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Superlattices and Microstructures

A complete set of transformation rules for quantum Boolean circuits with CNOT gates

Kazuo Iwama, Shigeru Yamashita

Published Feb 1, 2002• 2 citations
Publisher
DOI
10.1006/spmi.2002.1039

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We introduce a complete transformation rule set and a minimal equation set for controlled-NOT (CNOT)-based quantum circuits. Using these rules, quantum circuits that compute the same Boolean function are reduced to the same normal form. We can thus easily check the equivalence of circuits by comparing their normal forms. By applying the Knuth-Bendix completion algorithm to a set of modified 18 equations introduced by Iwama et al. 2002, we obtain a complete transformation rule set (i.e., a set of transformation rules with the properties of `termination' and `confluence'). Our transformation rule set consists of 114 rules. Moreover, we discovered a minimal combination of equations for the initial equation set.
This paper gives a simple but nontrivial set of local transformation rules for control-NOT (CNOT)-based combinatorial circuits. It is shown that this rule set is complete, namely, for any two equivalent circuits, S/sub 1/ and S/sub 2/, there is a sequence of transformations, each of them in the rule set, which changes S/sub 1/ to S/sub 2/. Our motivation is to use this rule set for developing a design theory for quantum circuits whose Boolean logic parts should be implemented by CNOT-based circuits. As a preliminary example, we give a design procedure based on the transformation rules which reduces the cost of CNOT-based circuits.