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The solution of linear systems of equations is the basis of many otherquantum algorithms, and recent results provided an algorithm with optimalscaling in both the condition number $\kappa$ and the allowable error$\epsilon$ [PRX Quantum \textbf{3}, 0403003 (2022)]. That work was based on thediscrete adiabatic theorem, and worked out an explicit constant factor for anupper bound on the complexity. Here we show via numerical testing on randommatrices that the constant factor is in practice about 1,500 times smaller thanthe upper bound found numerically in the previous results. That means that thisapproach is far more efficient than might naively be expected from the upperbound. In particular, it is over an order of magnitude more efficient thanusing a randomised approach from [arXiv:2305.11352] that claimed to be moreefficient.

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