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arxiv: 2412.05856 · v3 · submitted 2024-12-08 · 🪐 quant-ph

Development of Neural Network-Based Optimal Control Pulse Generator for Quantum Logic Gates Using the GRAPE Algorithm in NMR Quantum Computer

Ebrahim Khaleghian , Arash Fath Lipaei , Abolfazl Bahrampour , Morteza Nikaeen , Alireza Bahrampour This is my paper

Pith reviewed 2026-05-23 07:49 UTC · model grok-4.3

classification 🪐 quant-ph
keywords neural networkGRAPE algorithmNMR quantum computersingle-qubit gatesoptimal control pulsesreal-time pulse generationuniversal gatesNISQ
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The pith

A neural network trained on GRAPE pulses maps any single-qubit gate directly to its optimal NMR control pulse shape.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper trains a neural network using control pulses from the GRAPE algorithm, all begun from one shared initial condition, to learn the mapping from parameters of any single-qubit gate to the corresponding optimal pulse sequence for an NMR quantum computer. Numerical tests and physical runs on a three-qubit benchtop NMR system show that the network outputs produce the target dynamics with high fidelity. The result replaces repeated optimization with a single forward pass, allowing arbitrary single-qubit pulses to be generated in real time. When the network is paired with a separately computed GRAPE pulse for the CNOT gate, the combination supplies a universal gate set ready for use on NISQ devices.

Core claim

By developing the neural network using the GRAPE algorithm, the authors discover the function that maps any single-qubit gate to its corresponding pulse shape. This model enables the real-time generation of arbitrary single-qubit pulses. When combined with the GRAPE-generated pulse for the CNOT gate, it creates a comprehensive and effective set of universal gates that can efficiently implement any algorithm in noisy intermediate-scale quantum computers.

What carries the argument

Neural network trained on GRAPE-generated pulses from a fixed initial point that approximates the direct mapping from single-qubit gate parameters to control pulse shapes.

If this is right

  • Arbitrary single-qubit pulses can be produced in real time without running optimization for each new gate.
  • Together with one fixed GRAPE CNOT pulse the network supplies a complete universal gate set for NISQ-era devices.
  • The same training procedure can be applied to other quantum platforms whose Hamiltonians are similar to the NMR case.
  • Numerical simulations and physical experiments on the benchtop system both confirm that the generated pulses reach the desired target unitaries.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Circuit compilation time could drop because a trained forward pass replaces per-gate numerical optimization.
  • Retraining or fine-tuning the network on pulses that include realistic noise or different initial conditions would test whether the mapping remains accurate under experimental imperfections.
  • Extending the input representation to include two-qubit interaction parameters could allow the same architecture to generate entangling gates as well.

Load-bearing premise

A network trained exclusively on GRAPE pulses that all start from the same initial point will generalize to produce accurate pulses for any target single-qubit gate.

What would settle it

On the three-qubit benchtop NMR system, apply a network-generated pulse for a single-qubit gate outside the training distribution and measure whether its achieved fidelity is substantially lower than the fidelity of the corresponding GRAPE-optimized pulse.

Figures

Figures reproduced from arXiv: 2412.05856 by Abolfazl Bahrampour, Alireza Bahrampour, Arash Fath Lipaei, Ebrahim Khaleghian, Morteza Nikaeen.

Figure 1
Figure 1. Figure 1: FIG. 1. The subspace of the GRAPE-generated pulse [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Cosine similarity analysis of 16 samples gen [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Cosine similarity analysis of 17,000 samples gen [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. The neural network receives a [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Neural Network Architecture Overview. This [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Fidelity of unitary transformations given in Eq. [PITH_FULL_IMAGE:figures/full_fig_p007_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. The gate fidelities corresponding to the unitary [PITH_FULL_IMAGE:figures/full_fig_p008_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. 2D representation of the Fig.8 [PITH_FULL_IMAGE:figures/full_fig_p008_9.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. The tomography results, obtained after the [PITH_FULL_IMAGE:figures/full_fig_p009_11.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. Experimental results of the NMR spectrum [PITH_FULL_IMAGE:figures/full_fig_p009_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. The phase defined in Eq.10 is plotted against [PITH_FULL_IMAGE:figures/full_fig_p011_14.png] view at source ↗
read the original abstract

In this paper, we introduce a neural network to generate optimal control pulses for general single-qubit quantum logic gates, within a Nuclear Magnetic Resonance (NMR) quantum computer. By utilizing a neural network, we can efficiently implement any single-qubit quantum logic gates within a reasonable time scale. The network is trained by control pulses generated by the GRAPE algorithm, all starting from the same initial point. After implementing the network, we tested it using numerical simulations. Also, we present the results of applying Neural Network-generated pulses to a three-qubit benchtop NMR system and compare them with simulation outcomes. These numerical and experimental results showcase the precision of the Neural Network-generated pulses in executing the desired dynamics. Ultimately, by developing the neural network using the GRAPE algorithm, we discover the function that maps any single-qubit gate to its corresponding pulse shape. This model enables the real-time generation of arbitrary single-qubit pulses. When combined with the GRAPE-generated pulse for the CNOT gate, it creates a comprehensive and effective set of universal gates. This set can efficiently implement any algorithm in noisy intermediate-scale quantum computers (NISQ era), thereby enhancing the capabilities of quantum optimal control in this domain. Additionally, this approach can be extended to other quantum computer platforms with similar Hamiltonians.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper trains a neural network on GRAPE-generated control pulses for single-qubit gates in an NMR quantum computer, with all training pulses starting from the identical initial condition. The authors claim that the resulting model discovers the mapping from any single-qubit unitary to its optimal pulse shape, enabling real-time generation; they report numerical simulations and three-qubit benchtop NMR experiments showing that the NN pulses execute the desired dynamics, and combine the approach with a GRAPE CNOT pulse to obtain a universal gate set for NISQ devices.

Significance. If the network were shown to generalize reliably beyond its training distribution, the method could accelerate pulse design for single-qubit operations in NMR and related platforms. The work does not, however, supply evidence that the learned mapping is independent of the shared initial condition used in GRAPE, nor does it quantify performance on out-of-distribution gates; therefore the claimed advance in real-time arbitrary-gate synthesis remains unsecured.

major comments (2)
  1. [Abstract] Abstract: the assertion that the network 'discovers the function that maps any single-qubit gate to its corresponding pulse shape' is load-bearing for the central claim, yet all training data are generated by GRAPE runs that share one fixed initial pulse. Because the quantum-control landscape is non-convex, distinct initial conditions routinely yield different but equally valid pulses for the same target; the learned mapping may therefore be an artifact of that single optimization trajectory rather than a general function.
  2. [Abstract] Abstract (numerical and experimental validation paragraph): the reported simulations and three-qubit NMR experiments are said to 'showcase the precision' of the NN pulses, but no information is given on the diversity of the test-set gates, the distribution of achieved fidelities, or whether any test gates lie outside the convex hull of the training set. Without these data the generalization step required by the central claim cannot be evaluated.
minor comments (1)
  1. [Abstract] The abstract states that the approach 'can be extended to other quantum computer platforms with similar Hamiltonians' without indicating which Hamiltonian features are essential or providing a concrete example.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for their careful reading and constructive feedback. We address each major comment below and indicate the revisions we will make to the manuscript.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the assertion that the network 'discovers the function that maps any single-qubit gate to its corresponding pulse shape' is load-bearing for the central claim, yet all training data are generated by GRAPE runs that share one fixed initial pulse. Because the quantum-control landscape is non-convex, distinct initial conditions routinely yield different but equally valid pulses for the same target; the learned mapping may therefore be an artifact of that single optimization trajectory rather than a general function.

    Authors: We acknowledge that all GRAPE training pulses were generated from a single fixed initial condition. This is a substantive limitation given the non-convex control landscape, and the network may indeed be learning a mapping specific to that optimization trajectory rather than a fully general function. We will revise the abstract to qualify the claim, stating that the network learns to reproduce GRAPE pulse shapes obtained from the fixed initialization used in training. We will also add a brief discussion of the non-convexity issue and the possibility of multiple optima. No new multi-initial-condition experiments are planned for this revision. revision: partial

  2. Referee: [Abstract] Abstract (numerical and experimental validation paragraph): the reported simulations and three-qubit NMR experiments are said to 'showcase the precision' of the NN pulses, but no information is given on the diversity of the test-set gates, the distribution of achieved fidelities, or whether any test gates lie outside the convex hull of the training set. Without these data the generalization step required by the central claim cannot be evaluated.

    Authors: We agree that the current manuscript lacks the quantitative details needed to assess generalization. In the revised version we will expand the numerical and experimental sections to report: (i) the number and selection criteria for test gates (e.g., rotation angles sampled uniformly over the Bloch sphere), (ii) the distribution of achieved fidelities (mean, standard deviation, and minimum), and (iii) an explicit check of whether test gates lie outside the convex hull of the training distribution. These additions will directly address the referee's concern. revision: yes

standing simulated objections not resolved
  • Evidence that the learned mapping is independent of the shared initial condition used for GRAPE (this would require generating and comparing pulses from multiple distinct initial conditions, which was not done in the original study).

Circularity Check

1 steps flagged

NN mapping function reduces to GRAPE fitting by construction

specific steps
  1. fitted input called prediction [Abstract]
    "Ultimately, by developing the neural network using the GRAPE algorithm, we discover the function that maps any single-qubit gate to its corresponding pulse shape. This model enables the real-time generation of arbitrary single-qubit pulses."

    The network is trained exclusively on control pulses generated by GRAPE (all from the same initial point). The claimed mapping is therefore obtained by fitting to these GRAPE outputs; the 'discovered function' reduces by construction to an approximation of the training-set optimizer rather than an independent result.

full rationale

The paper's central claim is that training a neural network on GRAPE pulses discovers the general mapping from single-qubit gates to pulse shapes. This directly matches the fitted-input-called-prediction pattern: the network is supervised on GRAPE outputs (all from one fixed initial condition), so the asserted function is an approximation of those optimizer results rather than an independent derivation. No other circularity patterns are present; the work is otherwise self-contained as an empirical approximation technique.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The claim rests on the assumption that GRAPE pulses constitute reliable training targets and that the network generalizes from that finite set; no new physical entities are introduced and the only free parameters are the network weights fitted to the GRAPE data.

free parameters (1)
  • neural network weights and biases
    Fitted during supervised training on the GRAPE pulse dataset to learn the gate-to-pulse mapping.
axioms (1)
  • domain assumption GRAPE algorithm outputs are suitable ground-truth targets for the desired single-qubit unitaries
    The paper treats GRAPE pulses as the reference for training without independent verification of optimality for every gate.

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