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IEEE International Symposium on Circuits and Systems proceedings

Quantum circuit design of 8 /spl times/ 8 discrete Hartley transform

Chien‐Cheng Tseng, Tsung-Ming Hwang

Published Jan 1, 2004• 10 citations
Publisher

Abstract

In this paper, the 8 /spl times/ 8 discrete Hartley transform will be implemented by using quantum elementary gates such as controlled NOT gates and Hadamard gates. Two steps involved are as follows: first, the discrete Hartley transform matrix is decomposed into the product of several sparse unitary matrices using the structure of fast Hartley transform algorithm. Then, each sparse matrix is implemented by elementary quantum gates and cascade them to obtain the final circuit. The design results show that the proposed design method has less circuit complexity than the conventional Cybenko's method.

DOI
10.1109/iscas.2004.1328767

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In this paper, the 8 /spl times/ 8 discrete Hartley transform will be implemented by using quantum elementary gates such as controlled NOT gates and Hadamard gates. Two steps involved are as follows: first, the discrete Hartley transform matrix is decomposed into the product of several sparse unitary matrices using the structure of fast Hartley transform algorithm. Then, each sparse matrix is implemented by elementary quantum gates and cascade them to obtain the final circuit. The design results show that the proposed design method has less circuit complexity than the conventional Cybenko's method.
In this paper, the quantum circuit design of a discrete Hartley transform (DHT) is investigated. The proposed design procedure can be divided into the following two steps. First, the N-point transform matrix is decomposed into the combination of N/2-point transform matrices and several sparse matrices such that any 2/sup n/-point transform can be computed recursively from the 2-point transform. Second, starting from the 2-point Hadamard transform, the quantum circuit of 2/sup n/-point transform can be designed recursively by using elementary quantum gates. The design results are compared with those of the conventional factorization method to demonstrate the effectiveness of proposed method.
In this paper, the quantum circuit design of an 8/spl times/8 discrete cosine transform (DCT) is investigated. The proposed design procedure can be divided into the following three steps. First, the DCT matrix is decomposed into the product of sparse matrices, based on its fast computation flow graph. Second, each sparse matrix is implemented by elementary quantum gates. Third, the sparse matrix circuits are cascaded to obtain the final circuit. The proposed method makes the fast DCT computation algorithm in digital signal processing suitable for implementation in a quantum computer.
In this paper, quantum circuit design of 8x8 discrete cosine transform (DCT) is investigated. The proposed design procedure can be divided into the following three steps. First, the DCT matrix is decomposed into the product of sparse matrices based on its fast computation flow graph. Second, each sparse matrix is implemented by elementary quantum gates. Third, cascade the circuits of sparse matrices to obtain the final circuit. The proposed method makes fast DCT computation algorithm in digital signal processing be suitable for implementation in quantum computer.
The design of efficient quantum circuits is an important issue in quantum computing. It is in general a formidable task to find a highly optimized quantum circuit for a given unitary matrix. We propose a quantum circuit design method that has the following unique feature: It allows to construct efficient quantum circuits in a systematic way by reusing and combining a set of highly optimized quantum circuits. Specifically, the method realizes a quantum circuit for a given unitary matrix by implementing a linear combination of representing matrices of a group, which have known fast quantum circuits. We motivate and illustrate this method by deriving extremely efficient quantum circuits for the discrete Hartley transform and for the fractional Fourier transforms. The sound mathematical basis of this design method allows to give meaningful and natural interpretations of the resulting circuits. We demonstrate this aspect by giving a natural interpretation of known teleportation circuits.
This paper focuses on quantum algorithms for three key matrix operations: Hadamard (Schur) product, Kronecker (tensor) product, and elementary column transformations each. By designing specific unitary transformations and auxiliary quantum measurement, efficient quantum schemes with circuit diagrams are proposed. Their computational depths are: O(1) for Kronecker product; O(max(m,n)) for Hadamard product (linked to matrix dimensions); and O(m) for elementary column transformations of (2^n X 2^m) matrices (dependent only on column count).Notably, compared to traditional column transformation via matrix transposition and row transformations, this scheme reduces computation steps and quantum gate usage, lowering quantum computing energy costs.
The design of efficient quantum circuits is an important issue in quantum computing. It is in general a formidable task to find a highly optimized quantum circuit for a given unitary matrix. We propose a quantum circuit design method that has the following unique feature: It allows to construct efficient quantum circuits in a systematic way by reusing and combining a set of highly optimized quantum circuits. Specifically, the method realizes a quantum circuit for a given unitary matrix by implementing a linear combination of representing matrices of a group, which have known fast quantum circuits. We motivate and illustrate this method by deriving extremely efficient quantum circuits for the discrete Hartley transform and for the fractional Fourier transforms. The sound mathematical basis of this design method allows to give meaningful and natural interpretations of the resulting circuits. We demonstrate this aspect by giving a natural interpretation of known teleportation circuits.