible computing and parallelism. Spin-torque-based physical realization is the most suitable platform for reversible computing due to the electron spin analogous to the qubit. However, optimal quantum circuits are required to physically realize the complex Boolean logic due to spin-qubit decoherence and reduce the number of transistor switching activities for the spin generation and injection required for the spin-qubit rotation. Therefore, in this paper, optimal quantum circuit decompositions are presented with the help of developed elementary quantum library {Ry(θ), Rz(θ), √SWAP for the spin-torque-based QC architecture. The reversible Boolean logic performance is analyzed and compared for the conventional, reduced, and optimal decompositions on the first- and second-order transmission coefficient matrix-based spin-torque QC architecture. The results encourage to set a path toward QC-based reconfigurable complex computing systems in near future.
Quantum computing due to its ability of inherent parallel processing is emerged as one of the solutions to the complex computing problems. In connection with this, several proposals have been given for the quantum circuits-based design of the reversible combinational and sequential circuits. However, the implementation of the reversible sequential logic is challenging in comparison to the reversible combinational logic. Spin-torque-based reconfigurable architecture could be utilized to realize the quantum circuits. However, the architecture needs the optimized decomposition of the quantum circuits utilized for the reversible sequential logic due to required number of single qubit rotations and two-qubit entanglements. In this paper, the elementary decomposition of the quantum circuits representing the reversible D-Latch is optimized with the help of elementary quantum library{Ry(θ), Rz(θ), SWAP}\{R_{\mathrm{y}}(\theta),\ R_{\mathrm{z}}(\theta),\ \sqrt{\text{SWAP}}\}{Ry​(θ), Rz​(θ), SWAP​}. The number of elementary operations required to realize the D-Latch over 5 clock cycles is reduced by 43.56%. The average fidelity of the D-Latch considered for the implementation and analysis is obtained at the end of each clock cycle is well above 97%. The fidelity is further improved by approximating the present state output utilized for the next state input.
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.
The Quantum Fourier Transform (QFT)andQuantumPhaseEstimation (QPE)are twofundamental algorithmic frameworks in quantum computation, and they provide the mathematical basis of many exponential-speedup quan tum algorithms, such as Shor’s factoring algorithm and quantum simulation techniques. In this work we present a detailed theoretical and computational study of the Quantum Fourier Transform and its integration in the Quan tum Phase Estimation. The discussion begins with the linear algebraic structure of the Hilbert spaces, the unitary transformations, and the complex vector representations that enable quantum interference and superposition. The QFT is studied as the unitary transform of computational basis states in the amplitudes in the frequency domain by phase rotations and tensor-product decompositions. The efficient circuit implementation in terms of Hadamard and controlled phase gates of this transform is discussed. The advantage of the transform in terms of polynomial time complexity over the classical discrete Fourier transform in some quantum scenarios is discussed.The QFT is implemented and the Quantum Phase Estimation algorithm is proposed to obtain the eigenphase information of the unitary operators. In the report the mathematical representation of QPE is derived by using the controlled unitary evolution, the inverse Fourier transforms and the measurement theory in finite dimensional Hilbert spaces. We discuss the probabilistic interpretation of phase extraction in detail and the convergence properties of the bi nary phase approximation. We also discuss the relation of QPE to eigenvalue problems in quantum mechanics and its applications to Hamiltonian simulation, quantum chemistry, cryptography, and hidden subgroup prob lems.Implementation aspects of these algorithms are studied using quantum circuit representations and oracle based computation models. We also discuss error sensitivity, qubit precision requirements, decoherence limita tions and practical constraints of Noisy Intermediate-Scale Quantum (NISQ) hardware. The work synthesizes the mathematical foundations, computational complexity analysis and physical realizations of the QFT and the QPE, demonstrating how the Fourier-based quantum transformations provide the essential mechanism underlying modern quantum algorithmic speedups and scalable quantum information processing.
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.
Abstract The ability to implement the Quantum Fourier Transform (QFT) efficiently on a quantum computer facilitates the advantages offered by a variety of fundamental quantum algorithms, such as those for integer factoring, computing discrete logarithm over Abelian groups, solving systems of linear equations, and phase estimation, to name a few. The standard fault-tolerant implementation of an n-qubit unitary QFT approximates the desired transformation by removing small-angle controlled rotations and synthesizing the remaining ones into Clifford+T gates, incurring the T-count complexity of O(n log2 (n))O(n\,{\mathrm{log}}^{2}\,(n))O(nlog2(n)) O ( n log 2 ( n ) ) . In this paper, we show how to obtain approximate QFT with the T-count of O(n log (n))O(n\,{\mathrm{log}}\,(n))O(nlog(n)) O ( n log ( n ) ) . For brevity, the above figures omit the dependence on the approximation error ε, assuming the error is fixed. Our approach relies on quantum circuits with measurements and feedforward, and on reusing a special quantum state that induces the phase gradient transformation. We report asymptotic analysis as well as concrete circuits, demonstrating significant advantages in both theory and practice.