I'm a little late to the party but hopefully someone can still answer this question.

In order to solve the linear system of equations in this framework, you need to integrate a measurement of the system over some time greater than t_0 and tau. However in equation 12 you can see that t_0 and tau are functions of the eigenvalues and the norm of the matrix.

AFAIK the best runtime algorithms we have for computing matrix eigenvalues is still O(d²), so even if the thermodynamic part of algorithm is linear in d, computing how long you would need to run the algorithm for is still quadratic in d, so there's no real gain.

Or am I missing something here?