High-fidelity control of spin ensemble dynamics via artificial intelligence: from quantum computing to NMR spectroscopy and imaging

Abstract

Introduction

High-fidelity control of quantum spin systems is at the foundation of many applications such as quantum computing, coherent and optical spectroscopies, NMR, and MRI (1–5). Spin operations such as excitation, inversion, refocusing, etc., are central to these techniques and are achieved by applying radio-frequency (RF) pulses of finite length and amplitude. However, the RF and external field inhomogeneities and finite pulse length effects make the coherent manipulation of spin ensemble dynamics challenging (6). In the experimental implementation of quantum computing, inhomogeneities affect the experimental fidelity of quantum gates, on-demand entangled state generation, and coherent control (1, 7, 8). These experimental errors also affect NMR and MRI at high and ultra-high magnetic fields as they require high-fidelity levels for coherent and high-efficiency control of heterogeneous spin ensembles (9, 10). Moreover, these imperfections accumulate in multipulse experiments, leading to low-fidelity operations and sizable signal losses (11). Although advanced computational techniques have been instrumental for designing compensated RF pulses such as composite, adiabatic, and numerically optimized pulses (6, 12–18, 19, 20–29), high- and ultra-high–field NMR and MRI spectroscopy require RF pulses with larger bandwidths, higher fidelity, and compensation for instrumental inhomogeneities. Here, we introduce a novel strategy to achieve high-fidelity control of spin ensemble dynamics with a high-level compensation for inhomogeneity and offset effects, reaching a fidelity for several spin operations up to 0.99999. Instead of tuning RF amplitude and duration for a given pulse shape, we search the phase space and let an evolutionary algorithm generate a library of ∼2,00,000 phase shapes with constant amplitude. We then trained an artificial intelligence (AI) algorithm with this library to generate the optimal solution for a given problem. Searching the phase space enabled the algorithm to achieve optimal solutions and reach an operational fidelity of 0.9999. Our versatile approach enables the design of several spin operations for various applications, including quantum computing, biomolecular NMR spectroscopy, and MRI techniques.

Results and discussion

The architecture of the GENETICS-AI algorithm

To design high-fidelity RF pulses, we combined an evolutionary algorithm with AI into a modular software, GENErator of TrIply Compensated RF pulSes via Artificial Intelligence (GENETICS-AI). The overall architecture of GENETICS-AI is reported in Figure S1. The core module of the algorithm consists of a network of interconnected optimization routines, including an iterative forward search (IFSA) and iterative self-correcting algorithms (ISCA) (Figure S2). These routines search the RF phase space and generate an Optimal Phase Surface (OPS) library through an iterative evolution of the solutions. A neural network (PhaseNET) is trained on this library and predicts new RF shapes with customizable flip angles, bandwidth (BW) with a specified fidelity, RF inhomogeneity compensation, and operational fidelity (Figure S3). A standout feature of GENETICS-AI is the multiple bidirectional validations of each RF shape through the optimization network, where each solution is connected to its neighbor as a generator or corrector.

High-fidelity spin entanglement using GENETICS-AI

The creation and preservation of high-fidelity spin entanglement are central to several quantum computing applications (30, 31) and are ideal for testing the performance of GENETICS-AI (32–35). A reliable physical implementation of entangling two quantum systems requires unitary operations, which compensate for instrumental inhomogeneity within the experimentally feasible range. For NMR, dipolar/scalar coupled heteronuclear spins with I = ½ (such as ¹H and ¹³C in ¹³CHCl₃, Figure 1A) form a two-qubit (quantum bit) quantum computer. External field inhomogeneities and errors in determining the coupling constant lower the fidelity of unitary operations, affecting the faithful implementation of quantum gate or quantum state creation. To improve the fidelity of unitary operations, we used GENETICS-AI to create a robust entangling operator (GEN-Entangler) for the weakly coupled two-spin-1/2 system of ¹³CHCl₃ (Figure 1A to E). Each pulse in GEN-Entangler is triply compensated (36) to minimize inhomogeneities of RF field (Δω₁), offset (Ω), and weak scalar coupling interactions (ΔJ). The modular nature of the GEN-Entangler makes it possible to individually customize the level of compensation for Δω₁, Ω, and ΔJ, conferring a higher level of control for the desired operation. The J-coupling compensation level is controlled by the number of spin-echo (SE) elements (n) in the entangler sequence. For n = 1, GEN-Entangler consists of a single SE element with three global pulses and a J-coupling compensation level up to ±10%. For n = 2, the tolerance increases to ±30% and for n = 4 to ±50% (Figure 1C). To test the performance of GEN-Entangler, we used a pulse sequence with n = 2 and inhomogeneity compensation levels of ±20% and ±2 kHz for Δω₁ and BW, respectively (Figure 1C and Table S1). We first created a |αα⟩ pseudo-pure state (37) for the ¹³C–¹H spin pair (Figure 1F). Starting from the |αα⟩ state, we generated a maximally entangled state, 1⁄√2 (|αα⟩ + |ββ⟩), between the ¹H and ¹³C spins. Figure S4 shows the representative ¹H NMR spectra of thermal equilibrium, |αα⟩ and 1⁄√2 (|αα⟩ + |ββ⟩ states created by a π/2 pulse with phase zero. As a metric for robustness, we simulated a 0.99 fidelity iso-surface of the quantum states obtained by GEN-Entangler with the simultaneous compensation of Δω₁, Ω, and ΔJ, and compared it with the corresponding iso-surface generated using a standard Controlled-NOT (C-NOT) gate entangler using rectangular pulses (Figure 1G). We then experimentally generated six entangled states by detuning the pulses calibrated for optimal values. The yellow outer iso-surface in Figure 1G represents the fidelity volume obtained with the GEN-Entangler for n = 2, while the inner red surface represents the fidelity volume corresponding to the standard C-NOT entangler. Considering the maximum levels of inhomogeneity for all three parameters, the fidelity volume obtained by GEN-Entangler is approximately 100 times larger than the corresponding rectangular shape pulse. For an iso-surface fidelity of 0.98, the volume ratio, V_GEN/V_RECT, increases to 200 times (Figure 1H), demonstrating that the new entangler has a significantly higher tolerance for experimental errors. Additionally, the GEN-Entangler sequence is ∼62% shorter than the ΔJ compensated C-NOT gate sequence (38), offering an additional advantage in terms of time duration.

Figure 1.

Generation of spin entanglement for ¹³CHCl₃. (A) ¹³CHCl₃ used for validating the GEN-Entangler pulse sequence. (B) GEN-Entangler pulse sequence. All the pulses are global for both ¹H and ¹³C nuclei. Pulses and delay parameters for different values of n are given in Table S1. (C) J response curve of entanglement fidelity for n = 1, 2, and 4. (D) and (E) RF shapes and corresponding fidelity profiles used for GEN-Entangler with n = 2. (F) Quantum state tomography for the real and imaginary part of the |$|{\rm{\alpha \alpha }}\rangle $| pseudo pure state created for ¹H and ¹³C nuclear spins. (G) Simulated iso-surface at 0.99 fidelity for the entanglement created with n = 2 (outer surface—yellow) and standard C-NOT entangler with rectangular-shaped pulse (inner surface—red). The quantum state tomography results for simulated inhomogeneities on the 0.99 fidelity iso-surface for both extreme values of all three variables are shown in blue. (H) Fidelity volume fraction of entanglement created with GEN-Entangler (V_GEN) over standard C-NOT with rectangular pulses (V_RECT) for fidelities of 0.995, 0.99, and 0.98. All experiments were performed on a Bruker 600 MHz spectrometer at 300 K.

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As an additional test, we generated high-fidelity broadband (BB) RF pulses with arbitrary flip angles. Generalizing BB operations for an arbitrary spin-flip angle is quite challenging, as it requires individual optimization for the desired final state (39). The BB pulses for arbitrary flip angle transitions generated by our algorithm are reported in Figure 2A and B. Remarkably, the neural network trained on this OPS library output BB RF with smooth phase shapes with a fidelity greater than 0.9999 for any flip angle. We then tested this RF pulse generator by creating a programmable single qubit quantum state creator and a quantum gate generator. The former assumes an initial state |α⟩, whereas the latter consists of a BB universal flipping operation. Notably, the duration of the pulses does not depend on the flip angle or spin state. This feature is critical for quantum computing as the fixed length of these pulses removes any inhomogeneity associated with changes in time duration. The Bloch sphere trajectories of selected pulse operations with various flip angles are shown in Figure 2C and D. For example, we created the RF pulse of a 113.5° flip angle, with a smooth shape and an average fidelity of 0.9999 over a bandwidth of 50 kHz (Figure 2E-F). These high-fidelity single-qubit operations and the GEN-Entangler form a complete set of operations for universal quantum computing (40). Taken together these results show that the newly AI-generated pulses can perform any spin operation with high compensation and fidelity.

Figure 2.

BB high-fidelity arbitrary flip angle pulses generated with GENETICS-AI. (A) OPS of a BB arbitrary state preparation with fidelity > 0.99999. The Y-axis represent the zenith angle (ζ) of the final state. The OPS was generated with an initial state fixed at |$|\alpha \rangle $| and varying the ζ from 0 to π iteratively. The total nutation angle of the pulse was fixed at 8.68 π for all states. (B) OPS for arbitrary flipping operation with fidelity > 0.9999. The Y-axis represents the flip angle (θ) of the pulse. The OPS was created with θ as a target variable. The total nutation angle for the operation was fixed at 11.42 π. (C) Bloch sphere trajectories for the preparation of four different states (A, B, C, and D) using RF shapes from the OPS shown in (A). (D) Bloch sphere trajectories of two different flipping operations from A and A’ to B and B’ using RF shape from the OPS shown in (B). All trajectories are trimmed to show only 10% of the initial and final part. (E) RF shape of 113.5° universal flipping for a bandwidth of 50 kHz. The RF amplitude is constant (25 kHz). (F) 2D-fidelity response with offset and RF amplitude of the RF shape shown in (E).

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GENETICS-AI pulses for NMR spectroscopy at high and ultra-high magnetic fields

A critical requirement for NMR spectroscopy at high and ultra-high magnetic fields is the uniform excitation, inversion, and refocusing of nuclear magnetization over a broader BW. With this in mind, we programmed GENETICS-AI to generate new universal π/2 and π pulses to be used for multidimensional biological NMR (41). Two prototypical GENETICS-AI π/2 and π pulses are shown in Figures S5 and S6. These new pulses cover a BW of 40 kHz with pulse lengths less than 300 µs, with a constant RF amplitude (ω₁ = γB₁/2π) of 25 kHz, and an average operational fidelity of 0.9999, outperforming the standard rectangular pulses. The comparison between the universal GENETICS-AI π pulses and the rectangular pulses for different BWs show that the latter pulses display a limited high-fidelity region (Figure S7A), while the former pulses show quite large fidelity regions, which extend to reach approximately 500 kHz for a pulse duration of 1360 µs (Figure S7B to E). To estimate the tolerance of these pulses to instrumental noise, we added random numerical noise to the phase and amplitude of a GENETICS-AI universal π pulse with a BW of 100 kHz and RF amplitude (ω₁) of 25 kHz and evaluated changes in its operational fidelity (Figure S8A to C). In the absence of noise, this pulse performs spin operations with a fidelity of 0.999. Upon addition of numerical noise with an amplitude of 5 kHz (20%), the fidelity drops by only 0.014 (i.e. 0.985), demonstrating a high tolerance to random noise sources. A similar scenario is predicted for errors in both amplitude and phase shape. Lastly, we assessed the performance of GENETICS-AI refocusing pulses against the best-performing pulses currently used for NMR and MRI. (16, 17, 36, 42–45) We compared their duration, operational BW, and average fidelity (Figure 3A, Table S2 and Table S3). The minimum fidelity for all GENETICS-AI pulses within the BW range is 0.99. For all the shapes analyzed, our new pulses show higher fidelity levels and shorter duration. Unlike the published shapes, our pulses are also time-optimal and customizable for any operational fidelity.

Figure 3.

BB universal pulses and comparison of GENETICS-AI refocusing pulses with previously reported shapes. (A) Plot of the “Refocusing bandwidth-pulse length” for best performing GENETICS-AI pulses versus reported refocusing pulses. Refocusing BW of each pulse is calculated by averaging the offset responses of Mx and My magnetizations with a cutoff operational fidelity of 0.95. Each pulse is labeled with a short form and color-coded for the same family. Supplementary Table S2 shows the full names and relative reference for each pulse analyzed. The size of each dot represents the average operational fidelity of the pulse. Offset responses of a few selected RF shapes are shown in Figure S9. (B) Ultra-BB universal π/2 and (C), π pulse of BW 30 × ω₁. Amplitude shapes of these pulses are constant (ω₁) with phase shapes shown on the left. Offset response of π and π/2 are shown for initial states Mz (middle). Simulated fidelity response with offset and RF amplitude are shown on the right panel.

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Moreover, we programmed GENETICS-AI to create ultra-BB pulses for exciting nuclei with extensive chemical shift breadth. Specifically, we designed universal π/2 and π pulses with a BW of 30 × ω₁, which corresponds to 750 kHz for an RF amplitude, ω₁, of 25 kHz (Figure 3B and C). The pulse length of this ultra-BB pulse with 2,000 points generated was ∼1.5 ms. To the best of our knowledge, this is the shortest universal pulse obtained for a 750 kHz BW. In theory, it is possible to generate ultra-BB pulses for a range of very large BWs (9). For instance, we designed an ultra-BB excitation pulse of 500 points to excite a BW of 22 × ω₁ (Figure S15). This pulse is equivalent to an in-phase excitation BW of 550 kHz with a pulse duration of 440 µs (ω₁ = 25 kHz). We experimentally tested this BB pulse by recording a proton 1D spectrum of uniformly ¹⁵N labeled K48C mutant of ubiquitin (U-¹⁵N UBI^K48C) with an RF amplitude of 0.5 kHz on an 850 MHz spectrometer. The excitation BW of the pulse is 22 × 0.5 kHz = 11 kHz, sufficient for the excitation of the entire ¹H spectrum of a protein. A comparison between a rectangular pulse with ω₁ of 16.67 kHz and the BB pulse of 0.5 kHz is reported in Figure S15C. As expected, we observed a loss of signal intensity for the ultra-BB pulse due to the relaxation during the pulse execution. Nonetheless, the power required for these BB pulses is only 6.57 mW, while a corresponding rectangular pulse would require 7.3 W to excite the same BW. The latter shows that the algorithm has the potential to generate ultra low-power pulses for MRI, where power deposition becomes a problem for human subjects (9).

We then designed and applied BB pulses to multipulse NMR experiments. Specifically, we implemented BB π/2 and π pulses into the 2D [¹H-¹⁵N] Heteronuclear Single Quantum Correlation (HSQC) pulse scheme, a central building block for biomolecular solution NMR spectroscopy (51–53). We first tested the tolerance of the BB pulses to RF inhomogeneity. Figure 4A shows the phase shapes of BB π/2 and π pulses designed to compensate for inhomogeneity in RF up to ±20% and an excitation BW covering the full chemical shift range of ¹H and ¹⁵N for a Larmor frequency up to 1.2 GHz. The fidelity response of these pulses with offset and RF amplitude is shown in Figure 4B. We compared the 2D spectra of the amide fingerprint of U-¹⁵N UBI^K48C (54) acquired with the standard PFG-HSQC experiment (55, 56) with coherence selection and rectangular pulses and the corresponding BB GENETICS-AI version (GEN-HSQC). Figure 4C shows the 2D GEN-HSQC spectra recorded on U-¹⁵N UBI^K48C with calibrated RF amplitudes of 12.5 and 5.8 kHz for ¹H and ¹⁵N, respectively. To assess the BB pulses’ tolerance for RF inhomogeneity on both ¹H and ¹⁵N channels, we scaled the RF pulse amplitudes in steps of 5% up to ±20% while keeping their pulse duration constant. The insets of Figure 4C show the comparison of representative resonance intensities of the GEN-HSQC spectrum (⁠|${I}_{GEN}$|⁠) versus the standard PFG-HSQC experiment (⁠|${I}_{RECT}$|⁠). Overall, the spectrum acquired with the GEN-HSQC sequence displays higher S/N ratios, with a significantly higher tolerance for RF inhomogeneity across the entire spectrum. The gain in sensitivity observed with the GEN-HSQC sequence can be attributed to a combination of the high-fidelity pulse operation and RF inhomogeneity compensation. For PFG-HSQC, the intensity of the residues near the ¹H carrier frequency is ∼0.8 and drops to ∼0.5 for off-resonance irradiation relative to GEN-HSQC experiment. For amplitude attenuation of ±20%, the amide resonances in the PFG-HSQC are barely observable, whereas the signal intensities in the GEN-HSQC show a constant response. We performed the GEN-HSQC experiment on the catalytic subunit of cAMP-dependent protein kinase A (PKA-C, Figure 4D). For this larger protein (42 kDa), the GEN-HSQC spectrum shows a higher S/N ratio and detects additional amide peaks that are not observable in the standard PFG-HSQC experiment. The drop in peak intensities observed in the PFG-HSQC experiment is due to the accumulation of pulse imperfections over multiple 180^o pulses on both channels, i.e. five pulses for ¹H and four for ¹⁵N. In fact, the GEN-HSQC experiment has a significantly higher RF inhomogeneity compensation relative to our previous implementation of HSQC using the universal broadband pulse G5 (11). Lastly, we generated universal π/2 and π pulses for ¹H, ¹³C, and ¹⁵N nuclei for biomolecular NMR experiments up to 5 GHz (Figure S16). These RF pulses cover a BW of 20, 200, and 30 ppm for ¹H (B₁ = 16.67 kHz), ¹³C (B₁ = 16.67 kHz), and ¹⁵N (B₁ = 5 kHz) channels, respectively, for a static magnetic field B₀ = 5 GHz. Even at this high magnetic field, the average fidelity reached by these pulses is 0.999. Although this is a hypothetical scenario as the commercially available ultra-high magnetic fields are limited to 1.2 GHz, it shows that GENETICS-AI-generated pulses are suitable for biomolecular NMR spectroscopy at ultra-high magnetic fields greater than 1.2 GHz. Moreover, we generated new band-selective RF pulses for both excitation and inversion operations. These pulses are central to several triple-resonance experiments for liquid- or solid-state NMR techniques. However, the level of fidelity for the standard pulses is low, constituting a significant source of signal losses, particularly for multiple simultaneous operations (11, 56, 57). A typical application of selective excitation (or inversion) is the irradiation of ¹³C carbonyl and aliphatic carbons resonances (58). Using GENETICS-AI, we generated selective pulses for excitation and inversion operations with a duration less than 1/4 of the traditional band-selective pulses (41, 59). The shapes and offset responses in terms of excitation (or inversion) regions of these band-selective excitation and inversion pulses are shown in Figure 5. We experimentally tested the pulses on the ¹³C resonances of U-¹³C labeled maltose-binding protein. From the 1D spectra, it is apparent that selective excitation (or inversion) occurs with almost no signal loss (Figure 5). Taken together, the simulations and experimental applications of BB and selective pulses designed by GENETICS-AI support their application to high and ultra-high magnetic field NMR spectroscopy on biomacromolecules.

Figure 4.

RF inhomogeneity compensated high-resolution multipulse NMR experiment. (A) GENETICS-AI shapes for universal π and π/2 pulses for ¹H (blue) and ¹⁵N (red) channels. RF amplitudes were constant at 12.5 and 5.8 kHz for ¹H and ¹⁵N channels, respectively. The pulse lengths for π/2 and π pulses were 488.6 and 509.8 µs for ¹H and 483.8 and 524 µs for ¹⁵N. (B) Simulated fidelity response with offset and RF amplitude for ¹H (blue) and ¹⁵N (red) for the pulse shapes in (A). The inner contour level indicates a fidelity of 0.99. (C) GENETICS-AI version of the HSQC (GEN-HSQC) spectrum of ¹⁵N labeled K48C mutant of Ubiquitin recorded using calibrated amplitude values for ¹H (12.5 kHz) and ¹⁵N (5.8 kHz) channels on a Bruker 850 MHz spectrometer at 300 K. The insets show the intensity comparison of the standard PFG-HSQC (using the Bruker pulse sequence—“hsqcetfpf3gpsi2”) (⁠|${I}_{RECT}$|⁠) and GEN-HSQC (⁠|${I}_{GEN}$|⁠) at different RF amplitudes for selected residues. Left Y-axis shows the normalized peak intensities of standard (blue) and GEN-HSQC (red) relative to the calibrated reference spectra (⁠|$I_{RECT}^0$| and |$I_{GEN}^0$|⁠), whereas the right Y-axis represents the normalized intensity of standard HSQCs (blue) with respect to |$I_{GEN}^0$|⁠. (D) GEN-HSQC spectra of 200 µM ¹⁵N labeled catalytic subunit of protein kinase A (42 kDa). The figure insets compare the intensity of various resonances recorded using GEN-HSQC (red) and PFG-HSQC (blue) of the HSQC experiment. Both experiments are recorded on a Bruker 850 MHz spectrometer at 273 K using 128 complex indirect points were acquired with 16 scans per FID and a 1.5 s relaxation delay. Maximum RF amplitudes were 16.67 kHz for ¹H and 7.7 kHz for ¹⁵N.

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Figure 5.

Band-selective excitation and inversion pulses with (A) excitation on the left half of BW and (B) Inversion on left half of the BW. Note that to observe M_z and -M_z, we added a reading pulse (90^o) prior to acquisition. (C) Excitation on left half and inversion on right half of the BW, simultaneously. The excitation/inversion half of the offset response can be swapped by inverting the phase array. An RF amplitude of 25 kHz was used for the simulation. Experimental verification was performed using a uniformly ¹³C labeled maltose binding protein (MBP) on an 850 MHz spectrometer. The band-selective excitation/inversion of aliphatic/carbonyl regions of the MBP are shown in the right column. The reference of the ¹³C spectra is shown in blue, whereas the spectra obtained with different GENETICS-AI band-selective operations are shown in red.

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Prototypical pulses for MRI

RF inhomogeneity represents a severe problem for MRI both at low and high magnetic fields, limiting both the resolution and sensitivity of images (9). In an effort to reduce RF inhomogeneity, we generated GENETICS-AI pulses for an SE imaging sequence, which is typically used for MRI experiments (4). Figure 6A shows the RF inhomogeneity-compensated SE sequence (GEN-SE), where π/2 and π were substituted with the GENETICS-AI pulses. The pulse shapes and their fidelity response with offset and RF amplitude are shown in Figure S17. As a phantom, we utilized two 4 mm glass beads into a 5 mm Shigemi® tube filled with a solution of 10% H₂O and 90% ²H₂O (Figure 6B). We then generated 1D images of the phantom along the Z-direction using the SE sequence with standard rectangular-shaped pulses and compared them with the GEN-SE sequence (Figure 6C). To simulate the effect of RF inhomogeneity, we scaled the RF pulse amplitudes up to ±100%, keeping the pulse length constant. The intensity of the images acquired with the standard rectangular pulses becomes significantly lower with scaling the RF amplitude (Figure 6D). In contrast, the GEN-SE creates high-quality images with fewer artifacts, even at ω₁ inhomogeneity of ±80% (Figure 6E) (60). Overall, these tests show a superior tolerance of the GENETICS-AI pulses to RF field inhomogeneity than the rectangular pulses, making them more suitable for in vivo imaging and spectroscopy as well as for analyzing materials.

Figure 6.

SE imaging under ultra-high RF inhomogeneity. (A) SE pulse sequence used for imaging. The π/2 and π pulses are replaced with GENETICS-AI pulses for GEN-SE sequence. The amplitude of the pulses is constant (16.67 kHz) and pulse lengths for π/2 and π pulses are 437 and 567 µs, respectively. The maximum RF amplitude tolerance level (⁠|${\rm{\Delta \omega }}_1^{max}$|⁠) of these pulses were ±80%. The gradient strengths (G₁ and G₂) for all the experiments were set at 50% (3.3 G cm^–1). (B) Diagram of phantom and coil geometry. The phantom consists of two glass beads of diameter 4 mm in a 5 mm Shigemi tube filled with 10% D₂O and 90% H₂O. (C) 1D image of the phantom using the pulse sequence in (A). (D) RF inhomogeneity response of 1D imaging using standard rectangular-shaped pulses of amplitude 16.67 kHz. (E) Same as (D) using GENETICS-AI pulses. The RF shapes and their fidelity profiles are shown in Figure S17. The experiments were performed in a Bruker 900 MHz NMR spectrometer.

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The first application of artificial neural networks to pulse design was pioneered by Gezelter and Freeman, who trained a basic algorithm and generated JANUS pulses able to create antiphase × magnetization (61). More recently, deep reinforcement learning or neural networks have been used to optimize and refine RF pulse shapes for imaging (62–65). In contrast, GENETICS-AI generates pulse shapes for general spin operations with customizable BW, inhomogeneity compensation, and fidelity. Notably, during the execution of the GENETICS-AI pulses, there is no chemical shift evolution (Figure S19), and the final state of magnetization is unaffected by J-coupling (Figure S20). The GENETICS-AI-generated pulses possess smooth phase shapes and constant amplitude, thanks to the neural network trained on an extensive library of realistic high-fidelity RF shapes and performance profiles. The forward and reverse optimization algorithms coupled with a neural network generate time-optimal phase shapes in a matter of milliseconds. Unlike all previously reported pulse shapes, the operational fidelity, BW, RF inhomogeneity compensation, and flip angle of the new pulses are customizable and enable the user to meet the requirements of various magnetic resonance applications. Remarkably, we generated BB pulses suitable for high and ultra-high magnetic field spectroscopy. We also reached a high level of control for the spin dynamics in multipulse NMR experiments, which involve large ensembles of nuclear spin quantum processors in a highly mixed state. At this time, the GENETICS-AI algorithm can design only pulses with constant amplitude or preselected amplitude shapes. Future developments will include coding the RF pulse amplitude in the search space. The latter will potentially improve the design of band-selective pulses, minimizing the power required for specific operations.

In conclusion, we showed that combining an evolutionary algorithm and AI makes it possible to control the spin dynamics with high fidelity and design inhomogeneity compensated RF pulses directly applicable to NMR, MRI, and quantum computing. The proof-of-concept applications illustrated here lay the groundwork for more complex implementations of AI-driven RF design. These technical advancements will impact several fields, including structural biology, biomedicine, material sciences, and quantum information.

Experimental section

Architecture of GENETICS-AI

The architecture of GENETICS-AI is reported in Figure S1. Input module: The first module consists of a customizable input interface in which the user defines the specific problem to be solved. The input parameters are as follows:

• Operator type: desired pulse operation, i.e. excitation, inversion, universal π, π/2, π/3, and π/4 pulses;

• Maximum RF amplitude: maximum allowed peak amplitude of the shape;

• Operational BW: desired BW for the pulse operation (kHz);

• RF amplitude compensation level: amount of RF inhomogeneity/miscalibration to be compensated;

• Average fidelity for the operation: average fidelity for the pulse, where the average is calculated over the range of specified BW and/or amplitude compensation level.

Iterative Forward Search and Self Correction Algorithms (IFSA-ISCA): generation of OPS

Neural Network (PhaseNET): Generation of new optimal solutions from the OPS library

Extracting a specific pulse shape with desired properties from the OPS library is computationally challenging. Therefore, we used the AI module coded in MATLAB® to speed up the selection of the best phase shape and generate solutions that are not part of the original OPS library. Once IFSA-ISCA generates the OPS library, all the shapes are indexed using the “Contour Indexing Algorithm” (CIA). CIA evaluates all the possible rectangles fit within the various contour levels of the “offset-Δω₁” pulse profile, as shown in Figure S3A. The index values of each RF shape (CIA files) and the corresponding shape are used for training the neural network (PhaseNET), which predicts new RF shapes that are not directly available in the OPS library. To accomplish this task, we used the traincgp function provided in MATLAB (Conjugate gradient back-propagation with Polak–Ribiére updates). A schematic of PhaseNET is reported in Figure S3B. The validation of each shape from the PhaseNET is performed by simulating the responses of the RF pulse predicted by the trained neural network. If the fidelity, BW, and inhomogeneity compensation match the requirements, PhaseNET outputs a new shape in the specified format. In the case of the arbitrary flip angle generator, the PhaseNET was validated by evaluating the RF shapes for different flip angles using 1,000 different runs. For each run, the algorithm found the optimal solution for the specified RF shape.

All GENETICS-AI algorithms are written in MATLAB® (R2018a) and executed on a personal computer with Intel® Core™ i7-7700 K processor. The generation of a single OPS of 100 points takes approximately 12 hours of calculation on a single core. For a 500-point shape, the calculation time grows exponentially to approximately a week of computer time. So far, we implemented up to 2,000-point RF shape, generating a database with more than 2,00,000 RF shapes for various operators, BWs, and RF inhomogeneity compensation levels.

The details of the GENETICS-AI software can be found at: https://patentcenter.uspto.gov/#!/applications/16861506/ifw/docs.

NMR sample preparations

For the spin entanglement experiments, neat ¹³C labeled chloroform (¹³CHCl₃) was added to a solution of CDCl₃ (1:10 ratio) and loaded into a 5 mm Shigemi® tube. Two qubit quantum states of ¹H and ¹³C spins were measured using quantum state tomography, as explained in Mitra et al. (37). All experiments were recorded on a Bruker 600 MHz spectrometer at 290 K.

Recombinant expression and purification of U-¹⁵N ubiquitin K48C mutant (UbiK^48C) were carried out as described previously (68). The sample consisted of 0.80 mg of U-¹⁵N UbiK^48C was solubilized in 10 mM sodium acetate buffer (pH 6.0) and 100 mM NaN₃. The final concentration of the sample was 500 µM. All experiments were performed on a Bruker 850 MHz spectrometer at 300 K.

The catalytic subunit of PKA (PKA-C, Mus musculus gene) was expressed in Escherichia coli BL21 (DE3) cells in M9 minimal medium at 24°C. After cell lysis, PKA-C purification was carried out with affinity chromatography using the His6-RIIa(R213K) subunit (Hemmer et al., 1997). The pellets of cell overexpressing Rlla (R213K) and PKA-C were combined and suspended in a buffer containing 30 mM 3-(N-morpholino) propane sulfonic acid (MOPS), 200 µM ATP, 15 mM MgCl₂, 0.15 mg mL⁻¹ lysozyme, 1 mM PMSF; 100 U mL⁻¹ DNAse I (Roche Applied Science), and 5 mM 2-mercaptoethanol (pH 8.0). The cells were lysed using a French press at 1,000 psi. After centrifugation, the supernatant was added to a slur of Ni⁺²-NTA nitrilotriacetic acid resin (Thermo Scientific, 1 mL of resin per liter of culture) and shaken at 4°C for approximately 3 hours. After washing the resin with 30 mM MOPS, 25 mM KCl, 15 mM MgCl₂, and 5 mM 2-mercaptoethanol (pH 8.0), PKA-C was eluted using 30 mM MOPS, 25 mM KCl, 15 mM MgCl₂, 5 mM 2-mercaptoethanol, and 1 mM cAMP (pH 8.0). The fractions containing PKA-C were collected and dialyzed overnight in 20 mM KH₂PO₄, 25 mM KCl, and 5 mM 2-mercaptoethanol (pH 6.5). Cation exchange chromatography was performed using a HiTrap SP column (GE Healthcare Life Sciences) to separate the different phosphorylated isoforms of PKA-C. A linear gradient of KCl in 20 mM KH₂PO₄ at pH 6.5 (Yonemoto et al., 1993) was applied, and the purified protein was collected and stored in phosphate buffer with 10 mM DTT, 10 mM, MgCl2, and 1.0 mM NaN3, at 4°C. For perdeuterated protein expression, the cells were growth 80% ²H₂O in M9 minimal medium using a 2.0 L fermenter. The most abundant isoform of PKA-C, corresponding to phosphorylation at S338, T197, and S10 residues (isoform II) (Walsh and Ashby, 1973), was utilized for acquiring the NMR experiments.

MBP was expressed in E. coli BL21 DE3 cells (NEB) containing the pMAL plasmid. After an overnight growth in LB media, the cells were spun down and resuspended in M9 media in 100% ²H₂O with ¹⁵NH₄Cl as the only source of nitrogen. MBP expression was induced at an OD₆₀₀ of 0.8 using isopropyl β-d-1-thiogalactopyranoside (IPTG). Upon reaching an OD of ∼5, the cells were harvested and lysed in a buffer containing 20 mM phosphate buffer (PBS), 120 mM NaCl, 1 mM EDTA, mM 0.15 mg mL⁻¹ lysozyme, 0.05% glycerol, 2 mM dithiothreitol (DTT), one tablet of protease inhibitor (cOmplete™, Roche Applied Science), and 100 U mL⁻¹ DNAse I (Roche Applied Science). Amylose resin (NEB) was added to the supernatant and stirred overnight. After washing with a buffer containing 20 mM PBS, 120 mM NaCl, and 1 mM EDTA, the resin was loaded into a mini-column, and MBP was eluted using 60 mM maltose. The elution containing MBP was concentrated and loaded into a size exclusion Superdex 200 (GE Healthcare Life Sciences) to eliminate impurity. The correct mass was assessed using mass spectrometry. The final sample consisted of 1 mM MBP in 10 mM Na₂HPO₄, 0.1 mM EDTA, and 1 mM NaN₃ buffer. The PFG-HSQC and GEN-HSQC experiments were collected on a Bruker 850 MHz spectrometer at 300 K.

ACKNOWLEDGEMENTS

The authors would like to acknowledge Dr. Michael Garwood for helpful discussions.

Funding

The work was supported by the National Institutes of Health (HL144130, GM64742, and GM100310 to G.V.).

Authors’ Contributions

M.V.S and G.V. designed the research. M.V.S. wrote the code. M.V.S., K.P., and C.O. performed the experiments. G.V. and M.V.S. analyze the data. All authors contributed to writing and reviewing the paper.

Data Availability

All RF shapes and pulse sequences are publicly available through the University of Minnesota repository site: https://doi.org/10.13020/f9pb-ma04 and GitHub https://github.com/manuvsub/GENETICSAI.

Notes

Competing Interests: The authors declare no competing interests.

References