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Computer Engineering and Applications Journal

Circuits and simulation of multi-qubit quantum wavelet transforms

Zhu Xuan-yi

Published Jan 1, 2008• 0 citations
OpenAlex

Abstract

Because of the prominent advantages of quantum computation compared to classic computation,implementation of quantum wavelet transforms has profound significance to its completion and application,which based on the circuits of those transforms.In this paper,the circuits of 3-qubit Haar and D(4) wavelet transforms are designed by way of multiple-quantum operator algebra theory.Furthermore,the circuits are evolved into pulse sequences,which can be implemented in a real NMR system.Finally,based on the simulation on a Quantum Computer Emulator,the circuits are verified to be reasonable.

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Because of the prominent advantages of quantum computation compared to classic computation,implementation of quantum wavelet transforms has profound significance to its completion and application,which based on the circuits of those transforms.In this paper,the circuits of 3-qubit Haar and D(4) wavelet transforms are designed by way of multiple-quantum operator algebra theory.Furthermore,the circuits are evolved into pulse sequences,which can be implemented in a real NMR system.Finally,based on the simulation on a Quantum Computer Emulator,the circuits are verified to be reasonable.
Because of the prominent advantages of quantum computation compared to classic computation,implementation of quantum wavelet transforms has profound significance to its completion and application.After the circuit for perfect shuffle permutation matrices is finished,the unitary matrices for Harr functions are decomposed into a sequence of W-H and perfect shuffle permutation matrices,based on the generalized Kronecker product.Then the quantum circuit for Harr matrices is provided,its realizable complexity and pos-sibility of physical implementation is analyzed.
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.
The quantum Fourier transform (QFT), a quantum analog of the classical Fourier transform, has been shown to be a powerful tool in developing quantum algorithms. However, in classical computing there is another class of unitary transforms, the wavelet transforms, which are every bit as useful as the Fourier transform. Wavelet transforms are used to expose the multi-scale structure of a signal and are likely to be useful for quantum image processing and quantum data compression. In this paper, we derive efficient, complete, quantum circuits for two representative quantum wavelet transforms, the quantum Haar and quantum Daubechies D(4)D^{(4)}D(4) transforms. Our approach is to factor the operators for these transforms into direct sums, direct products and dot products of unitary matrices. In so doing, we find that permutation matrices, a particular class of unitary matrices, play a pivotal role. Surprisingly, we find that operations that are easy and inexpensive to implement classically are not always easy and inexpensive to implement quantum mechanically, and vice versa. In particular, the computational cost of performing certain permutation matrices is ignored classically because they can be avoided explicitly. However, quantum mechanically, these permutation operations must be performed explicitly and hence their cost enters into the full complexity measure of the quantum transform. We consider the particular set of permutation matrices arising in quantum wavelet transforms and develop efficient quantum circuits that implement them. This allows us to design efficient, complete quantum circuits for the quantum wavelet transform.
Currently,the NMR technology is one of the effective physical systems to realize quantum computation.As the multiple-quantum operator algebra theory mentions,any unitary transformation can be decomposed into a sequence of a limited number of one-qubit quantum gates and two-qubit diagonal gates.Based on these backgrounds,we propose a method to form NMR pulse sequences to realize the multi-qubit Grover quantum search algorithm.Experiments finish the algorithm with three-qubit,on a Quantum Computer Emulator.
The Haar wavelet algorithm based on quantum computing and its application for color image compression are discussed.Firstly,classical Haar wavelet operator is decomposed into direct sums,direct products and dot products of unitary matrices,so that shuffle permutation matrices is found.Secondly,based on the generalized Kronecker product,Harr functions are decomposed into a sequence of W-H and shuffle permutation matrices.Thirdly,the efficient,complete,quantum circuits for representative quantum wavelet transform are given.Finally,algorithm for color image compression by quantum Haar wavelet is presented,and experiment results are given.The experiment results show that the quantum Haar wavelet has very good feasibility in image compression.
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.