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