Measuring Trotter error and its application to precision-guaranteed Hamiltonian simulations

Trotterization is the most common and convenient approximation method for Hamiltonian simulations on digital quantum computers, but estimating its error accurately is computationally difficult for large quantum systems. Here, we develop a method for measuring the Trotter error without ancillary qubits on quantum circuits by combining the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mrow><a:mi>m</a:mi><a:mtext>th</a:mtext></a:mrow></a:math>- and <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mrow><b:mi>n</b:mi><b:mtext>th</b:mtext></b:mrow></b:math>-order (<c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mrow><c:mi>m</c:mi><c:mo><</c:mo><c:mi>n</c:mi></c:mrow></c:math>) Trotterizations rather than consulting with mathematical error bounds. Using this method, we make Trotterization precision guaranteed, developing an algorithm named Trotter<d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mrow><d:mo>(</d:mo><d:mi>m</d:mi><d:mo>,</d:mo><d:mi>n</d:mi><d:mo>)</d:mo></d:mrow></d:math>, in which the Trotter error at each time step is within an error tolerance <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mi>ε</e:mi></e:math> preset for our purpose. Trotter<f:math xmlns:f="http://www.w3.org/1998/Math/MathML"><f:mrow><f:mo>(</f:mo><f:mi>m</f:mi><f:mo>,</f:mo><f:mi>n</f:mi><f:mo>)</f:mo></f:mrow></f:math> is applicable to both time-independent and -dependent Hamiltonians, and it adaptively chooses almost the largest step size <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"><g:mrow><g:mi>δ</g:mi><g:mi>t</g:mi></g:mrow></g:math>, which keeps quantum circuits shallowest, within the error tolerance. Benchmarking it in a quantum spin chain, we find the adaptively chosen <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"><h:mrow><h:mi>δ</h:mi><h:mi>t</h:mi></h:mrow></h:math> to be about 10 times larger than that inferred from known upper bounds of Trotter errors. Published by the American Physical Society 2024