QMDD and Spectral Transformation of Binary and Multiple-Valued Functions*

The use of decision diagrams (DD) for the computation and representation of binary function spectra has been well studied [2,3,5]. Computations and representation of spectra for multiplevalued logic (MVL) functions have also been considered [6]. For binary functions, this approach can be implemented using one of a number of the highly efficient publicly available binary decision diagram (BDD) packages, e.g. CUDD [13]. Work with MVL functions requires a package suited to the MVL case, e.g. [7,8]. Quantum multiple-valued decision diagrams (QMDD) were introduced in [9-12] as a means to represent and manipulate the matrices required for binary or multiple-valued reversible and quantum gates and circuits. In this paper, we show how QMDD can also be applied to the computation of spectral transformations of binary and multiple-valued functions. A major motivation for this work is that it introduces an approach to the spectral analysis of reversible and quantum circuits in a representation that is applicable to simulation and synthesis of such circuits. It is also of interest in that it is a consistent approach for a variety of transformations of binary and multiple-valued functions. 1. Spectral Transformation We only consider issues here that are involved in computing spectral transformations and not the reasons for doing so or how spectral representations assist in analysis and synthesis problems. The reader interested in those matters should consult the literature, e.g. [4,14]. It is also important to note that we only show three representative spectral transformations. The approach is directly applicable to many other transformations found in the literature. Consider a binary or MVL n-variable function represented by a truth column vector F. We are interested in spectral transformations defined as in eqn. 1 where n T is a n n r r ! transformation matrix and r is the radix of the function. The transformation matrices of interest are defined by the Kronecker product shown in eqn. 2 where 1 T is a r r ! ‘base’ matrix defining the transformation. n S T F " (1) 1 1 n n i T T " " # (2) Rademacher-Walsh: 1 1 1 1 1 T $ % " & ' ( ) * (3a) Reed-Muller: 1 1 0 1 1 T $ % " & ' ) * (3b)