Optimised Trotter decompositions for classical and quantum computing
Johann Ostmeyer
Abstract
Abstract Suzuki–Trotter decompositions of exponential operators like <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>exp</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>H</mml:mi> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> are required in almost every branch of numerical physics. Often the exponent under consideration has to be split into more than two operators <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>H</mml:mi> <mml:mo>=</mml:mo> <mml:mrow> <mml:munder> <mml:mo>∑</mml:mo> <mml:mi>k</mml:mi> </mml:munder> </mml:mrow> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:math> , for instance as local gates on quantum computers. We demonstrate how highly optimised schemes originally derived for exactly two operators <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mi>A</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> </mml:math> can be applied to such generic Suzuki–Trotter decompositions, providing a formal proof of correctness as well as numerical evidence of efficiency. A comprehensive review of existing symmetric decomposition schemes up to order <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>⩽</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:math> is presented and complemented by a number of novel schemes, including both real and complex coefficients. We derive the theoretically most efficient unitary and non-unitary 4th order decompositions. The list is augmented by several exceptionally efficient schemes of higher order <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>n</mml:mi> <mml:mo>⩽</mml:mo> <mml:mn>8</mml:mn> </mml:math> . Furthermore we show how Taylor expansions can be used on classical devices to reach machine precision at a computational effort at which state of the art Trotterization schemes do not surpass a relative precision of 10 −4 . Finally, a short and easily understandable summary explains how to choose the optimal decomposition in any given scenario.