  This paper develops practical summation techniques in ZXW calculus to reason
about quantum dynamics, such as unitary time evolution. First we give a direct
representation of a wide class of sums of linear operators, including arbitrary
qubits Hamiltonians, in ZXW calculus. As an application, we demonstrate the
linearity of the Schroedinger equation and give a diagrammatic representation
of the Hamiltonian in Greene-Diniz et al, which is the first paper that models
carbon capture using quantum computing. We then use the Cayley-Hamilton theorem
to show in principle how to exponentiate arbitrary qubits Hamiltonians in ZXW
calculus. Finally, we develop practical techniques and show how to do Taylor
expansion and Trotterization diagrammatically for Hamiltonian simulation. This
sets up the framework for using ZXW calculus to the problems in quantum
chemistry and condensed matter physics.
