Quantum Engineering of Qudits with Interpretable Machine Learning

Akram Youssry; Alberto Peruzzo; Gerardo A. Paz-Silva; Ritik Sareen; Yule Mayevsky

REVIEW 4 major objections 8 minor 70 references

Quantum Engineering of Qudits with Interpretable Machine Learning

T0 review · 4 major / 8 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper claims that a physics-informed graybox machine-learning model can synthesize high-fidelity qudit gates under non-Markovian noise by representing the noise operator with a learnable unitary eigendecomposition.

desk verdict A credible, incremental qudit extension of the graybox control framework, but the noise-operator parameterization does not cover all operators allowed by the paper's own constraints, so the generality claim overreaches. read the letter →

arxiv 2506.13075 v1 pith:GZP3HNGX submitted 2025-06-16 quant-ph

Yule Mayevsky , Akram Youssry , Ritik Sareen , Gerardo A. Paz-Silva , Alberto Peruzzo This is my paper

classification quant-ph

keywords quditcontrolgrayboxmachinelearningnon-Markoviannoiseoperatorformalisminterpretablequtritgatesrotating-waveapproximationquantumsystemidentification

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

This paper claims that a physics-informed 'graybox' machine-learning model can control and characterize qudits—quantum systems with dimension $d>2$—under realistic conditions that standard control methods struggle with: non-Markovian dephasing noise, non-ideal pulse shapes, and dynamics beyond the rotating-wave approximation. The paper extends a previously single-qubit graybox approach to arbitrary dimension by parameterizing the noise operator $V_O(T)$ with a unitary eigendecomposition whose parameters are emitted by a recurrent neural network. In numerical experiments on a qutrit ($d=3$) driven by Hanning pulses and subject to $1/f$ plus high-frequency noise, the optimized global gates reach infidelities below 0.08 under strong noise and subspace gates below 0.085, with near-ideal closed-system performance. The paper also introduces a local Taylor expansion of the learned noise operator, giving an interpretable analytic picture of how pulse perturbations interact with noise. A sympathetic reader would take the contribution to be a scalable route to noise-aware qudit gate design and system identification.

What carries the argument

The load-bearing object is the parameterized noise operator $V_O(T)$, defined through $\langle O(T)\rangle = \mathrm{tr}(V_O(T) U_0(T)\rho U_0^\dagger(T) O)$. Its Hermitian core $W_O = Q D Q^\dagger$ is built from a $2\times 2$ subunitary decomposition of the unitary $Q$ and diagonal entries $D_{ii} = d\, p_i x_i$, so the neural network only needs to output bounded rotation parameters, tanh values $x_i$, and softmax probabilities $p_i$. A second mechanism is the local analytic expansion: after training, the model's $V_O(\epsilon; P_i)$ predictions are fit to $X_0 + \epsilon X_1 + \epsilon^2 X_2 + \cdots$, turning the latent noise response into matrix-valued Taylor coefficients. These coefficients feed an interpretable cost function $J(\epsilon; P_i)$ that combines noise deviation from identity with gate infidelity, and the paper uses it to show why optimized pulses outperform random ones.

What would settle it

Run the paper's own dephasing model for many random pulse sequences, compute the exact noise operator $W_O$, and check whether any realization has eigenvalues $\lambda_i$ with $(1/d)\sum_i |\lambda_i| > 1$ while still satisfying $|\lambda_i| \le d$ and $|\sum_i \lambda_i| \le d$. Such a spectrum cannot be represented as $d\,p_i x_i$, since $\sum_i |d\, p_i x_i| = d \sum_i p_i |x_i| \le d$. Finding one would show the graybox is not expressive enough to learn that noise operator, breaking the claimed generality.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the graybox model—composed of a whitebox that simulates the known Hamiltonian evolution and a blackbox recurrent network that predicts the parameters of the noise operator—generalizes from qubits to qudits without changing the MSE training objective. The noise operator of each observable is written as $V_O = O^{-1} W_O$ with $W_O = Q D Q^\dagger$, where $Q$ is a product of $d(d-1)/2$ two-level unitaries and $D = d\,\mathrm{diag}(p_i x_i)$ with $p_i$ a probability distribution and $x_i\in[-1,1]$; this bounded parameterization is what the neural network outputs. Trained on Monte Carlo simulations of a truncated anharmonic oscillator qutrit with non-RWA coupling and non-Markovian dephasing ($1/f$ plus proportional noise), the model predicts measurement outcomes to MSE on the order of $10^{-3}$ under strong noise and $10^{-5}$ in the closed system. Optimizing pulse amplitudes with the trained model yields process infidelities below $6\times 10^{-5}$ for all tested gates in the closed system, around $10^{-2}$ under weak noise, and below 0.08 (global) or 0.085 (subspace) under strong noise. The authors interpret these results as evidence that high-fidelity qudit control is achievable without recourse to rotating-wave or Markovian approximations.

Load-bearing premise

The load-bearing premise is that every physically allowed noise operator for these qudit systems can be written as $W_O = Q D Q^\dagger$ with $D = d\,\mathrm{diag}(p_i x_i)$, where $p_i$ is a probability vector and each $x_i$ lies in $[-1,1]$; the paper does not prove that this form covers all Hermitian operators that satisfy its stated eigenvalue and trace bounds.

Editorial extensions

If this is right

If the central claim is right, the same graybox pipeline should synthesize high-fidelity gates for qudits of dimension $d>3$ by only resizing the basis and the neural network output layer.
Non-RWA dynamics and non-ideal pulse envelopes need not be approximated away; the whitebox simulates them directly, so the method applies to hardware where these effects are significant.
The local Taylor expansion turns a trained blackbox into a physically readable model: the coefficient matrices $X_k$ show how strongly noise responds to pulse-amplitude perturbations, which can guide pulse design and robustness analysis.
The cost function $J(\epsilon; P_i)$ can serve as a gate-fidelity proxy that is cheaper to evaluate than process tomography and is consistent with achieved fidelities.
MSE training without a custom loss suffices for qudit control, which simplifies extension of the approach to other platforms and gate families.

Reading between the lines

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

If the parameterization of $W_O$ is complete, the architecture should learn the noise operator for any dephasing model with the same $1/f+\alpha f$ spectrum; a direct numerical test would be to generate a dataset from a different noise PSD (for example, a Lorentzian) and check whether the trained graybox predictions retain the reported accuracy.
The Taylor coefficient matrices give a fingerprint of the noise environment; comparing coefficients across noise models could allow distinguishing noise sources that produce similar power spectra, an application the paper mentions but does not carry out.
The gate-dependent minima in $J(\epsilon; P^*)$ suggest that the optimal pulse amplitude and the easiest universal gate set depend on the noise environment; using this cost to select a gate set is a natural next step the paper leaves open.
An experimental test on a real superconducting or trapped-ion qutrit could compare the predicted $V_O$ expansion with measured noise sensitivity; a mismatch would localize the failure to the noise-operator parameterization rather than the optimization.

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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

Quick take: the paper is a genuine, incremental extension of the graybox control framework to qutrits, with a new noise-operator parameterization and a local Taylor expansion for interpretability. The simulations are careful, and the gate fidelities are credible. But the central claim of generality to arbitrary qudit noise is undercut by the parameterization gap the stress-test note identifies, and the experimental validation is entirely synthetic.

What's new: extending the noise operator formalism to d>2 is a natural but non-trivial step, and the local expansion for V_O is a nice addition that gives physical insight into why certain pulses work. The closed-system case recovering V_O ≈ I validates the model. The training details and system parameters are reported in the supplement, which is good practice.

Where it's soft: The parameterization W_O = Q D Q† with D = d diag(p_i x_i) is not surjective onto the set of Hermitian operators with eigenvalues bounded by d and trace bounded by d. The stress-test counterexample diag(1,1,-1.5) in d=3 satisfies the paper's stated bounds but has ∑|λ_i| = 3.5 > d, so it cannot be represented. This doesn't invalidate the specific simulations, but it means the "arbitrary qudit noise" claim is an assumption, not a consequence. A missing theorem would be needed.

Beyond that: all results are single realizations on synthetic data with no error bars, no comparison to standard control baselines like GRAPE, and no public code or data (available upon request is weak). The interpretability demonstration is limited to one pulse, so its generality is unproven. These are fixable but should be addressed.

My verdict: this paper deserves peer review. The core idea is sound, the simulations look reasonable, and the parameterization flaw is a technical gap that a good referee could catch and the authors could fix—either by proving the stronger trace-norm bound for physical noise or by softening the claims. I'd probably cite it for the qudit extension once it's cleaned up.

Referee Report

4 major / 8 minor

Summary. The paper extends the graybox machine-learning framework for quantum control from qubits to qudits. The method combines a whitebox model of the known Hamiltonian (including non-RWA dynamics and non-ideal pulse shapes) with a blackbox neural network that parameterizes the noise operator V_O via an eigendecomposition, and trains on synthetic expectation-value data from a simulated qutrit (and qubit) under non-Markovian dephasing. After training, control pulses are optimized for global clock-shift gates and two-level subspace gates, with reported infidelities below 0.08 and 0.085, respectively, under strong noise. The paper also introduces a local Taylor expansion of the noise operator as an interpretability tool and uses it to define a control cost function J(epsilon).

Significance. If the framework performs as claimed, it would be a useful contribution to qudit control under realistic noise and to interpretable quantum system identification. The paper has concrete strengths: it explicitly respects physical constraints in the model architecture, demonstrates the approach on a non-RWA qutrit Hamiltonian with non-Markovian noise, and proposes a structured local expansion that goes beyond pure blackbox explainability. The main significance, however, is tempered by three issues: the noise-operator parameterization is not proven to cover the stated class of noise operators, the numerical results are single synthetic runs without error bars or baselines, and the interpretability analysis is not validated against ground truth. These issues affect the strength of the central claims but appear addressable in a revision.

major comments (4)

[§II.A.1, Eqs. (8)–(15)] The parameterization W_O = Q D Q^† with D = d diag(p_i x_i) is not surjective onto the set of Hermitian operators satisfying the stated constraints (Hermitian, |λ_i| ≤ d, |tr(W_O)| ≤ d). For any representable spectrum, ∑_i |λ_i| = d ∑_i |p_i x_i| ≤ d, but the stated constraints allow spectra with ∑_i |λ_i| > d, e.g., d=3 with eigenvalues (1,1,−1.5). No theorem is given showing that the physically admissible W_O for the noise models considered satisfies this stronger nuclear-norm bound. This is load-bearing for the claim that the framework handles 'arbitrary' qudit noise, and it also affects the interpretation of the learned V_O as a faithful physical noise operator. Please either prove the stronger condition for the relevant noise class or replace the eigenvalue construction with a parameterization that is surjective onto the stated constraint set.
[§III.E.1, Tables I–II] All reported infidelities are single numbers from a single synthetic dataset, with no error bars, no repeated noise realizations, and no comparison to alternative control methods. In strong noise the infidelities cluster in the 10^{-2} range, and the text claims 'reliable generalization across the SU(3) Hilbert space' on the basis of these single values. Without confidence intervals, multiple Monte Carlo seeds, and at least one baseline (e.g., pulse optimization without the graybox, or a standard optimal-control routine), the performance claim is not statistically supported. Please add statistical repeats and baselines, or soften the claims accordingly.
[§III.D, Eq. (21)–(22)] The local Taylor expansion is fitted to the trained graybox output only; there is no validation against the true noise operator W_O or V_O computed from the simulator, nor a check against an independent physical model. The statement in the Discussion that J(epsilon) 'reliably tracks gate performance' is based on a single pulse (index 1595) at a single value of epsilon (Fig. 3), without a quantitative correlation over the random-pulse ensemble. Please provide a ground-truth comparison of the fitted V_O(epsilon) and a correlation statistic for J(epsilon) versus actual fidelity across the tested pulses.
[§II.A.3, §III.B] The abstract and introduction claim a framework for qudits of arbitrary dimension and for scalable control, but the experiments cover only d=2 and d=3, and the stated basis-scaling O(d^5) is not derived or demonstrated. The theoretical generalization argument in Eq. (19) is conditional on the graybox predicting G(i,j,k) for all basis elements, which is not shown for d>3. Please either provide a scalability analysis (e.g., d=4 or d=5 results, or a parameter-count/scaling study) or temper the 'arbitrary dimension' and 'scalable' claims.

minor comments (8)

[Abstract] The phrase 'allows us to to understand' contains a duplicated word; please fix.
[§II.A.1] The equation numbering skips Eq. (12), and the proof that −1 ≤ z_i ≤ 1 should explicitly invoke p_i ∈ [0,1] as well as x_i ∈ [−1,1]; also 'Eignevalues' is a typo.
[§II.B] The final sentence says the output ⃗E has d(d^2−1)^2 elements, which contradicts the d(d^2−1) count in Fig. 1 and §III.B; please clarify the intended dimension.
[Eq. (22)] The unitary U in the fidelity term is never defined; please specify whether it is the full noisy evolution U(T), the ideal U0(T), or the optimized actual evolution.
[§II.A.3] The claim that the Gell-Mann basis scaling grows as O(d^5) is asserted without derivation; please add a short justification, as the scaling is non-obvious.
[Data and code availability] The statement 'available upon request from the corresponding author' is insufficient for a reproducibility-focused journal; please provide a permanent repository or a more detailed access procedure.
[Figure 3(c)] The x-axis labels are a long concatenated string of pulse indices; use a table or a readable axis to allow identification of individual pulses.
[§III.E.1] There is a typo 'as shwon' in the sentence referring to Table II; please correct it.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the reported gate fidelities are evaluated against a held-out simulator benchmark, and the fitted local expansion is a diagnostic rather than the source of the central claim.

full rationale

The paper's core loop is to simulate a noisy qutrit with non-RWA dynamics and non-Markovian dephasing, train a graybox model on randomized pulses with a separate test split, optimize pulses against the model, and then evaluate the resulting gates using process fidelity computed from the simulator. This is an external benchmark relative to the trained model: the reported infidelities in Tables I and II come from evaluating the optimized pulses in the simulator, not from the model's own predictions. The local Taylor expansion in Eq. 21 is explicitly fitted to the trained model's outputs and is used only for interpretability and for the auxiliary cost function J in Eq. 22; it is not used to derive the high-fidelity claim, so fitting it does not make the main result circular. The noise-operator parameterization W_O = Q D Q^dagger with D = d diag(p_i x_i) is introduced as a sufficient construction that satisfies the stated constraints; whether it covers every physically admissible noise operator is a legitimate expressiveness and correctness concern, as the skeptic's diag(1,1,-1.5) example shows, but it is not a circular reduction of the paper's central claims to its inputs, because the high-fidelity claim is empirical and is benchmarked independently. References to the authors' earlier graybox work supply motivation and the noise-operator formalism, but the qudit extension and the numerical results are new and are not obtained by citing that work as the result itself. No step in the derivation chain reduces, by construction or by self-citation, to its own inputs.

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

The central claim rests on the assumption that the noise operator formalism and the specific parameterization can capture arbitrary qudit noise, which is not proven. The simulation parameters and pulse shape are modeling choices.

assumptions (5)

domain assumption The noise operator formalism of Refs. [37,38] is assumed valid for dimensions d>2.
The paper extends the single-qubit noise operator representation to qudits without a new derivation, relying on prior work by the same authors.
ad hoc to paper The eigendecomposition parameterization in Eqs. (9)-(15) is assumed to cover the relevant noise operators.
No proof or citation is given that every Hermitian matrix with bounded eigenvalues and trace can be expressed as Q D Q† with D = d diag(p_i x_i); the construction may be restrictive.
domain assumption The noise is classical, dephasing, with power spectral density given by Eq. (26).
This is a modeling choice for the simulated system, not derived from first principles or hardware measurements.
domain assumption Control pulses are restricted to the Hanning envelope form of Eq. (25).
This is a practical but non-general choice; the method's independence of pulse shape is claimed but only this shape is tested.
domain assumption The Monte Carlo simulator accurately represents the true dynamics of the qudit.
Training and evaluation data both come from this simulator, so the results are only as valid as the simulator's fidelity.

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Cite this review

Pith. "Pith review of Quantum Engineering of Qudits with Interpretable Machine Learning." pith.science (2026). https://pith.science/paper/GZP3HNGX

@misc{pith2026250613075,
  author       = {Pith},
  title        = {Pith review of: Quantum Engineering of Qudits with Interpretable Machine Learning},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GZP3HNGX}},
  note         = {Machine review of arXiv:2506.13075}
}

read the original abstract

Higher-dimensional quantum systems (qudits) offer advantages in information encoding, error resilience, and compact gate implementations, and naturally arise in platforms such as superconducting and solid-state systems. However, realistic conditions such as non-Markovian noise, non-ideal pulses, and beyond rotating wave approximation (RWA) dynamics, pose significant challenges for controlling and characterizing qudits. In this work, we present a machine-learning-based graybox framework for the control and noise characterization of qudits with arbitrary dimension, extending recent methods developed for single-qubit systems. Additionally, we introduce a local analytic expansion that enables interpretable modelling of the noise dynamics, providing a structured and efficient way to simulate system behaviour and compare different noise models. This interpretability feature allows us to to understand the mechanisms underlying successful control strategies; and opens the way for developing methods for distinguishing noise sources with similar effects. We demonstrate high-fidelity implementations of both global unitary operations as well as two-level subspace gates. Our work establishes a foundation for scalable and interpretable quantum control techniques applicable to both NISQ devices and finite-dimensional quantum systems, enhancing the performance of next-generation quantum technologies.

Figures

Figures reproduced from arXiv: 2506.13075 by the authors.

**Figure 2.** FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]

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Reference graph

Works this paper leans on

70 extracted references · 58 canonical work pages

[1]

In this paper we propose a new parameter- ization that can generalize to arbitrary system of dimension d

Parameterization of the noise operators The proposal in [37] is limited to single-qubit sys- tems. In this paper we propose a new parameter- ization that can generalize to arbitrary system of dimension d. The noise operator of an observable O can be expressed as [37] VO = O−1WO. (7) The operator WO has to satisfy the constraints, that is Hermitian, with b...

work page
[2]

The eigenvalues ¼i of the operator WO satisfy that |¼i| f d, where d is the dimensionality of the Hilbert space

work page
[3]

To satisfy those constraints we can use the eigende- composition of WO operator, WO = QDQ

|tr(WO)| f d =⇒ ∑ d i=1 ¼i f d. To satisfy those constraints we can use the eigende- composition of WO operator, WO = QDQ . (8) The matrix Q is a general d × d unitary. This can be parametrized using 2 × 2 subunitary decomposition [39]. This decomposition has a lot of applications in optics [40, 41], and the idea behind it is that we can decompose the gen...

work page
[4]

Handling non-invertible observables A non-invertible observable O will result in an un- deﬁned VO operator. However, this can be addressed by applying a linear transformation with parameters a, b to the observable O yielding, ˜O = aO + bId×d, (16) The parameter b shifts the eigenvalues of O such that ˜O becomes invertible [42]. Speciﬁcally, b is chosen to...

work page
[5]

Two commonly used bases for this purpose are the clock-shift basis and the Gell-Mann basis, both of which generalize the Pauli matrices to higher-dimensional systems

Choice of basis The graybox framework requires an informationally-complete basis for accurately estimating then noise operators. Two commonly used bases for this purpose are the clock-shift basis and the Gell-Mann basis, both of which generalize the Pauli matrices to higher-dimensional systems. The Gell-Mann basis provides a traceless, Hermitian represent...

work page
[6]

Symmetric Gell-Mann Matrices: ¼(S) jk = (|jð ïk| + |kð ïj|) , 1 f j f k f d

work page
[7]

Anti-symmetric Gell-Mann Matrices: ¼(A) jk = −i (|jð ïk| − |kð ïj|) , 1 f j f k f d

work page
[8]

Diagonal Gell-Mann Matrices: ¼l = √ 2 l(l + 1) ( l∑ m=1 |mð ïm| − l |l + 1ð ïl + 1| ) , l = 1, . . . , d− 1. Together with the identity operator IN , these ma- trices form a complete Hermitian basis for all d × d operators. The scaling of all observables in this case grows as O(d5), (there are ( d2 − 1) basis, each has d eigenstates, and we have d2 measur...

work page

Show all 70 references

[9]

Let the measurement operator O be expressed as a linear combination of an orthonormal basis (ONB) {Ai}, such that O = ∑ i aiAi, and let the initial state Ä = ∑ j bjAj

Generalization power of the graybox The graybox framework provides a powerful tool for predicting the dynamics of arbitrary observables and initial states in quantum systems. Let the measurement operator O be expressed as a linear combination of an orthonormal basis (ONB) {Ai}...
[10]

The choice can be used de- pending on the application and control pulse shape

Interpretability via local expansions of the noise operator In order to construct local expansions of VO(T ), we ﬁrst deﬁne neighbourhoods around points in the space of control pulses. The choice can be used de- pending on the application and control pulse shape. Particularly,...
[11]

Explainable (post-hoc interpretable) control cost function Using the local Taylor expansion, we deﬁne a con- trol cost function J(ϵ; Pi) that quantiﬁes the devia- tion from ideal noiseless evolution: J(ϵ; Pi) = d2−1∑ k=1 ∥VOk (ϵ; Pi) − I∥ + ( 1 − |Tr(U G)|2 d2 ) . (22) The ﬁrs...
[12]

Gate optimization To asses the performance of our proposed method, we ﬁnd the optimal pulses to implement a universal set of qutrit gates utilizing the cost function shown as equation 4). We demonstrate the results for two families of gates: Global qutrit gates represented in ...
[13]

Figure 3(a–c) compare random pulses against the graybox- optimized pulse P ∗ for target gate R01

Explainable control cost function To understand the performance of pulse the op- timization, we evaluate their robustness (robust against variations in amplitudes ) using Equation 22, as an alternative physics-based cost function. Figure 3(a–c) compare random pulses against th...
[14]

6 J(ϵ) a) J(ϵ, P i) = ∑ d2− 1 k=1 ‖VOk(ϵ, P i) − I‖ + ( 1 − |Tr(U †G)|2 d2 ) − 1. 00 − 0. 75 − 0. 50 − 0. 25 0. 00 0. 25 0. 50 0. 75 1. 00 ϵ
[15]

70 N (ϵ) b) N (ϵ, P i) = ∑ d2− 1 k=1 ‖VOk(ϵ, P i) − I‖ Pulse 1484 Pulse 1117 Pulse 1674 Pulse 1595 Pulse 1286 Pulse 956 Pulse 604 Pulse 141 Pulse 1778 Pulse 1709 Pulse 484 Pulse 117 Pulse 174 Pulse 295 Pulse 86 Pulse 756 Pulse 4 Pulse 941 Pulse 478 Pulse 709 P ∗ 14841117167415...
[16]

9 Fidelity0.35 0.44 0.52 0.81 0.46 0.340.32 0.61 0.36 0.21 0.420.41 0.57 0.42 0.23 0.53 0.40 0.60 0.36 0.47 c) Fidelity of random pulses atϵ = − 0. 44
[17]

00 0. 05 0. 10 0. 15 0. 20 0. 25 t (µs) − 8 − 6 − 4 − 2 0 2 4 6 Envelope(MHz) d) Optimised pulse P∗ for R01 f (I) 1 f (Q) 1 f (I) 2 f (Q) 2 FIG. 3: Interpretable pulse parameter robustness analysis of qutrit control pulses for implementing target gate, G = R01. (a) Evaluation ...
[18]

5 J(ϵ; p∗) for X01 a) Closed Weak Strong − 1. 0 − 0. 5 0. 0 0. 5 1. 0 ϵ
[19]

5 J(ϵ; p∗) for H01 b) Closed Weak Strong − 1. 0 − 0. 5 0. 0 0. 5 1. 0 ϵ
[20]

5 J(ϵ; p∗) for R01 c) Closed Weak Strong − 1. 0 − 0. 5 0. 0 0. 5 1. 0 ϵ
[21]

5 J(ϵ; p∗) for σ0, 0 d) Closed Weak Strong − 1. 0 − 0. 5 0. 0 0. 5 1. 0 ϵ
[22]

5 J(ϵ; p∗) for σ1, 0 e) Closed Weak Strong − 1. 0 − 0. 5 0. 0 0. 5 1. 0 ϵ
[23]

whitening

5 J(ϵ; p∗) for σ2, 1 f) Closed Weak Strong FIG. 4: Control cost landscapes J(ϵ; P ∗) for six optimized qutrit gates under varying noise conditions. Each panel depicts the cost function J(ϵ) as a function of the amplitude scaling parameter ϵ ∈ [−1, 1], evaluated using a ﬁxed op...
[24]

Erhard, M

M. Erhard, M. Krenn, and A. Zeilinger, Nature Re- views Physics 2, 365 (2020)

2020
[25]

Neeley et al

M. Neeley et al. , Science 325, 722 (2009)

2009
[26]

N. Goss, A. Morvan, B. Marinelli, B. K. Mitchell, L. B. Nguyen, R. K. Naik, L. Chen, C. J¨ unger, J. M. Kreikebaum, D. I. Santiago, et al. , Nature commu- nications 13, 7481 (2022)

2022
[27]

Blatt and D

R. Blatt and D. Wineland, Nature 453, 1008 (2008)

2008
[28]

Mourik, S

V. Mourik, S. Asaad, H. Firgau, J. J. Pla, C. Holmes, G. J. Milburn, J. C. McCallum, and A. Morello, Physical Review E 98, 042206 (2018)

2018
[29]

M. W. Doherty, N. B. Manson, P. Delaney, F. Jelezko, J. Wrachtrup, and L. C. Hollenberg, Physics Reports 528, 1 (2013)

2013
[30]

Fern´ andez de Fuentes, T

I. Fern´ andez de Fuentes, T. Botzem, M. A. I. John- son, A. Vaartjes, S. Asaad, V. Mourik, F. E. Hud- son, K. M. Itoh, B. C. Johnson, A. M. Jakob, J. C. McCallum, D. N. Jamieson, A. S. Dzu- rak, and A. Morello, Nature Communications 15, 10.1038/s41467-024-45368-y (2024)

2024 doi
[31]

B. B. Zhou, P. C. Jerger, V. O. Shkolnikov, F. J. Heremans, G. Burkard, and D. D. Awschalom, Physical review letters 119, 140503 (2017)

2017
[32]

X. Yu, B. Wilhelm, D. Holmes, A. Vaartjes, D. Schwienbacher, M. Nurizzo, A. Kringhøj, M. R. van Blankenstein, A. M. Jakob, P. Gupta, F. E. Hudson, K. M. Itoh, R. J. Murray, R. Blume- Kohout, T. D. Ladd, A. S. Dzurak, B. C. Sanders, D. N. Jamieson, and A. Morello, arXiv preprin...

2024 arXiv
[33]

J. Wang, F. Sciarrino, A. Laing, and M. G. Thomp- son, Nature Photonics 14, 273 (2020)

2020
[34]

N. J. Cerf, M. Bourennane, A. Karlsson, and N. Gisin, Physical Review Letters 88, 127902 (2002)

2002
[35]

Bechmann-Pasquinucci and A

H. Bechmann-Pasquinucci and A. Peres, Physical Review Letters 85, 3313 (2000)

2000
[36]

E. T. Campbell, Physical review letters 113, 230501 (2014)

2014
[37]

Gottesman, arXiv preprint quant-ph/9807006 https://doi.org/10.48550/arXiv.quant-ph/9807006 (1998)

D. Gottesman, arXiv preprint quant-ph/9807006 https://doi.org/10.48550/arXiv.quant-ph/9807006 (1998)

1998 doi arXiv
[38]

Motzoi, J

F. Motzoi, J. M. Gambetta, S. T. Merkel, and F. K. Wilhelm, Physical Review Letters 103, 110501 (2009)

2009
[39]

Schutjens, F

R. Schutjens, F. A. Dagga, D. J. Egger, and F. K. Wilhelm, Physical Review A 88, 052330 (2013)

2013
[40]

W. S. Warren, H. Rabitz, and M. Dahleh, Science 259, 1581 (1994)

1994
[41]

Morris, F

J. Morris, F. A. Pollock, and K. Modi, Open Sys- tems & Information Dynamics 29, 2250007 (2022)

2022
[42]

G. A. White, C. D. Hill, F. A. Pollock, L. C. Hol- lenberg, and K. Modi, Nature Communications 11, 6301 (2020)

2020
[43]

Giarmatzi, T

C. Giarmatzi, T. Jones, A. Gilchrist, P. Pakkiam, A. Fedorov, and F. Costa, arXiv preprint arXiv:2308.00750 https://doi.org/10.48550/arXiv.2308.00750 (2023)

2023 doi
[44]

G. Yang, M. Gluza, S. Y. Koh, C. P. Y. Wong, K. E. J. Goh, B. Weber, H. K. Ng, and T. S. Koh, High-ﬁdelity cz gates in double quantum dot – cir- cuit qed systems beyond the rotating-wave approx- imation (2024)

2024
[45]

Burgarth, P

D. Burgarth, P. Facchi, R. Hillier, and M. Ligab` o, 14 Quantum 8, 1262 (2024)

2024
[46]

Richter, D

L. Richter, D. Burgarth, and D. Lon- igro, arXiv preprint arXiv:2410.18694 https://doi.org/10.48550/arXiv.2410.18694 (2024)

2024 doi arXiv
[47]

H. A. Rabitz, M. M. Hsieh, and C. M. Rosenthal, Science 303, 1998 (2004)

2004
[48]

F¨ osel, P

T. F¨ osel, P. Tighineanu, T. Weiss, and F. Mar- quardt, Physical Review X 8, 031084 (2018)

2018
[49]

Youssry, G

A. Youssry, G. A. Paz-Silva, and C. Ferrie, New Journal of Physics 25, 073004 (2023)

2023
[50]

Youssry and H

A. Youssry and H. I. Nurdin, Quantum Science and Technology 8, 015018 (2022)

2022
[51]

A. Auza, A. Youssry, G. Paz-Silva, and A. Peruzzo, arXiv preprint arXiv:2404.19251 https://doi.org/10.48550/arXiv.2404.19251 (2024)

2024 doi arXiv
[52]

Pira and C

L. Pira and C. Ferrie, Quantum Machine Intelli- gence 6, 52 (2024)

2024
[53]

L. Pira, A. Antony, N. Prathap, D. Peace, and J. Romero, arXiv preprint arXiv:2505.09266 https://doi.org/10.48550/arXiv.2505.0926 (2025)

2025 doi arXiv
[54]

Ran and G

S.-J. Ran and G. Su, Intelligent Computing 2, 0061 (2023)

2023
[55]

Power and K

L. Power and K. Guha, arXiv preprint arXiv:2405.08917 https://doi.org/10.48550/arXiv.2405.08917 (2024)

2024 doi arXiv
[56]

W. J. Murdoch, C. Singh, K. Kumbier, R. Abbasi- Asl, and B. Yu, Proceedings of the National Academy of Sciences 116, 22071 (2019)

2019
[57]

Ferry, G

J. Ferry, G. Laberge, and U. A ¨ ıvodji, arXiv preprint arXiv:2303.04437 (2023)

2023 arXiv
[58]

Wang and Q

T. Wang and Q. Lin, Journal of Machine Learning Research 22, 1 (2021)

2021
[59]

Baier, Z

A. Baier, Z. Boukhers, and S. Staab, arXiv preprint arXiv:2103.06727 https://doi.org/10.48550/arXiv.2103.06727 (2021)

2021 doi arXiv
[60]

Youssry, G

A. Youssry, G. A. Paz-Silva, and C. Ferrie, npj Quantum Information 6, 95 (2020)

2020
[61]

G. A. Paz-Silva, L. M. Norris, and L. Viola, Phys. Rev. A 95, 022121 (2017)

2017
[62]

M. A. Nielsen and I. L. Chuang, Quantum computa- tion and quantum information (Cambridge univer- sity press, 2010)

2010
[63]

M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, Phys. Rev. Lett. 73, 58 (1994)

1994
[64]

Youssry and A

A. Youssry and A. Peruzzo, Quantum Science and Technology 10.1088/2058-9565/add802 (2025)

2025 doi
[65]

Quiroz, P

G. Quiroz, P. Titum, P. Lotshaw, P. Lougovski, K. Schultz, E. Dumitrescu, and I. Hen, Physical Re- view Research 7, 023240 (2025)

2025
[66]

Theis, F

L. Theis, F. Motzoi, and F. Wilhelm, Physical Re- view A 93, 012324 (2016)

2016
[67]

J. M. Martinis and M. R. Geller, Phys. Rev. A 90, 022307 (2014)

2014
[68]

Krantz, M

P. Krantz, M. Kjaergaard, F. Yan, T. P. Orlando, S. Gustavsson, and W. D. Oliver, Applied physics reviews 6, 10.1063/1.5089550 (2019)

2019 doi
[69]

G. B. Team, Tensorﬂow: An end-to-end open-source machine learning platform (2024), accessed: 2025- 03-27

2024
[70]

shift” matrix) and Σ 3 (the “clock

G. A. ´Alvarez and D. Suter, Phys. Rev. Lett. 107, 230501 (2011). Supplementary Materials for Quantum Engineering of Qudits with Interpretable Machine Learning Yule Mayevsky,1 Akram Youssry, 1 Ritik Sareen, 1 Gerardo A. Paz-Silva, 2 and Alberto Peruzzo 1, 3 1Quantum Photonics ...

2011

Pith tools

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

[1] [1]

In this paper we propose a new parameter- ization that can generalize to arbitrary system of dimension d

Parameterization of the noise operators The proposal in [37] is limited to single-qubit sys- tems. In this paper we propose a new parameter- ization that can generalize to arbitrary system of dimension d. The noise operator of an observable O can be expressed as [37] VO = O−1WO. (7) The operator WO has to satisfy the constraints, that is Hermitian, with b...

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[2] [2]

The eigenvalues ¼i of the operator WO satisfy that |¼i| f d, where d is the dimensionality of the Hilbert space

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[3] [3]

To satisfy those constraints we can use the eigende- composition of WO operator, WO = QDQ

|tr(WO)| f d =⇒ ∑ d i=1 ¼i f d. To satisfy those constraints we can use the eigende- composition of WO operator, WO = QDQ . (8) The matrix Q is a general d × d unitary. This can be parametrized using 2 × 2 subunitary decomposition [39]. This decomposition has a lot of applications in optics [40, 41], and the idea behind it is that we can decompose the gen...

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[4] [4]

Handling non-invertible observables A non-invertible observable O will result in an un- deﬁned VO operator. However, this can be addressed by applying a linear transformation with parameters a, b to the observable O yielding, ˜O = aO + bId×d, (16) The parameter b shifts the eigenvalues of O such that ˜O becomes invertible [42]. Speciﬁcally, b is chosen to...

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[5] [5]

Two commonly used bases for this purpose are the clock-shift basis and the Gell-Mann basis, both of which generalize the Pauli matrices to higher-dimensional systems

Choice of basis The graybox framework requires an informationally-complete basis for accurately estimating then noise operators. Two commonly used bases for this purpose are the clock-shift basis and the Gell-Mann basis, both of which generalize the Pauli matrices to higher-dimensional systems. The Gell-Mann basis provides a traceless, Hermitian represent...

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[6] [6]

Symmetric Gell-Mann Matrices: ¼(S) jk = (|jð ïk| + |kð ïj|) , 1 f j f k f d

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[7] [7]

Anti-symmetric Gell-Mann Matrices: ¼(A) jk = −i (|jð ïk| − |kð ïj|) , 1 f j f k f d

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[8] [8]

Diagonal Gell-Mann Matrices: ¼l = √ 2 l(l + 1) ( l∑ m=1 |mð ïm| − l |l + 1ð ïl + 1| ) , l = 1, . . . , d− 1. Together with the identity operator IN , these ma- trices form a complete Hermitian basis for all d × d operators. The scaling of all observables in this case grows as O(d5), (there are ( d2 − 1) basis, each has d eigenstates, and we have d2 measur...

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