Quantum-Trajectory-Inspired Lindbladian Simulation

Simulating the dynamics of open quantum systems is a crucial task in quantum computing, offering wide-ranging applications but remaining computationally challenging. In this paper, we propose two quantum algorithms for simulating the dynamics of open quantum systems governed by Lindbladians. We introduce a new approximation channel for short-time evolution, inspired by the quantum trajectory method, which underpins the efficiency of our algorithms. The first algorithm achieves a gate complexity independent of the number of jump operators, <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>m</a:mi></a:math>, marking a significant improvement in efficiency. The second algorithm achieves near-optimal dependence on the evolution time <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>t</c:mi></c:math> and precision <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>ϵ</e:mi></e:math> and introduces only an additional <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mrow><g:mover><g:mi>O</g:mi><g:mo stretchy="false">~</g:mo></g:mover></g:mrow><g:mo stretchy="false">(</g:mo><g:mi>m</g:mi><g:mo stretchy="false">)</g:mo></g:math> factor, which strictly improves upon state-of-the-art gate-based quantum algorithm that has an <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" display="inline"><l:mrow><l:mover><l:mi>O</l:mi><l:mo stretchy="false">~</l:mo></l:mover></l:mrow><l:mo stretchy="false">(</l:mo><l:msup><l:mi>m</l:mi><l:mn>2</l:mn></l:msup><l:mo stretchy="false">)</l:mo></l:math> factor. The improvement stems from the integration of the new approximation channel with a novel structured linear combination of unitaries method. In both our algorithms, the reduction of dependence on <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:mi>m</q:mi></q:math> significantly enhances the efficiency of simulating practical dissipative processes characterized by a large number of jump operators.