Algebraic compression of quantum circuits for Hamiltonian evolution

Unitary evolution under a time dependent Hamiltonian is key component of simulation on quantum hardware. Synthesizing the corresponding circuit typically done by breaking into small steps, also known as Trotterization, which leads to circuits whose depth scales with number steps. When elements are limited subset SU(4) -- or equivalently, when may be mapped onto free fermionic models several identities exist that combine and simplify circuit. Based this, we present an algorithm compresses Trotter steps single block gates. This results in fixed for certain classes Hamiltonians. We explicitly show how this works spin models, demonstrate its use adiabatic state preparation transverse field Ising model.