  The Pauli-based Circuit Optimization, Analysis and Synthesis Toolchain
(PCOAST) was recently introduced as a framework for optimizing quantum
circuits. It converts a quantum circuit to a Pauli-based graph representation
and provides a set of optimization subroutines to manipulate that internal
representation as well as methods for re-synthesizing back to a quantum
circuit. In this paper, we focus on the set of subroutines which look to
optimize the PCOAST graph in cases involving unitary and non-unitary operations
as represented by nodes in the graph. This includes reduction of node cost and
node number in the presence of preparation nodes, reduction of cost for
Clifford operations in the presence of preparations, and measurement cost
reduction using Clifford operations and the classical remapping of measurement
outcomes. These routines can also be combined to amplify their effectiveness.
  We evaluate the PCOAST optimization subroutines using the Intel Quantum SDK
on examples of the Variational Quantum Eigensolver (VQE) algorithm. This
includes synthesizing a circuit for the simultaneous measurement of a mutually
commuting set of Pauli operators. We find for such measurement circuits the
overall average ratio of the maximum theoretical number of two-qubit gates to
the actual number of two-qubit gates used by our method to be 7.91.
