Modified Newton-Raphson GRAPE methods for optimal control of spin systems

Abstract

Quadratic convergence throughout the active space is achieved for the gradient ascent pulse engineering (GRAPE) family of quantum optimal control algorithms. We demonstrate in this communication that the Hessian of the GRAPE fidelity functional is unusually cheap, having the same asymptotic complexity scaling as the functional itself. This leads to the possibility of using very efficient numerical optimization techniques. In particular, the Newton-Raphson method with a rational function optimization (RFO) regularized Hessian is shown in this work to require fewer system trajectory evaluations than any other algorithm in the GRAPE family. This communication describes algebraic and numerical implementation aspects (matrix exponential recycling, Hessian regularization, etc.) for the RFO Newton-Raphson version of GRAPE and reports benchmarks for common spin state control problems in magnetic resonance spectroscopy.

Practical aspects of real‐time pure shift HSQC experiments

“…Simulations of real‐time pure shift HSQC experiments were performed using the Spinach package in MATLAB to complement the experimental results (see Figure , right). A three‐spin ( AMX ) system was defined for the simulation, where A and M correspond to the two J ‐coupled protons and X is a 13 C spin coupled only to spin A , corresponding approximately to the isotopomer responsible for the experimental data shown in Figure .…”

“…[29] In practice, for small molecules, the impact of the extra broadening is often small given the relatively short total acquisition times commonly used in HSQC, but in systems such as proteins that have short T 2 s, and where τ is relatively long, as for example dictated by the 1 J NH coupling constant in 15 N HSQC or by long selective pulses in Zangger-Sterk or band-selective experiments, the extra line broadening can significantly degrade the resolution obtainable. [26] Simulations of real-time pure shift HSQC experiments were performed using the Spinach package [43,44] in MATLAB to complement the experimental results (see Figure 3, right). A three-spin (AMX) system was defined for the simulation, where A and M correspond to the two J-coupled protons and X is a 13 C spin coupled only to spin A, corresponding approximately to the isotopomer responsible for the experimental data shown in Figure 3.…”

A new approach to Z‐spectrum acquisition: prospective baseline enhancement (PROBE) for CEST/Nuclear Overhauser Effect

“…The calculation of was significantly accelerated by the optimal‐control approach . In applications in which fast prototyping for contrast improvement may be relevant, this should be particularly beneficial.…”

“…The calculation of 1,max ( i ) was significantly accelerated by the optimal-control approach. 76 In applications in which F I G U R E 9 Results from Lorentzian peak area analysis of the lactate methyl resonance at −3.5 ppm (see Figure 6A for the arrangement of samples in the phantom). Metabolite maps extracted from the densely sampled data (single average) (A) in comparison to artificially sparsely sampled data (four averages) (B).…”

Engineering effective Hamiltonians

“…Gradient evaluation for Dyson terms is crucial for ensuring fast convergence to high accuracy solutions, as gradient information increases the efficiency of the search. Here, we have only used first order derivatives, but as is described in [66], incorporating second order derivative information into standard quantum control problems can ensure fast convergence rates near the optimum, and lead to better performance than first order methods [45]. To reiterate, as we rephrase the effective Hamiltonian control problem into a bilinear control theory problem, all of the higher order algorithms and methods outlined in [45] may be applied.…”