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Mediterranean Conference on Embedded Computing (MECO)

Implementation and Analysis of Quantum Fourier Transform Gates Over FPGA Framework

Tin Moe Aye, Masahiro Iwahashi

Published Jun 1, 2019• 3 citations
Publisher

Abstract

Since the emergence of the Shor's factoring algorithm, the research over Quantum Fourier Transform (QFT) has motivated the broad attention in the dilemma of the efficient quantum computation. For the purpose of exploring the sizable extent for their exponential growth and the massive computational steps, this paper confers the Field Programmable Gate Arrays (FPGA) implementation of the QFT and, investigate the association between the logic density, complexity and power consumption. In this context, reusability as well as compensating resource utilization is discussed together with the observing issues.

DOI
10.1109/meco.2019.8760171

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Since the emergence of the Shor's factoring algorithm, the research over Quantum Fourier Transform (QFT) has motivated the broad attention in the dilemma of the efficient quantum computation. For the purpose of exploring the sizable extent for their exponential growth and the massive computational steps, this paper confers the Field Programmable Gate Arrays (FPGA) implementation of the QFT and, investigate the association between the logic density, complexity and power consumption. In this context, reusability as well as compensating resource utilization is discussed together with the observing issues.
Quantum computation shows its great advantage on exponential speed-up for some complex algorithms. Compared to software simulation with sequential execution, hardware emulation based on Field Programmable Gate Array (FPGA) can help to realize the intrinsic quantum parallelism. Quantum Fourier Transform (QFT) is the core of several quantum algorithms like Shor's algorithm. This paper proposes a recursive decomposition of QFT-n, which can be emulated with a pipeline architecture based on Intel Stratix IV FPGA device. Although QFT-8 is much more complex than QFT-5, experimental results show that our QFT-8 architecture can reduces 91% DSP blocks and 76% combinational ALUTs at the cost of 11x registers compared with QFT-5 in [4] on average.
Hardware emulation based on field programmable gate array (FPGA) platform is vital to harness the power of quantum parallelism. As resource requirement grows exponentially in classical modeling quantum system, an optimum hardware architecture is crucial to emulate practical quantum circuits. Quantum Fourier transform (QFT) finds application in several critical quantum algorithms. In this work, experiments are conducted based on QFT to identify suitable qubit representation and hardware design technique. Experimental results show that 24-bit fixed point representation and serial architecture achieve optimal computation accuracy and resource utilization in QFT circuit emulation.
This survey presents a comprehensive analysis of the quantum Fourier transform (QFT), a fundamental tool in quantum computing, in comparison to its classical counterpart, the fast Fourier transform (FFT). The study begins with an introduction to the classical Fourier transform, which is widely used in different disciplines such as signal and image processing. The article introduces the QFT as a quantum version of the Fourier transform, detailing how it leverages quantum parallelism and superposition to reduce the time complexity of the operation, and highlighting its crucial role in quantum algorithms like Shor’s algorithm for integer factorization. The analysis also addresses the mathematical foundations of the QFT, its implementation in quantum circuits, and the key advantages and challenges associated with its use, such as measurement precision and quantum decoherence. Finally, it concludes with an exploration of the current and potential applications of QFT in quantum computing.
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With the development of quantum algorithms and practical quantum computing devices, how to efficiently simulate emerging quantum algorithms via quantum circuit simulators on classical parallel-computing hardware platforms has raised more and more interest. The superior customized parallelism and pipeline efficiency over general-purpose processors make FPGAs ideal for large scale quantum algorithms/circuits simulation. Since the quantum Fourier transform (QFT) has been used in many quantum algorithms, such as quantum linear algebra, quantum phase estimation and quantum machine leaning, this paper presented a novel precision analysis and an efficient customized-precision QFT simulator accelerated on AMD Alveo U55C FPGAs. Customized-precision representations via high-level synthesis are employed for QFT implementation with up to 16 qubits, demonstrating up-to 8.3× speedup versus implementations in floating-point arithmetic within an error tolerance of 0.1%.
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.