The discrete adiabatic quantum linear system solver has lower constant factors than the randomized adiabatic solver

The solution of linear systems of equations is the basis of many other quantum algorithms, and recent results provided an algorithm with optimal scaling in both the condition number <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&amp;#x03BA;</mml:mi></mml:math> and the allowable error <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&amp;#x03F5;</mml:mi></mml:math> \cite{CostaAnYuvalEtAl2022}. That work was based on the discrete adiabatic theorem, and worked out an explicit constant factor for an upper bound on the complexity. Here we show via numerical testing on random matrices that the constant factor is in practice about 1,200 times smaller than the upper bound found numerically in the previous results. That means that this approach is far more efficient than might naively be expected from the upper bound. In particular, it is about an order of magnitude more efficient than using a randomised approach from \cite{jennings2023efficient} that claimed to be more efficient.