Fully efficient time-parallelized quantum optimal control algorithm

Abstract

We present a time-parallelization method that enables to accelerate the computation of quantum optimal control algorithms. We show that this approach is approximately fully efficient when based on a gradient method as optimization solver: the computational time is approximately divided by the number of available processors. The control of spin systems, molecular orientation and BoseEinstein condensates are used as illustrative examples to highlight the wide range of application of this numerical scheme.

PARAOPT: A Parareal Algorithm for Optimality Systems

“…For α = 1000, the bounding disc is identical to the previous case; however, since we have L < αL 0 for all cases except for σ = −16, we observe no eigenvalue outside the disc, except for the very last case. In that very last case, we have |δγ| = 0.0107, so (48) gives the lower bound µ * > −1.07 × 10 −5 , which again is quite accurate when compared with the bottom right panel of Figure 4. Re(µ) Re(µ) Note that the inequality (48) is valid for all L > 0, i.e., regardless of whether the isolated eigenvalue µ * exists.…”

“…Therefore, a multiple shooting approach allows us to deal with local subproblems on shorter time horizons, where we obtain faster convergence. Such convergence enhancement has also been observed in [3,37,38], and also more recently in [48]. Secondly, if we use parareal to parallelize the forward and backward sweeps, then the speedup ratio will be bounded above by L/K, where L is the number of sub-intervals and K is the number of parareal iterations required for convergence.…”

“…Table 1 shows the values of the relevant parameters. For α = 1, we see that there is always one isolated eigenvalue on the negative real axis, since L > L 0 in all cases, and its location is predicted rather accurately by the formula (48). The rest of the eigenvalues all lie within the disc D σ defined in (42).…”

“…[52]. Another method, which has been proposed in [36,48] in the context of quantum control, consists of decomposing the time interval into sub-intervals and defining intermediate states at sub-interval boundaries; this allows one to construct a set of independent optimization problems associated with each sub-interval in time. Each iteration of the method then requires the solution of these independent sub-problems in parallel, followed by a cheap update of the intermediate states.…”

Fokker-Planck formalism in magnetic resonance simulations

“…It follows from Equation (26) (45) Pulse amplitudes and frequencies enter this equation linearly and may therefore be optimised by the GRAPE procedure [82], as well as its recent enhancements [81,84,85]. The first group of control operators consists of…”