  Simulating the time evolution of quantum field theories given some
Hamiltonian $H$ requires developing algorithms for implementing the unitary
operator e^{-iHt}. A variety of techniques exist that accomplish this task,
with the most common technique used so far being Trotterization, which is a
special case of the application of a product formula. However, other techniques
exist that promise better asymptotic scaling in certain parameters of the
theory being simulated, the most efficient of which are based on the concept of
block encoding.
  In this work we study the performance of such algorithms in simulating
lattice field theories. We derive and compare the asymptotic gate complexities
of several commonly used simulation techniques in application to Hamiltonian
Lattice Field Theories. Using the scalar \phi^4 theory as a test, we also
perform numerical studies and compare the gate costs required by Product
Formulas and Signal Processing based techniques to simulate time evolution. For
the latter, we use the the Linear Combination of Unitaries construction
augmented with the Quantum Fourier Transform circuit to switch between the
field and momentum eigenbases, which leads to immediate order-of-magnitude
improvement in the cost of preparing the block encoding.
  The paper also includes a pedagogical review of utilized techniques, in
particular Product Formulas, LCU, Qubitization, QSP, as well as a technique we
call HHKL based on its inventors' names.
