Simplifying the simulation of local Hamiltonian dynamics

Local Hamiltonians <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:msub><a:mi>H</a:mi><a:mi>k</a:mi></a:msub></a:math> describe nontrivial <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mi>k</b:mi></b:math>-body interactions in quantum many-body systems. Here, we address the dynamical simulatability of a <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mi>k</c:mi></c:math>-local Hamiltonian by a simpler one <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:msub><d:mi>H</d:mi><d:msup><d:mi>k</d:mi><d:mo>′</d:mo></d:msup></d:msub></d:math> with <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mrow><e:msup><e:mi>k</e:mi><e:mo>′</e:mo></e:msup><e:mo><</e:mo><e:mi>k</e:mi></e:mrow></e:math>, under the realistic constraint that both Hamiltonians act on the same Hilbert space. When it comes to exact simulation, we build upon known methods to derive examples of <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"><f:msub><f:mi>H</f:mi><f:mi>k</f:mi></f:msub></f:math> and <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"><g:msub><g:mi>H</g:mi><g:msup><g:mi>k</g:mi><g:mo>′</g:mo></g:msup></g:msub></g:math> that simulate the same physics. We also address the most realistic case of approximate simulation. There, we upper-bound the error up to which a Hamiltonian can simulate another one, regardless of their internal structure, and show an example suggesting that the accuracy of a <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"><h:mrow><h:mo>(</h:mo><h:msup><h:mi>k</h:mi><h:mo>′</h:mo></h:msup><h:mo>=</h:mo><h:mn>2</h:mn><h:mo>)</h:mo></h:mrow></h:math>-local Hamiltonian to simulate <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"><i:msub><i:mi>H</i:mi><i:mi>k</i:mi></i:msub></i:math> with <j:math xmlns:j="http://www.w3.org/1998/Math/MathML"><j:mrow><j:mi>k</j:mi><j:mo>></j:mo><j:mn>2</j:mn></j:mrow></j:math> increases with <k:math xmlns:k="http://www.w3.org/1998/Math/MathML"><k:mi>k</k:mi></k:math>. Finally, we propose a method to search for the <l:math xmlns:l="http://www.w3.org/1998/Math/MathML"><l:msup><l:mi>k</l:mi><l:mo>′</l:mo></l:msup></l:math>-local Hamiltonian that simulates, with the highest possible precision, the short time dynamics of a given <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:msub><m:mi>H</m:mi><m:mi>k</m:mi></m:msub></m:math> Hamiltonian.