Nearly tight Trotterization of interacting electrons

Abstract

We consider simulating quantum systems on digital quantum computers. We show that the performance of quantum simulation can be improved by simultaneously exploiting commutativity of the target Hamiltonian, sparsity of interactions, and prior knowledge of the initial state. We achieve this using Trotterization for a class of interacting electrons that encompasses various physical systems, including the plane-wave-basis electronic structure and the Fermi-Hubbard model. We estimate the simulation error by taking the transition amplitude of nested commutators of the Hamiltonian terms within the<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>η</mml:mi></mml:math>-electron manifold. We develop multiple techniques for bounding the transition amplitude and expectation of general fermionic operators, which may be of independent interest. We show that it suffices to use<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mfrac><mml:msup><mml:mi>n</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mn>5</mml:mn><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>/</mml:mo></mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mi>η</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mn>2</mml:mn><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>/</mml:mo></mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup></mml:mfrac><mml:mo>+</mml:mo><mml:msup><mml:mi>n</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mn>4</mml:mn><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>/</mml:mo></mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mi>η</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mn>2</mml:mn><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>/</mml:mo></mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:msup><mml:mi>n</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi>o</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math>gates to simulate electronic structure in the plane-wave basis with<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>n</mml:mi></mml:math>spin orbitals and<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>η</mml:mi></mml:math>electrons, improving the best previous result in second quantization up to a negligible factor while outperforming the first-quantized simulation when<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>n</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mi>η</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mn>2</mml:mn><mml:mo>−</mml:mo><mml:mi>o</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math>. We also obtain an improvement for simulating the Fermi-Hubbard model. We construct concrete examples for which our bounds are almost saturated, giving a nearly tight Trotterization of interacting electrons.

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