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IEEE Transactions on Circuits and Systems I Regular Papers

Quantum Implementation Circuits of Quantum Signal Representation and Type Conversion

Hai-Sheng Li, Ping Fan, Haiying Xia, Huiling Peng, Shuxiang Song

Published Jul 23, 2018• 110 citations
Publisher

Abstract

Quantum signal processing offers a possible solution to store and process massive signals efficiently. As the foundation of quantum signal processing, quantum signal representation (QSR) that stores color information using basis states, is convenient to retrieve accurately signals from quantum systems. But the existing QSRs are only focus on integer signals (i.e., images) and are difficult to be directly operated by quantum Fourier transform (QFT) and quantum wavelet transform (QWT). Thus, we propose four models of QSR for integer, real, and complex signals. Furthermore, we design quantum implementation circuits of type conversions of QSR with complexity$O$(1). Type conversions of QSR are illustrated by general processes of QFT and Haar QWT, and demonstrate that the proposed QSRs are suitable for QFT and QWT.

DOI
10.1109/tcsi.2018.2853655

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Quantum signal processing offers a possible solution to store and process massive signals efficiently. As the foundation of quantum signal processing, quantum signal representation (QSR) that stores color information using basis states, is convenient to retrieve accurately signals from quantum systems. But the existing QSRs are only focus on integer signals (i.e., images) and are difficult to be directly operated by quantum Fourier transform (QFT) and quantum wavelet transform (QWT). Thus, we propose four models of QSR for integer, real, and complex signals. Furthermore, we design quantum implementation circuits of type conversions of QSR with complexityOOO(1). Type conversions of QSR are illustrated by general processes of QFT and Haar QWT, and demonstrate that the proposed QSRs are suitable for QFT and QWT.
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The quantum Fourier transform and quantum wavelet transform have been cornerstones of quantum information processing. However, for non-stationary signals and anomaly detection, the Hilbert transform can be a more powerful tool, yet no prior work has provided efficient quantum implementations for the discrete Hilbert transform. This letter presents a novel construction for a quantum Hilbert transform in polylogarithmic size and logarithmic depth for a signal of length NNN, exponentially fewer operations than classical algorithms for the same mapping. We generalize this algorithm to create any ddd-dimensional Hilbert transform in depth O(dlog⁡N)O(d\log N)O(dlogN). Simulations demonstrate effectiveness for tasks such as power systems control and image processing, with exact agreement with classical results.
This work focuses into the emerging field of quantum image representation, which is a key for processing grayscale and color images on quantum computing platforms. We explore foundational methods like the qubit lattice model, the flexible representation for quantum images, the novel enhanced quantum representation, and the generalized quantum image representation, and categorize them into state-based, amplitude-based, and quaternion-based representations - the latter offering holistic color processing. A novel discrete signal representation mapping data on the unit circle will be introduced to simplify the Fourier transform concepts in quantum systems. We will examine crucial quantum operations, including quantum gradients, multiplication, square roots, Fourier transform, and filtering. Challenges in implementing the circular convolution on quantum hardware will be addressed with practical examples such as quantum low-pass and high-pass filters and 3-qubit convolution circuits. Additionally, we will discuss ongoing research into quantum algorithms inspired by classical transform processing, including the Fourier transforms, that leverage quantum states’ unique properties to achieve efficient encoding, transformation, and decoding of images.
The classical wavelet transform has been widely applied in the information processing field. It implies that quantum wavelet transform (QWT) may play an important role in quantum information processing. This chapter firstly describes the iteration equations of the general QWT using generalized tensor product. Then, Haar QWT (HQWT), Daubechies D4 QWT (DQWT), and their inverse transforms are proposed respectively. Meanwhile, the circuits of the two kinds of multi-level HQWT are designed. What's more, the multi-level DQWT based on the periodization extension is implemented. The complexity analysis shows that the proposed multi-level QWTs on 2n elements can be implemented by O(n3) basic operations. Simulation experiments demonstrate that the proposed QWTs are correct and effective.
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Algorithms such as quantum wavelet transform (QWT) can be utilized on quantum processors to gain significant speedup compared to their classical counterparts. Domains such as high-energy-physics and remote-sensing hyperspectral imagery require high compute capabilities and can benefit from applying QWT for dimension reduction of multidimensional data. However, quantum circuits for QWT include permutations that contribute to large overall circuit depth. Deep circuits pose a critical challenge for state-of-the-art quantum computers because of quantum decoherence. In this work, we propose QWT circuits, which are optimized in terms of circuit depth, to account for the effects of decoherence, and resulting in high fidelity and efficient implementation on quantum processors. We present the circuits in generalized forms and show that they can be used for multilevel decomposable, multidimensional wavelet operations. Experimental evaluation of the proposed circuits is performed through simulation using MATLAB and IBM-Q qasm, and implemented on a real 15-qubit quantum processor from IBM.