Exponentially Reduced Circuit Depths Using Trotter Error Mitigation
James D. Watson, Jacob Watkins
Abstract
Product formulas are a popular class of digital quantum simulation algorithms due to their conceptual simplicity, low overhead, and performance, which often exceeds theoretical expectations. Recently, Richardson extrapolation and polynomial interpolation have been proposed to mitigate the Trotter error incurred by the use of these formulas. This work provides a rigorous, general analysis of these techniques for computing time-evolved observables, simplifying the interpolation algorithm in the process, and shows that extrapolation generically improves the performance of product formulas for this task. We demonstrate that, to achieve error <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>ϵ</a:mi> </a:math> in a simulation of time <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mi>T</c:mi> </c:math> using a <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mi>p</e:mi> <e:mi>th</e:mi> </e:math> -order product formula with extrapolation, circuit depths of <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:mi>O</g:mi> <g:mo stretchy="false">(</g:mo> <g:msup> <g:mi>T</g:mi> <g:mrow> <g:mn>1</g:mn> <g:mo>+</g:mo> <g:mn>1</g:mn> <g:mo>/</g:mo> <g:mi>p</g:mi> </g:mrow> </g:msup> <g:mi>polylog</g:mi> <g:mo></g:mo> <g:mo stretchy="false">(</g:mo> <g:mn>1</g:mn> <g:mo>/</g:mo> <g:mi>ϵ</g:mi> <g:mo stretchy="false">)</g:mo> <g:mo stretchy="false">)</g:mo> </g:math> are sufficient—an exponential improvement in the precision over product formulas alone. Furthermore, we prove that these algorithms achieve commutator scaling, and improve the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:mi>T</m:mi> </m:math> complexity for the interpolation algorithm. By relaxing the requirement of performing exact Chebyshev interpolation, our simplified algorithm eliminates the need for fractional implementations of Trotter steps, reducing computational overhead. Finally, we show these techniques can be combined with the classical shadows method to estimate many time-evolved local observables. Taken together, our findings provide the strongest evidence yet for the utility of Trotter error-mitigation techniques in algorithmic applications.