  Previous work has provided methods for decomposing unitary matrices to series
of quantum multiplexers, but the multiplexers created in this way are highly
non-minimal. This paper presents a new approach for optimizing quantum
multiplexers with arbitrary single-qubit quantum target functions. For quantum
multiplexers, we define standard forms and two types of new forms: fixed
polarity quantum forms (FPQF) and Kronecker quantum forms (KQF), which are
analogous to Minterm Sum of Products forms, Fixed Polarity Reed-Muller (FPRM)
forms, and Kronecker Reed-Muller (KRM) forms, respectively, for classical logic
functions. Drawing inspiration from the usage of butterfly diagrams for FPRM
and KRM forms, we devise a method to exhaustively construct all FPQF and KQF
forms. Thus, the new forms can be used to optimize quantum circuits with
arbitrary target unitary matrices, rather than only multi-controlled NOT gates
such as CNOT, CCNOT, and their extensions. Experimental results on FPQF and KQF
forms, as well as FPRM and KRM classical forms, applied to various target gates
such as NOT, V, V+, Hadamard, and Pauli rotations, demonstrate that FPQF and
KQF forms greatly reduce the gate cost of quantum multiplexers in both randomly
generated data and FPRM benchmarks.
