Notice that the complexity depends on the error as eps^(-2), since the method is based on integrating a stochastic differential equation (SDE) over time. So this is an approximation algorithm, and not something that requires "arbitrarily precise measurements".
You could simulate the SDE digitally, but you would probably need d^2 time per iteration, where this approach just initializes the systems and waits for it to converge to a sufficient precision. Turns out the convergence time depends on sqrt(condition number) similar to the best iterative linear solvers, conjugate gradient (CG).
You can debate whether it's fair to assume a fully connected d^2 chip, since a similar size cpu or gpu could perhaps do each iteration of CG in constant time, and so would have the same (or better) complexity as the thermodynamic method. However, each cell the the proposed chip is way simpler than a cpu cell, so it should be cheaper/more energy efficient.
You could simulate the SDE digitally, but you would probably need d^2 time per iteration, where this approach just initializes the systems and waits for it to converge to a sufficient precision. Turns out the convergence time depends on sqrt(condition number) similar to the best iterative linear solvers, conjugate gradient (CG).
You can debate whether it's fair to assume a fully connected d^2 chip, since a similar size cpu or gpu could perhaps do each iteration of CG in constant time, and so would have the same (or better) complexity as the thermodynamic method. However, each cell the the proposed chip is way simpler than a cpu cell, so it should be cheaper/more energy efficient.