Composite Qdrift-product formulas for quantum and classical simulations in real and imaginary time

Recent study has shown that it can be advantageous to implement a composite channel that partitions the Hamiltonian <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mi>H</a:mi></a:math> for a given simulation problem into subsets <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mi>A</b:mi></b:math> and <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mi>B</c:mi></c:math> such that <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mrow><d:mi>H</d:mi><d:mo>=</d:mo><d:mi>A</d:mi><d:mo>+</d:mo><d:mi>B</d:mi></d:mrow></d:math>, where the terms in <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mi>A</e:mi></e:math> are simulated with a Trotter-Suzuki channel and the <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"><f:mi>B</f:mi></f:math> terms are randomly sampled via the Qdrift algorithm. Here we extend Qdrift and composite product formulas to imaginary time, formulating candidate classical algorithms for quantum Monte Carlo calculations. We upper bound the induced Schatten-<g:math xmlns:g="http://www.w3.org/1998/Math/MathML"><g:mrow><g:mn>1</g:mn><g:mo>→</g:mo><g:mn>1</g:mn></g:mrow></g:math> norm on both imaginary-time Qdrift and composite channels. Another recent result demonstrated that simulations of lattice Hamiltonians containing geometrically local interactions can be improved using a Lieb-Robinson argument to decompose <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"><h:mi>H</h:mi></h:math> into subsets that contain only terms supported on that subset of the lattice. Here, we provide a quantum algorithm by unifying this result with the composite approach into “local composite channels” and we upper bound the diamond distance. We provide exact numerical simulations of algorithmic cost by counting the number of gates of the form <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"><i:msup><i:mi>e</i:mi><i:mrow><i:mo>−</i:mo><i:mi>i</i:mi><i:msub><i:mi>H</i:mi><i:mi>j</i:mi></i:msub><i:mi>t</i:mi></i:mrow></i:msup></i:math> and <j:math xmlns:j="http://www.w3.org/1998/Math/MathML"><j:msup><j:mi>e</j:mi><j:mrow><j:mo>−</j:mo><j:msub><j:mi>H</j:mi><j:mi>j</j:mi></j:msub><j:mi>β</j:mi></j:mrow></j:msup></j:math> to meet a certain error tolerance <k:math xmlns:k="http://www.w3.org/1998/Math/MathML"><k:mi>ε</k:mi></k:math>. In doing so, we optimize the partitioning into sets <l:math xmlns:l="http://www.w3.org/1998/Math/MathML"><l:mi>A</l:mi></l:math> and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mi>B</m:mi></m:math> using gradient boosted tree models from machine learning. These numerical studies are important given that product formulas have been historically known to outperform analytic upper bounds. We show constant factor advantages for a variety of interesting Hamiltonians, the maximum of which is a <n:math xmlns:n="http://www.w3.org/1998/Math/MathML"><n:mrow><n:mo>≈</n:mo><n:mn>20</n:mn></n:mrow></n:math>-fold speedup that occurs in the simulation of Jellium.