Abstract
We provide an explicit recursive divide-and-conquer approach for simulating quantum dynamics and derive a discrete first-quantized nonrelativistic 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 for grid size , particles, simulation time , field cutoff , and error . Our qubitization algorithm scales as . This shows that even a naive partitioning and low-order splitting formula can yield, through our divide-and-conquer formalism, superior scaling to qubitization for large . 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 multicontrolled- gates that can be used for better analysis of circuit cost.
- Received 27 August 2023
- Accepted 21 February 2024
DOI:https://doi.org/10.1103/PRXQuantum.5.010345
Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.
Published by the American Physical Society
Physics Subject Headings (PhySH)
Popular Summary
Modern technology is built on our understanding of how light and matter interact at a fundamental level, from designing computer hardware to video display technologies to efficient solar panels. However, as these technologies advance, quantum effects of electrons and photons can dominate on the nanoscale. Even in the nonrelativistic regime, simulating light-matter interactions to full accuracy is intractable on the most advanced classical computers. However, when larger quantum computers become available, full quantum simulations in this regime become possible. Despite quantum computers’ potential, their limited speed necessitates efficient algorithms for practical simulations. In this work, we derive two efficient approaches to simulate nonrelativistic light-matter interactions on a quantum computer.
Specifically, we derive a general first-quantized spin- particle Pauli-Fierz Hamiltonian to simulate with two approaches. First, in a divide-and-conquer approach, we divide the Hamiltonian into fragments using the Trotter-Suzuki formula and simulate each fragment using different algorithms and then combine the results. Second, we analyze using qubitization for the full simulation. By comparing the relative costs of these methods, we observe that divide and conquer performs better with respect to certain parameter regimes.
Overall, we find that our approaches are efficient on a quantum computer. Further open questions involve whether these ideas can be generalized to a broader class of physical models including non-Abelian gauge theories needed in the standard model. Designing approaches that go beyond the limitations of this model will be an important step toward the quantum simulation of nature.
