Quantum Mass Production Theorems

We prove that for any n-qubit unitary transformation U and for any r = 2^{o(n / log n)}, there exists a quantum circuit to implement U^{? r} with at most O(4?) gates. This asymptotically equals the number of gates needed to implement just a single copy of a worst-case U. We also establish analogous results for quantum states and diagonal unitary transformations. Our techniques are based on the work of Uhlig [Math. Notes 1974], who proved a similar mass production theorem for Boolean functions

Realizing Boolean Functions on Disjoint Sets of Variables

For switching functions $f$ let $C(f)$ be the combinatorial complexity of $f$. We prove that for every $\varepsilon greater than 0$ there are arbitrarily complex functions $f:\{0,1\}^{n} \rightarrow \{0,1\}^{n}$ such that $C(fxf) \leq (1+ \varepsilon) C(f)$ and arbitrarily complex functions $g:\{0,1\}$ such that $C(v c(fxf)) \leq (1 + \varepsilon)C(f)$. These results and the techniques developed to obtain them are used to show, that Ashenhurst decomposition of switching functions does not always yield optimal circuits and to prove a new result concerning the trade-off between circuit size and monotone circuit size

Advances in Functional Decomposition: Theory and Applications

Functional decomposition aims at finding efficient representations for Boolean functions. It is used in many applications, including multi-level logic synthesis, formal verification, and testing. This dissertation presents novel heuristic algorithms for functional decomposition. These algorithms take advantage of suitable representations of the Boolean functions in order to be efficient. The first two algorithms compute simple-disjoint and disjoint-support decompositions. They are based on representing the target function by a Reduced Ordered Binary Decision Diagram (BDD). Unlike other BDD-based algorithms, the presented ones can deal with larger target functions and produce more decompositions without requiring expensive manipulations of the representation, particularly BDD reordering. The third algorithm also finds disjoint-support decompositions, but it is based on a technique which integrates circuit graph analysis and BDD-based decomposition. The combination of the two approaches results in an algorithm which is more robust than a purely BDD-based one, and that improves both the quality of the results and the running time. The fourth algorithm uses circuit graph analysis to obtain non-disjoint decompositions. We show that the problem of computing non-disjoint decompositions can be reduced to the problem of computing multiple-vertex dominators. We also prove that multiple-vertex dominators can be found in polynomial time. This result is important because there is no known polynomial time algorithm for computing all non-disjoint decompositions of a Boolean function. The fifth algorithm provides an efficient means to decompose a function at the circuit graph level, by using information derived from a BDD representation. This is done without the expensive circuit re-synthesis normally associated with BDD-based decomposition approaches. Finally we present two publications that resulted from the many detours we have taken along the winding path of our research