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Part of: Synthesis digital library of engineering and computer science.

Includes bibliographical references (pages 99-104) and index.

1. Introduction -- 1.1 Conventional computing -- 1.2 Boolean functions of one variable -- 1.3 Boolean functions of two variables -- 1.4 Boolean functions of n variables -- 1.4.1 The minterm expansion -- 1.4.2 The Reed-Muller expansion -- 1.4.3 The minimal ESOP expansion -- 1.5 Group theory -- 1.6 Reversible computing -- 1.7 Permutation groups -- 1.8 A permutation decomposition -- 1.9 Matrix groups -- 1.10 Subgroups -- 1.11 Young subgroups -- 1.12 Quantum computing -- 1.13 Bottom-up vs. top-down --

2. Bottom -- 2.1 The group S2 -- 2.2 Two important young subgroups of S2w -- 2.2.1 Controlled circuits -- 2.2.2 Controlled NOT gates -- 2.2.3 Controlled circuits vs. controlled gates -- 2.3 Primal decomposition -- 2.4 Dual decomposition -- 2.5 Synthesis efficiency -- 2.6 Refined synthesis algorithm -- 2.7 Examples -- 2.8 Variable ordering --

3. Bottom-up -- 3.1 The square root of the NOT -- 3.1.1 One-(qu)bit calculations -- 3.1.2 Two and multi-(qu)bit calculations -- 3.2 More roots of NOT -- 3.3 NEGATORs -- 3.4 NEGATOR circuits -- 3.5 The group ZU(n) -- 3.6 The group XU(n) -- 3.7 A matrix decomposition -- 3.8 Group hierarchy --

4. Top -- 4.1 Preliminary circuit decomposition -- 4.2 Primal decomposition -- 4.3 Group structure -- 4.4 Dual decomposition -- 4.5 Detailed procedure -- 4.6 Examples -- 4.7 Synthesis efficiency -- 4.8 Further synthesis -- 4.9 An extra decomposition --

5. Top-down -- 5.1 Top vs. bottom -- 5.2 Light matrices -- 5.3 Primal decomposition -- 5.4 Group hierarchy -- 5.5 Dual decomposition --

6. Conclusion -- A. Polar decomposition -- Bibliography -- Authors' biographies -- Index.

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At first sight, quantum computing is completely different from classical computing. Nevertheless, a link is provided by reversible computation. Whereas an arbitrary quantum circuit, acting on w qubits, is described by an n x n unitary matrix with n = 2w, a reversible classical circuit, acting on w bits, is described by a 2w x 2w permutation matrix. The permutation matrices are studied in group theory of finite groups (in particular the symmetric group Sn); the unitary matrices are discussed in group theory of continuous groups (a.k.a. Lie groups, in particular the unitary group U(n)). Both the synthesis of a reversible logic circuit and the synthesis of a quantum logic circuit take advantage of the decomposition of a matrix: the former of a permutation matrix, the latter of a unitary matrix. In both cases the decomposition is into three matrices. In both cases the decomposition is not unique.

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