Quantum Simulation of the First-Quantized Pauli-Fierz Hamiltonian

Understand

We provide an explicit recursive divide and conquer approach for simulating quantum dynamics and derive a discrete first quantized non-relativistic QED Hamiltonian based on the many-particle Pauli Fierz Hamiltonian.

• We apply this recursive divide and conquer algorithm to this Hamiltonian and compare it to a concrete simulation algorithm that uses qubitization.
• Our divide and conquer algorithm, using lowest order Trotterization, scales for fixed grid spacing as $\widetilde{O}(\Lambda N^2\eta^2 t^2 /\epsilon)$ for grid size $N$, $\eta$ particles, simulation time $t$, field cutoff $\Lambda$ and error $\epsilon$.
• Our qubitization algorithm scales as $\widetilde{O}(N(\eta+N)(\eta +\Lambda^2) t\log(1/\epsilon)) $.