Quantum Simulation of the First-Quantized Pauli-Fierz Hamiltonian

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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><a:mrow><a:mover><a:mi>O</a:mi><a:mo>~</a:mo></a:mover></a:mrow><a:mo stretchy="false">(</a:mo><a:mi mathvariant="normal">Λ</a:mi><a:msup><a:mi>N</a:mi><a:mn>2</a:mn></a:msup><a:msup><a:mi>η</a:mi><a:mn>2</a:mn></a:msup><a:msup><a:mi>t</a:mi><a:mn>2</a:mn></a:msup><a:mo>/</a:mo><a:mi>ϵ</a:mi><a:mo stretchy="false">)</a:mo></a:math> for grid size <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><g:mi>N</g:mi></g:math>, <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><j:mi>η</j:mi></j:math> particles, simulation time <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><m:mi>t</m:mi></m:math>, field cutoff <p:math xmlns:p="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><p:mi mathvariant="normal">Λ</p:mi></p:math>, and error <t:math xmlns:t="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><t:mi>ϵ</t:mi></t:math>. Our qubitization algorithm scales as <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><w:mrow><w:mover><w:mi>O</w:mi><w:mo>~</w:mo></w:mover></w:mrow><w:mo stretchy="false">(</w:mo><w:mi>N</w:mi><w:mo stretchy="false">(</w:mo><w:mi>η</w:mi><w:mo>+</w:mo><w:mi>N</w:mi><w:mo stretchy="false">)</w:mo><w:mo stretchy="false">(</w:mo><w:mi>η</w:mi><w:mo>+</w:mo><w:msup><w:mi mathvariant="normal">Λ</w:mi><w:mn>2</w:mn></w:msup><w:mo stretchy="false">)</w:mo><w:mi>t</w:mi><w:mi>log</w:mi><w:mo></w:mo><w:mo stretchy="false">(</w:mo><w:mn>1</w:mn><w:mo>/</w:mo><w:mi>ϵ</w:mi><w:mo stretchy="false">)</w:mo><w:mo stretchy="false">)</w:mo></w:math>. 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 <ib:math xmlns:ib="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><ib:mi mathvariant="normal">Λ</ib:mi></ib:math>. 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-<mb:math xmlns:mb="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><mb:mi>X</mb:mi></mb:math> gates that can be used for better analysis of circuit cost. Published by the American Physical Society 2024

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