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Chinese Science Bulletin

t-bit semiclassical quantum Fourier transform

Xiang-Qun Fu, Wan‐Su Bao, Chun Hua Zhou, Zhen Song

Published Jan 1, 2012• 3 citations
Publisher

Abstract

Because of the difficulty of building a high-dimensional quantum register, this paper presents an implementation of the high-dimensional quantum Fourier transform (QFT) based on a low-dimensional quantum register. First, we define the t-bit semiclassical quantum Fourier transform. In terms of probability amplitude, we prove that the transform can realize quantum Fourier transformation, illustrate that the requirement for the two-qubit gate reduces obviously, and further design a quantum circuit of the transform. Combining the classical fixed-window method and the implementation of Shor’s quantum factorization algorithm, we then redesign a circuit for Shor’s algorithm, whose required computation resource is approximately equal to that of Parker’s. The requirement for elementary quantum gates for Parker’s algorithm is O(⌈ log N⌉3, and the quantum register for our circuit requires t−1 more dimensions than Parker’s. However, our circuit is t 2 times as fast as Parker’s, where t is the width of the window.

DOI
10.1007/s11434-011-4692-8

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Because of the difficulty of building a high-dimensional quantum register, this paper presents an implementation of the high-dimensional quantum Fourier transform (QFT) based on a low-dimensional quantum register. First, we define the t-bit semiclassical quantum Fourier transform. In terms of probability amplitude, we prove that the transform can realize quantum Fourier transformation, illustrate that the requirement for the two-qubit gate reduces obviously, and further design a quantum circuit of the transform. Combining the classical fixed-window method and the implementation of Shor’s quantum factorization algorithm, we then redesign a circuit for Shor’s algorithm, whose required computation resource is approximately equal to that of Parker’s. The requirement for elementary quantum gates for Parker’s algorithm is O(⌈ log N⌉3, and the quantum register for our circuit requires t−1 more dimensions than Parker’s. However, our circuit is t 2 times as fast as Parker’s, where t is the width of the window.
Since the difficulty in preparing the equal superposition state of amplitude is 1/√ N , we construct a quantile transform of quantum Fourier transform (QFT) over Z N based on the elementary transforms, such as Hadamard transform and Pauli transform. The QFT over Z N can then be realized by the quantile transform, and used to further design its quantum circuit and analyze the requirements for the quantum register and quantum gates. However, the transform needs considerable quantum computational resources and it is difficult to construct a high-dimensional quantum register. Hence, we investigate the design of t -bit quantile transform, and introduce the definition of t -bit semiclassical QFT over Z N . According to probability amplitude, we prove that the transform can be used to realize QFT over Z N and further design its quantum circuit. For this transform, the requirements for the quantum register, the one-qubit gate, and two-qubit gate reduce obviously when compared with those for the QFT over Z N .
To overcome the difficulty of realizing large-scale quantum Fourier transform (QFT) within existing technology, this paper presents a resource-saving method, namely t-bit semiclassical QFT over (Z_(2^n)), which could realize large-scale QFT using arbitrary-scale quantum register. Using our method, the scale of quantum register can be determined flexibility according to the scale of quantum system, enabling the quantum resource and speed of realizing QFT to be optimal. By developing a feasible method to realize the control quantum gate R_k, we experimentally demonstrate the 2-bit semiclassical QFT over (Z_(2^3)) on IBM's quantum cloud computer, showing the feasibility of our proposed method. Then, we compare the actual performance of 2-bit semiclassical QFT and standard QFT in the experiments. Experimental data show that the fidelity of the result of 2-bit semiclassical QFT is higher than that of standard QFT, which is mainly due to less two-qubit controlled gates are required in the semiclassical QFT. Furthermore, based on the proposed method, we successfully implement the Shor's algorithm to factorize N=15.
Abstract We propose an implementation of the algorithm for the fast Fourier transform (FFT) as a quantum circuit consisting of a combination of some quantum gates. In our implementation, a data sequence is expressed by a tensor product of vector spaces. Namely, our FFT is defined as a transformation of the tensor product of quantum states. It is essentially different from the so-called quantum Fourier transform (QFT) defined to be a linear transformation of the amplitudes for the superposition of quantum states. The quantum circuit for the FFT consists of several circuits for elementary arithmetic operations such as a quantum adder, subtractor and shift operations, which are implemented as effectively as possible. Namely, our circuit does not generate any garbage bits. The advantages of our method compared to the QFT are its high versatility, and data storage efficiency in terms, for instance, of the quantum image processing.
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.
It is shown that the Fourier transform preceding the final measurement in Shor's algorithm for factorization on a quantum computer can be carried out in a semiclassical way by using the ``classical'' (macroscopic) signal resulting from measuring one bit to determine the type of measurement carried out on the next bit, and so forth. In this way all the two-bit gates in the Fourier transform can be replaced by a smaller number of one-bit gates controlled by classical signals. This suggests that it may be worthwhile looking for other uses of semiclassical methods in quantum computing.
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.