Fast Machine Learning for Quantum Control of Microwave Qudits on Edge Hardware

Andy C. Y. Li; Flor Sanders; Gabriel N. Perdue; Gaurav Agarwal; Giuseppe Di Guglielmo; Luca Carloni

REVIEW 3 major objections 4 minor 38 references

Fast Machine Learning for Quantum Control of Microwave Qudits on Edge Hardware

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A neural network can predict SNAP-gate control pulses for a five-level microwave cavity qudit with mean trace infidelity near 10^-3, and a quantized version fits on an FPGA.

desk verdict Solid ML-surrogate engineering paper for SNAP gates on a simulated 5-level qudit, but the 'fast' in the title is never measured — expect latency gaps and some post-hoc model selection if you referee it. read the letter →

arxiv 2506.03323 v1 pith:SZT3HRAG submitted 2025-06-03 quant-ph

Flor Sanders , Gaurav Agarwal , Luca Carloni , Giuseppe Di Guglielmo , Andy C. Y. Li , Gabriel N. Perdue This is my paper · ORCID

classification quant-ph

keywords quantumoptimalcontrolSNAPgatesmicrowavecavityquditsneuralnetworkregressionmixtureofexpertsknowledgedistillationquantizationFPGAinference

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper tries to establish that a small neural network can compute control-pulse parameters for SNAP gates on a microwave cavity qudit quickly enough and cheaply enough to run on programmable logic, matching the fidelity of classical numerical optimal control. The authors generate a training set by optimizing 16 in-phase and 16 quadrature B-spline coefficients for 10,000 target phase angles on a five-level qudit, then train and fine-tune several architectures, culminating in a mixture-of-experts model with mean trace infidelity near $10^{-3}$. They then show that a compressed, quantized version of the model fits on an FPGA using under 90 percent of available lookup tables while keeping mean infidelity around $2x10^{-3}$. If the simulation-based fidelity transfers to hardware, this would enable on-the-fly, low-latency gate compilation in cryogenic environments where CPUs and GPUs are impractical.

What carries the argument

The load-bearing object is the learned function $\alpha \mapsto (\theta_1,\ldots,\theta_{32})$, where the $\theta$ are quadratic B-spline coefficients for the in-phase and quadrature microwave drives that realize a SNAP gate, a diagonal phase gate acting on one selected level of a multilevel cavity. The training pipeline first regresses these coefficients from the target phase angle using a multilayer perceptron or a mixture-of-experts network, the latter with soft or hard switching to handle discontinuities in the optimal-pulse surface. A second stage fine-tunes the network by backpropagating through the differentiable trace-infidelity cost of the simulated gate, which improves mean infidelity by more than an order of magnitude over mean-squared-error training alone. Knowledge distillation then transfers the mixture-of-experts behavior into a single small MLP, and quantization-aware training plus high-level synthesis converts that MLP into register-transfer logic for an FPGA.

What would settle it

Measure the deployed quantized model's SNAP gate on a real cavity-transmon device: if the mean trace infidelity is substantially above $10^{-2}$ under realistic decoherence and pulse distortions, or even in an open-system simulation with $T_1$ and $T_2$ included, the central fidelity claim would not transfer.

Watch

Extended reading notes

Core claim

The central claim is that machine-learned models can replace the numerical optimizer in a control pipeline for microwave cavity qudits. Concretely, a neural network maps a requested SNAP phase angle to the 32 B-spline coefficients of a 290 ns control pulse, and the resulting gate has mean trace infidelity near $10^{-3}$ for a five-level qudit in closed-system simulation. A five-expert mixture-of-experts model reaches a mean infidelity near $7\times10^{-4}$ and a maximum near $1.6\times10^{-3}$. The authors further claim that a distilled single-MLP version, trained by knowledge distillation from the mixture-of-experts model and quantized to 5-bit weights, synthesizes to FPGA logic with roughly 87 percent lookup-table utilization at a mean infidelity near $2.3\times10^{-3}$, making low-latency inference plausible on edge hardware.

Load-bearing premise

The reported infidelities are evaluated in the same closed-system simulation used to create the training data, so the claim stands on that noiseless model faithfully representing the physical cavity-transmon system.

Editorial extensions

If this is right

A five-expert mixture-of-experts model based on a roughly 4,300-parameter MLP achieves mean trace infidelity near $7\times10^{-4}$ for the five-level SNAP gate, with maximum infidelity near $1.6\times10^{-3}$.
Direct fine-tuning on the differentiable infidelity cost is what pushes performance below $10^{-3}$; MSE-only training leaves mean infidelity around $10^{-2}$ for the same model sizes.
Knowledge distillation transfers the mixture-of-experts behavior into a single MLP, so the hardware-friendly model retains near-Pareto infidelity without the multi-expert overhead.
Quantizing weights to five bits degrades mean infidelity only to about $2.3\times10^{-3}$ while reducing FPGA lookup-table usage to 87 percent, which fits the target programmable-logic device.
Because the optimized control pulses last 290 ns instead of the roughly 2000 ns standard SNAP implementation, the ML-computed pulses reduce decoherence exposure and raise the achievable gate clock rate.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Editorial: the same data-generation, distillation, and quantization pipeline should apply directly to SNAP gates on other levels and to qudits with more than five levels; the paper explicitly leaves this open, so the generality claim is an extrapolation, not a demonstrated result.
Editorial: the discovery that distilled models produce a continuous pulse-parameter surface suggests the discontinuities in the raw optimized data are largely an artifact of the optimizer's initialization and branch choices, not a physical feature of the gate; enforcing continuity during data generation could remove the need for mixture-of-experts entirely.
Editorial: the natural next benchmark is a lookup table mapping angles to pulse parameters on the same FPGA; the paper lists this as future work, but a comparative latency-and-fidelity study would show whether the ML route earns its complexity.
Editorial: since the infidelity cost is differentiable, one could fine-tune the deployed model against measured hardware outcomes rather than the closed-system simulation, turning the pipeline into a closed-loop calibration tool; the paper hints at this but does not test it.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Main take: this is a solid engineering paper that shows a plausible ML surrogate for SNAP gate pulse synthesis on a simulated 5-level cavity qudit, with a credible path to FPGA deployment. The thing to know is that the 'fast' in the title is not measured anywhere — there is no latency number, only LUT/FF utilization. So read the central claim as 'simulation-level approximation quality plus synthesizability,' not demonstrated real-time control.

What is actually new: previous ML control work on single-qubit gates, but this is the first to do SNAP gates on a 5-level qudit, and the use of mixture-of-experts to handle the discontinuous optimal-control landscape, followed by knowledge distillation to compress the MoE into a single MLP, is a nice piece of work. The design space exploration is thorough and the resource utilization table is real evidence that the quantized model can be synthesized to an FPGA with reasonable area. The paper is also transparent: it lists the hardware experiment and the lookup-table comparison as future work.

Where it is soft: (1) The infidelity of ~1e-3 is computed in the same closed-system simulator that generated the training data and that is used for fine-tuning. The paper does not hide this, but the abstract's 'near 10^-3' can be misread as hardware performance. (2) The MR models are assembled from the best fine-tuning checkpoints across runs, which is post-hoc selection; the comparison against MoE may be optimistic. (3) The 'random 3' quantized model is not fully specified in the text. (4) There are no error bars on the infidelity numbers. None of these are fatal — the core argument holds up as an engineering feasibility study — but they should be addressed.

Who it's for: people working on real-time quantum control hardware, ML surrogates for control, or qudit systems. It deserves a serious referee. I would recommend major revision that adds a latency estimate or at least a hardware-timing analysis, fixes the MR model selection protocol, and clarifies the simulation-only nature of the reported infidelity. A couple of quick additions would turn this from a useful tech report into a solid paper.

Referee Report

3 major / 4 minor

Summary. The paper presents a machine-learning-based pipeline for predicting optimal control-pulse parameters for SNAP gates on a five-level microwave cavity qudit. Data are generated by L-BFGS-B optimization of B-spline pulse parameters against a closed-system Hamiltonian (Eq. (2)) with chi=5 MHz and xi=200 MHz. Several architectures are compared: MLPs, mixture-of-experts (MoE), hand-segmented multi-region (MR) models, and knowledge-distilled compact MLPs. The best MoE achieves mean trace infidelity around 6.9e-4 in simulation. A knowledge-distilled model is quantized with QKeras, synthesized via hls4ml to a ZCU102 FPGA, and the 5-bit version achieves mean infidelity ~2.3e-3 with 87% LUT utilization. The paper reports no measured latency, only resource utilization.

Significance. The paper provides a concrete, end-to-end design flow for embedding a quantum-control surrogate in programmable logic, and the resource-utilization numbers show that a quantized model can fit on an FPGA. The DSE with Pareto analysis and the use of differentiable infidelity fine-tuning are useful contributions. However, the main quantitative claim is an in-sample simulation metric: training, fine-tuning, and evaluation all use the same closed-system simulator and the same infidelity cost. The 'fast' claim is not backed by any latency measurement. If the authors supply post-synthesis latency and quantify robustness to simulation error, the work would be a meaningful step toward real-time qudit control.

major comments (3)

[Sec. 2.4 and Table 4] The title and Sec. 1.1 motivate this work by the need for low-latency control, yet no latency, throughput, or achieved clock frequency is reported anywhere. Table 4 gives only LUT/FF utilization and infidelity; the target clock period of 3.225 ns is mentioned but the achieved timing is not. Without post-synthesis latency or a comparison to the classical optimizer's runtime (e.g., L-BFGS-B seconds per angle), the 'Fast Machine Learning' claim is unsupported. Please add latency estimates and a baseline comparison, or re-scope the title.
[Sec. 2.1, 2.2.3, and 2.3.1] The reported gate trace infidelity is computed with the same closed-system Hamiltonian (Eq. (2)) used to generate the training data and the same differentiable infidelity cost used in fine-tuning. Thus the numbers in Table 2 and Fig. 13 measure how well the network reproduces the classical optimizer's solutions within the idealized model, not the fidelity achievable on physical hardware. The abstract presents 'gate trace infidelity near 10^{-3}' without this caveat. Please add a robustness study (e.g., Lindblad decoherence, pulse distortion, finite bandwidth) or explicitly state in the abstract and conclusion that these are simulation-level results.
[Sec. 2.2.4 (Multi-Region)] For the MR models, the authors state that they 'evaluated the results of every intermediate fine-tuning step and manually composed a model with the best regressor found for each decision region.' It is not stated whether this selection used the validation set or the test set. If the test set was used for model selection, the MR infidelities reported in Table 2 are optimistically biased. Please clarify the selection procedure and, if necessary, use a nested validation split to report unbiased performance.

minor comments (4)

[Table 4] The first row of Table 4 lists a quantization value '18'; this appears to be a typo for '8', which would match the text saying that weight quantization bits 7 and 8 exceed 100% LUT utilization. Please correct the table and the column header.
[Sec. 2.3.1] The model selected for quantization is called 'random 3' but this name is never defined in the design space exploration of Sec. 2.2.4. Please describe which of the 100 random configurations corresponds to 'random 3' and its architecture.
[Sec. 2.2.2] The statement that the universal approximation theorem 'holds for discontinuous functions as well, as long as the network is at least three layers deep' is too broad for the cited reference; please quote the precise theorem from [28].
[Sec. 2.2.3] The fine-tuning procedure is described in terms of 'batches of each 16 angles per round' and 'iterations'; the relationship between a round and an iteration should be defined when first used (e.g., 'one round consists of N iterations').

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the ML surrogate is a standard supervised fit evaluated on held-out angles; FPGA synthesis evidence is independent.

full rationale

The paper's derivation chain is a conventional supervised-surrogate pipeline. Classical optimal control (L-BFGS-B minimizing infidelity on the closed-system Hamiltonian H0 of Eq. (2) with B-spline pulse parameters) generates 10,000 labeled samples, and the neural network learns the map from phase angle α to optimal pulse parameters θ*. The reported mean trace infidelity near 10^-3 is computed with the same simulator used for label generation, but this reuse is not circular: at inference the network has not been given the optimal parameters, and the data are split into training, validation, and testing sets (80/10/10, Sec. 2.2.1), so the test infidelity is a generalization measure. The fine-tuning step in Sec. 2.2.3 minimizes the same differentiable infidelity cost used in data generation, which makes the reported infidelity partly an optimization objective rather than an independent prediction; however, it is an honest statement about surrogate approximation quality inside the simulator, not a fitted parameter renamed as a prediction. The FPGA resource-utilization results in Table 4 are independent evidence of synthesizability. The paper is also transparent about its external-validation gaps: Sec. 4 (Future Work) states that a real-time qudit control experiment “should be conducted” and that the ML model “should ideally be compared to the more common lookup table implementation.” These are limitations on the practical “fast control” claim, not circularity. No load-bearing argument reduces to a self-citation, an imported uniqueness theorem, or an ansatz smuggled in by citation. The derivation is self-contained, so the circularity score is 0.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the fidelity of the simulation to physical hardware, the optimizer's ability to find near-global minima, and a set of hand-chosen preprocessing and architecture parameters (filter threshold, smoothing window, MR boundaries, expert count, quantization width, fine-tuning length). No new physical entities are introduced.

free parameters (6)

Infidelity filtering threshold = 1e-4
Samples with infidelity above 1e-4 are discarded; chosen empirically in Section 2.2.1, it shapes the training set and the discontinuities the model must learn.
Smoothing window width = 50 samples
Uniform windowed average along the angle axis, roughly 1.8 degrees; chosen by visual inspection of the heatmap to reduce discontinuities.
Multi-Region decision boundaries = [-0.490pi, -0.426pi, 0, 0.682pi]
Hand-selected from the mean first derivative of the pulse parameters to define the hard-switching subdomains in the Multi-Region architecture.
Number of MoE experts = 5
Selected from design space exploration; more than six experts gives minimal infidelity gain per added expert.
Weight quantization bit width = 5 bits
The 'random 3' distilled model is synthesized at 5-bit weight precision, giving 0.00234 mean infidelity with 87% LUT utilization on ZCU102.
Fine-tuning rounds = 25-30
The number of infidelity fine-tuning rounds is chosen per model; e.g., MoE-5 models receive 25-30 rounds until convergence.

assumptions (5)

domain assumption The dispersive Hamiltonian H0 = -chi a-dagger a b-dagger b - xi (b-dagger)^2 b^2, with chi=5 MHz and xi=200 MHz and qubit truncated to three levels, accurately models the real qudit-qubit system.
Section 2.1 adopts this model from reference [8]; all training data and all reported infidelities are computed from it.
domain assumption The L-BFGS-B optimizer returns pulse parameters that are globally near-optimal after filtering out infidelity above 1e-4.
Section 2.1 generates the ground-truth dataset with this optimizer; a global optimum is not proven, and local minima may survive the threshold.
domain assumption The quadratic B-spline parameterization with 16 coefficients per quadrature can represent the optimal 290 ns pulses.
Section 2.1 restricts the pulse space; if the true optimal pulse is not in this function space, the ground truth itself is suboptimal.
standard math Trace infidelity computed from the simulation is the correct metric to minimize for gate quality.
Standard definition in quantum optimal control; used both for data generation and fine-tuning.
standard math The universal approximation theorem justifies that a sufficiently wide MLP can represent the target pulse-parameter function.
Section 2.2.2 relies on references [27,28] to motivate MLP capacity; this is background math, not a domain-specific assumption.

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Cite this review

Pith. "Pith review of Fast Machine Learning for Quantum Control of Microwave Qudits on Edge Hardware." pith.science (2026). https://pith.science/paper/SZT3HRAG

@misc{pith2026250603323,
  author       = {Pith},
  title        = {Pith review of: Fast Machine Learning for Quantum Control of Microwave Qudits on Edge Hardware},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SZT3HRAG}},
  note         = {Machine review of arXiv:2506.03323}
}

abstract

Quantum optimal control is a promising approach to improve the accuracy of quantum gates, but it relies on complex algorithms to determine the best control settings. CPU or GPU-based approaches often have delays that are too long to be applied in practice. It is paramount to have systems with extremely low delays to quickly and with high fidelity adjust quantum hardware settings, where fidelity is defined as overlap with a target quantum state. Here, we utilize machine learning (ML) models to determine control-pulse parameters for preparing Selective Number-dependent Arbitrary Phase (SNAP) gates in microwave cavity qudits, which are multi-level quantum systems that serve as elementary computation units for quantum computing. The methodology involves data generation using classical optimization techniques, ML model development, design space exploration, and quantization for hardware implementation. Our results demonstrate the efficacy of the proposed approach, with optimized models achieving low gate trace infidelity near $10^{-3}$ and efficient utilization of programmable logic resources.

Figures

Figures reproduced from arXiv: 2506.03323 by the authors.

**Figure 1.** System Design Flow ity of hls4ml combined with the HLS tuning capabilities to optimize the design of a hardware implementation for programmable devices [21]. We will compute performance indicators such as model fidelity, latency, and resource utilization as programmable logic. Our core contribution then is a high-fidelity ML model compiled to a programmable device for controlling a qudit encoded in a three-dimensi… view at source ↗

**Figure 3.** Pulse Parameter Heatmap Note that the standard SNAP implementation requires the pulse duration time to be much longer than 1/ξ. The optimal pulse implementation we consider here has a much shorter duration (290 ns) than that required by the standard implementation (∼2000 ns). This shorter duration is important in terms of reducing decoherence, which can be viewed as an undesired quantum to classical collapse with a… view at source ↗

**Figure 5.** Training Data Infidelity • Perform a smoothing operation along the angle axis, using a uniform windowed average over 50 samples (equivalent to 1.8◦ ). The pulse parameters after pre-processing are shown in [PITH_FULL_IMAGE:figures/full_fig_p003_5.png] view at source ↗

Figures from the paper (12 more)

**Figure 6.** Figure 6: Mixture-of-Experts Model Architecture, from [29] [PITH_FULL_IMAGE:figures/full_fig_p004_6.png]

**Figure 7.** Figure 7: Model mlp_4308 Infidelity Performance fore (blue) and after fine-tuning (orange). These examples show that the proposed fine-tuning procedure effectively improves the infidelity of the trained models. Mixture-of-Experts Design Space Exploration Next, we assess the pot…

**Figure 9.** Figure 9: MoE expert count vs. model infidelity [PITH_FULL_IMAGE:figures/full_fig_p005_9.png]

**Figure 10.** Figure 10: MoE Fine-tuning for mlp_4308 [PITH_FULL_IMAGE:figures/full_fig_p006_10.png]

**Figure 13.** Figure 13: MoE mlp_4308 Infidelity Performance character in the table. 2.3 Quantization To translate a machine learning model for implementation in hardware, the model first needs to be quantized. This section discusses our approach of using QKeras for initial quantization and H…

**Figure 15.** Figure 15: Infidelity vs. Angle for 4-bit precision parameters [PITH_FULL_IMAGE:figures/full_fig_p007_15.png]

**Figure 16.** Figure 16: Infidelity vs. Angle for 6-bit precision parameters [PITH_FULL_IMAGE:figures/full_fig_p007_16.png]

**Figure 17.** Figure 17: Infidelity vs. Angle for 8-bit precision parameters [PITH_FULL_IMAGE:figures/full_fig_p008_17.png]

**Figure 18.** Figure 18: Trace Output with perfect result precision [PITH_FULL_IMAGE:figures/full_fig_p008_18.png]

**Figure 21.** Figure 21: Trace Output for Layer 6 [PITH_FULL_IMAGE:figures/full_fig_p009_21.png]

**Figure 23.** Figure 23: LUT Utilization vs. Mean Infidelity for better integration and more extensive model finetuning. • While the MLP model architectures achieve near-Pareto performance through knowledge distillation, the best performance is achieved with the MoE architectures. In future …

**Figure 22.** Figure 22: Trace Output for Layer 10 ers remains an open research problem. Below, we offer some specific questions that may inform future research directions. • The main issue holding back model infidelity performance without the need for further fine-tuning optimizations are …

Discussion (0). Continue with ORCID to comment.

Reference graph

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METHODOLOGY This section provides a more detailed explanation of the various steps of our design methodology shown in Fig. 1. In Sec. 2.1 we explain how we produced the dataset used to train the ML model for qudit-based quantum control. Next, Sec. 2.2 discusses the steps required to design and optimize a ML model for this task. In Sec. 2.3, we discuss the...

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

INTRODUCTION Quantum Computing (QC) has undergone rapid develop- ment over the past four decades [1]. Harnessing the quan- tum mechanical properties of the natural world, specialized algorithms such as Shor’s Algorithm [2] and Grover’s Algo- rithm [3] theoretically offer asymptotic scaling advantages over all known relevant classical algorithms. Although ...

work page Pith review arXiv 2025

[2] [2]

clock-rate

METHODOLOGY This section provides a more detailed explanation of the various steps of our design methodology shown in Fig. 1. In Sec. 2.1 we explain how we produced the dataset used to train the ML model for qudit-based quantum control. Next, Sec. 2.2 discusses the steps required to design and optimize a ML model for this task. In Sec. 2.3, we discuss the...

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hard switching

model selector and hand-defined regions as a clas- sifier, further labeled Multi-Region (MR). Owing to the hand-selected switching, we refer to this as “hard switching.” Note the contrast to soft switching where a weighted average is implicitly learned. The advantage of these architectures is that the network can explicitly model or learn the discontinuit...

work page

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In this case, we go for 10,000 Figure 9: MoE expert count vs

Create a dataset containing input-output samples pro- duced by the MoE model. In this case, we go for 10,000 Figure 9: MoE expert count vs. model infidelity Figure 10: MoE Fine-tuning for mlp_4308 Figure 11: MoE Fine-tuning for mlp_514 linearly spaced input angles

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Here we select all configurations from Table 1

Train a MLP model on this new dataset. Here we select all configurations from Table 1. After training, KD models with parameter counts exceed- ing 1000 surpass their fine-tuned counterparts in mean infi- delity performance, confirming the effectiveness of the KD technique. However, the results for models with lower pa- rameter counts do not achieve the sa...

work page

[6] [6]

Here, we explored the fusion of machine learning and quantum con- trol, with a focus on microwave cavity qudits

CONCLUSION Quantum computing may offer super-polynomial asymp- totic scaling advantages for some important problems, but its practical realization hinges on addressing critical chal- lenges such the development of optimal quantum control sys- tems with sufficiently low latency and high fidelity. Here, we explored the fusion of machine learning and quantum...

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Below, we offer some specific questions that may inform future research directions

FUTURE WORK While the contributions made here are concrete and promis- ing, optimal control for qudit devices in quantum comput- Figure 21: Trace Output for Layer 6 Figure 22: Trace Output for Layer 10 ers remains an open research problem. Below, we offer some specific questions that may inform future research directions. • The main issue holding back mod...

work page

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Quantum computing: A taxonomy, systematic review and future directions

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work page 2022