Structural Rewriting in XOR-Majority Graphs

In this paper, we present a structural rewriting method for a recently proposed XOR-Majority graph (XMG), which has exclusive-OR (XOR), majority-of-three (MAJ), and inverters as primitives. XMGs are an extension of Majority-Inverter Graphs (MIGs). Previous work presented an axiomatic system, Omega, and its derived transformation rules for manipulation of MIGs. By additionally introducing XOR primitive, the identities of MAJ-XOR operations should be exploited to enable powerful logic rewriting in XMGs. We first proposed two MAJ-XOR identities and exploit its potential optimization opportunities during structural rewriting. Then, we discuss the rewriting rules that can be used for different operations. Finally, we also address structural XOR detection problem in MIG. The experimental results on EPFL benchmark suites show that the proposed method can optimize the size/depth product of XMGs and its mapped look-up tables (LUTs), which in turn benefits the quantum circuit synthesis that using XMG as the underlying logic representations.