Quantum Simulation of the First-Quantized Pauli-Fierz Hamiltonian

TL;DR

This paper introduces a recursive divide and conquer quantum simulation algorithm for the first-quantized Pauli-Fierz Hamiltonian, demonstrating superior scaling in certain regimes compared to qubitization, with new gate optimization techniques.

Contribution

It presents a novel recursive divide and conquer approach for simulating quantum dynamics of the Pauli-Fierz Hamiltonian, outperforming qubitization in specific parameter regimes.

Findings

Divide and conquer algorithm scales as ( extLambda) N^2 ta^2 t^2 / \u03b5

Qubitization scales as (N(ta+N)(ta +mbda^2) t log(1/psilon))

Divide and conquer can be more efficient for large mbda regimes.

Abstract

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)) $. This shows that even a na\"ive partitioning and low-order splitting formula can yield, through our divide and conquer formalism, superior scaling to qubitization for large $\Lambda$. We compare the relative costs of these two algorithms on systems that are relevant for applications such as the spontaneous emission of photons, and the photoionization of electrons. We observe that for different parameter regimes, one method can be favored over the other. Finally, we give new algorithmic and circuit level techniques for gate optimization including a new way of implementing a group of multi-controlled-X gates that can be used for better analysis of circuit cost.