Quantum Circuit Design using Complex valued Neural Network in Stiefel Manifold

Mahesh Mohan M R; Sayan Manna

REVIEW 4 major objections 4 minor 19 references

The paper claims that a single-layer complex-valued neural network trained on the Stiefel manifold can learn the unitary of a quantum circuit and reach fidelity 1 on the tested 2-qubit and 5-qubit examples.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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A single-layer complex-valued neural network constrained to the Stiefel manifold via Cayley updates is used to learn and transpile quantum circuit unitaries, with fidelity reaching 1 on reported toy examples.

T0 review reviewed 2026-08-05 challenge →

load-bearing objection Standard optimizer applied to a new domain, with a local-decrease proof overclaimed as global and an experiment too underspecified to support the central claim. the 4 major comments →

arxiv 2509.02374 v1 pith:4TB3FQFO submitted 2025-09-02 quant-ph

Quantum Circuit Design using Complex valued Neural Network in Stiefel Manifold

Sayan Manna , Mahesh Mohan M R This is my paper

classification quant-ph MSC 81P6868T07 PACS 03.67.Lx

keywords quantum circuit designcomplex-valued neural networkStiefel manifoldCayley transformunitary matrixquantum state fidelitycircuit transpilationquantum machine learning

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

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 is trying to show that a single-layer complex-valued neural network, trained on pairs of input and output quantum states, can recover the unitary transformation of a quantum circuit while keeping the weight matrix unitary at every step. Training is constrained to the Stiefel manifold of unitary matrices, and each update uses a Cayley transform of a skew-Hermitian direction. The authors report that this gives a globally decreasing loss, avoids the loss spikes of periodic Gram-Schmidt re-orthogonalization, and yields a learned unitary with fidelity 1 on the tested examples. If the claim transfers to larger circuits, circuit synthesis could be done by supplying input-output states and letting the network find the unitary.

Core claim

The paper's central claim is that solving the constrained problem min f(W) subject to W†W=I by gradient descent on the Stiefel manifold preserves unitarity throughout training and produces a learned unitary that matches the target circuit. With W as the network weight matrix and G the Euclidean gradient, the paper constructs A=GW†−WG†, which is skew-Hermitian, and updates W(k+1)=(I+λA/2)^{-1}(I−λA/2)W(k). The appendix proves that each update stays on the Stiefel manifold and that, for small enough λ, the loss strictly decreases unless the Riemannian gradient vanishes. On a 5-qubit sample circuit the unitary error ‖WW†−I‖² reaches about 10⁻¹¹, and the fidelity F=(1/d)|Tr(U_true† U_learned)| r

What carries the argument

The central object is the Stiefel manifold V_N(C^N)={W∈C^{N×N}: W†W=I}, the set of unitary matrices. The carrying mechanism is the Cayley update W(k+1)=(I+λA/2)^{-1}(I−λA/2)W(k) with A=GW†−WG†, where G is the Euclidean gradient of the loss. Because A is skew-Hermitian, the Cayley factor is unitary and maps one unitary matrix to another; the proof shows the curve's tangent at λ=0 is −AW and the directional derivative of f is −‖AW‖²_c, so the loss descends along the manifold. This is what guarantees unitarity at every step rather than only after intermittent re-orthogonalization.

Load-bearing premise

The paper assumes the input-output state pairs used for training uniquely determine the target unitary, so that reaching fidelity 1 on those pairs means the learned circuit is the target circuit.

What would settle it

Take a fixed 2-qubit circuit U_true and provide only one state pair, say |00⟩→U_true|00⟩, to the network. Train until fidelity F=1 on that pair, then compute F on the held-out basis states |01⟩, |10⟩, and |11⟩. If the held-out fidelity is measurably below 1 because many unitaries agree on |00⟩, the paper's claim that F=1 implies U_learned=U_true fails in that setting.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

Because every updated weight matrix is unitary, the learned matrix is always physically valid as a quantum gate and can be handed to a transpiler to produce an executable circuit.
The loss decreases monotonically for sufficiently small learning rates, so training avoids the loss spikes that intermittent Gram-Schmidt orthogonalization introduces.
If the fidelity result generalizes, a user would only need known input-output quantum state pairs to obtain a circuit, bypassing manual gate decomposition.
The same update rule could be applied to any complex-valued neural network that needs a unitary weight matrix, not just quantum circuit synthesis.

Where Pith is reading between the lines

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

The paper leaves open how many input-output state pairs are used and does not test on held-out states; the claim that reaching fidelity 1 means reproducing the target circuit depends on the training set uniquely determining the unitary, which is not demonstrated.
A natural extension is to train on a single basis-state pair for a known random unitary and then evaluate fidelity on the remaining basis states; if many unitaries fit the one pair, the learned matrix will not match the target on the held-out states.
The method could be applied to learning unknown processes: given experimentally measured input-output state pairs, the same Stiefel-constrained optimization would return the unitary closest to the data, with unitarity acting as a built-in regularizer.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

Standard optimizer applied to a new domain, with a local-decrease proof overclaimed as global and an experiment too underspecified to support the central claim.

read the letter

Colleague,

This paper applies the Cayley-transform update on the Stiefel manifold—originally from Wisdom et al.'s unitary RNN work—to quantum circuit synthesis. The mathematical core is standard and, for the most part, correctly presented. The unitarity-preservation proof (Claim 1) is clean. And the idea of avoiding the intermittent re-orthogonalization of Gram-Schmidt by staying on the manifold throughout training is sensible and worth a sentence of credit.

The problems are in the claims and the experiments. Section 4 says the loss 'decreases globally,' but the appendix proof only establishes a decrease for sufficiently small learning rate—that is a local, not global, statement. Overclaiming to 'global' needs correcting. More important, the empirical section is underspecified: we don't know how many input-output pairs were used, what states they were, what loss was minimized, or whether any held-out states were evaluated. The fidelity (Eq. 5) is reported between U_learned and U_true, but without saying whether the training set uniquely determines the unitary, F=1 on training data is just convergence of a fit. The 2-qubit appendix example adds little: it is the same procedure without additional details. No code or data is provided, so nothing can be reproduced.

There is also a citation problem: reference [1] is 'Anonymous' and not verifiable, and the literature review is thin.

In short, the approach is reasonable but the paper does not make a convincing case that the learned unitary implements the target circuit. The core idea warrants a revision that states the training set size, the loss function, and a train/test split, and that corrects the 'global decrease' claim. As it stands, I'd desk-reject; a revised version with these details might deserve referee time.

Hope this helps.

Referee Report

4 major / 4 minor

Summary. The paper proposes learning a quantum circuit's unitary transformation by training a single-layer complex-valued neural network whose weight matrix is constrained to the Stiefel manifold (i.e., unitary) via the Cayley update of Eq. (4). The authors claim this update preserves unitarity at every step and guarantees a globally decreasing loss, and they report numerical experiments on a 5-qubit circuit and a 2-qubit entangling operation, with training loss decreasing and fidelity (Eq. 5) reaching 1, after which the learned unitary is transpiled into a quantum circuit. The appendix contains proofs of unitarity preservation and of a local decrease property for sufficiently small learning rates.

Significance. If fully validated, the proposal would be a simple baseline for learning unitary maps from input--output quantum state pairs, with the useful property that the learned weight matrix is exactly unitary throughout training. The mathematical appendix correctly proves the key feasibility of the Cayley/Stiefel update: the update (A7) stays on the Stiefel manifold, and the directional derivative at λ=0 is non-positive. The method also has the practical advantage of being directly expressible in existing transpilation toolchains. However, the central empirical claim—that the learned Wfinal implements the target quantum circuit—is not established by the reported experiments, and the claimed global monotone decrease is stronger than what the proof supports. These issues are load-bearing for the paper's main contribution.

major comments (4)

[§4 and Appendix, Claim 2] The paper states as a guaranteed property that "the loss function decreases globally, i.e., f(W(k+1)) ≤ f(W(k))" for the update (4). The appendix proof, Eqs. (A12)–(A15), only shows that if the Riemannian gradient AW is nonzero, then for sufficiently small λ one has f(Y(λ)) < f(W). At a stationary point (AW=0) the derivative is zero and no decrease is proved. Moreover, no rule is given for choosing λ that satisfies the bound (A15). The monotone-decrease claim used to motivate the method is therefore not established; please replace it with the local statement and provide a concrete step-size procedure or line search.
[§2, §5, Fig. 2] The empirical claim that the trained Wfinal reproduces the target unitary Utrue is not supported by the experiments as reported. A training set of k pairs (|ψ_i>, Utrue|ψ_i>) determines a unitary only on the span of the input states; for k < 2^n or for linearly dependent inputs, infinitely many unitaries fit the training data. Section 2 only says "many input and output quantum states are known," and Section 5 reports a 5-qubit experiment without stating the number of training pairs, their distribution, or their linear independence. Figure 2b shows fidelity reaching 1, but no held-out fidelity on states not used in training is reported. Please specify the training data and add held-out evaluation before the fidelity claim can be assessed.
[§5, Eqs. (1)–(4)] The loss f(W) and its Euclidean gradient G are never concretely specified for the reported experiments. Equation (1) defines the constrained optimization abstractly, but the experiments require a concrete loss—mean squared error over output states, trace fidelity, or another—to compute G in Eq. (3) and the update (4). The paper also omits the learning-rate schedule, initialization, number of epochs, and the exact 5-qubit target circuit. Without these details, the numerical results are not reproducible and the fidelity curves cannot be interpreted.
[Appendix, 'Additional Result'] The 2-qubit example in the appendix is also under-specified: the target entangling operation is not named, the training data are not described, and Figure 4 shows a transpiled circuit without any comparison to the target or a correctness metric beyond the stated unitary error. This example should either be integrated into a fully specified experimental section or removed.

minor comments (4)

[References] Reference [1] is listed as "Anonymous" and reference [6] contains "and 1 others"; these are incomplete citations and should be corrected.
[Throughout] There are numerous typos: "varius," "ouput," "Caley" (should be Cayley), "uniatry," and "Steifel" in figure captions. Please proofread carefully.
[§5, Fig. 2] The axes in Figures 2a and 2b are not fully described. In particular, the fidelity plot should state whether the fidelity is evaluated on the training data or on a separate test set, and the loss plot should state which loss function is displayed.
[§1, Literature Study] The claim that this is "one of the first applications of Cayley-transform-based manifold optimization in quantum computing" is vague, especially given reference [17] on unitary neural networks for quantum circuit design. Please clarify the precise novelty with a more systematic comparison to prior work.

Circularity Check

0 steps flagged

No construction-level circularity; the Cayley/Stiefel derivation is self-contained and the reported F=1 is an in-sample training metric, not a circular reduction.

full rationale

The paper's central derivation is the Cayley-transform update on the Stiefel manifold (Eq. 4) with the proof in the Appendix that W remains unitary (Claim 1) and that the loss decreases for small learning rates (Claim 2). These proofs are self-contained and rely only on standard Riemannian optimization facts; they do not depend on the paper's conclusions and there are no self-citations used as load-bearing premises. The fidelity F=1 reported in Fig. 2b is computed between the learned unitary U_learned (the optimized weight matrix W) and the target unitary U_true (Eq. 5). This is an in-sample training evaluation, not an out-of-sample prediction: the paper does not claim to predict held-out states, and it does not present F as an independent confirmation. The main weakness is that Sections 2 and 5 do not specify the number of training input-output pairs, the loss function, or a train/test split, so the identifiability of the target unitary from the training data is not demonstrated. This is a correctness/validation gap, not a circular reduction: no equation equates F to the training objective by construction, and no load-bearing step reduces to its own inputs. Therefore no significant circularity is identified.

Axiom & Free-Parameter Ledger

2 free parameters · 4 axioms · 0 invented entities

The paper's contribution rests on standard Stiefel-manifold mathematics plus an unverified data-sufficiency assumption. The only explicit free parameter is the learning rate; the initialization and training set are unspecified.

free parameters (2)

Learning rate lambda = unspecified (positive, sufficiently small)
The convergence proof requires lambda small enough; experiments do not report the value. This is a hand-chosen hyperparameter.
Initial unitary W(0) = random, seed unspecified
Results depend on random initialization; no seed or distribution is given.

axioms (4)

standard math The loss f(W) is a smooth real-valued function of the complex weight matrix with a well-defined Euclidean gradient.
Invoked in Methodology and Appendix to define G and the Taylor expansion; not stated as an assumption.
domain assumption A = G W dagger - W G dagger gives the Riemannian gradient direction under the canonical inner product (A3).
Stated in Appendix without full proof; standard for Stiefel manifolds but is an unproved background result in this paper.
ad hoc to paper The input-output state pairs used in training are sufficient to identify the target unitary U.
The paper assumes 'many input and output quantum states are known' but never specifies the number or whether F=1 on training states implies Ulearned=Utrue. This is load-bearing for the claim that the learned matrix is the target circuit.
domain assumption Qiskit's transpile function can convert the learned unitary into an executable quantum circuit.
Used in Sections 4 and 5 without discussion of decomposition overhead or hardware constraints.

reviewed 2026-08-05 · how reviews work

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Cite this review

Pith. "Pith review of Quantum Circuit Design using Complex valued Neural Network in Stiefel Manifold." pith.science (2026). https://pith.science/paper/4TB3FQFO

@misc{pith2026250902374,
  author       = {Pith},
  title        = {Pith review of: Quantum Circuit Design using Complex valued Neural Network in Stiefel Manifold},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4TB3FQFO}},
  note         = {Machine review of arXiv:2509.02374}
}

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read the original abstract

Quantum algorithms operate on quantum states through unitary transformations in high dimensional complex Hilbert space. In this work, we propose a machine learning approach to create the quantum circuit using a single-layer complex-valued neural network. The input and ouput quantum states are provided to the network, which is trained to approximate the output state of a given quantum algorithm. To ensure that the fundamental property of unitarity is preserved throughout the training process, we employ optimization in Stiefel Manifold.

Figures

Figures reproduced from arXiv: 2509.02374 by Mahesh Mohan M R, Sayan Manna.

**Figure 1.** Figure 1: Sample circuit and Gram-Schmidt loss As shown in Fig. 1b, the training loss increases whenever the Gram-Schmidt orthogonalization constraint is being applied to the weight matrix. Here unitarity is not being maintained during training. (a) Loss vs Epochs using optimization on Steifel manifold (b) Fidelity vs iteration on Stiefel manifold [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

**Figure 2.** Figure 2: Training loss and Fidelity According to our approach as shown in Fig. 2a the loss is decreasing when Eq.(4) is being used. Throughout the training, the unitary error is observed to be of the order [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗

**Figure 3.** Figure 3: Training loss and Fidelity for a 2-qubit quan [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

**Figure 4.** Figure 4: 2-qubit optimized quantum circuit after tran [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

discussion (0)

Reference graph

Works this paper leans on

19 extracted references · 18 canonical work pages · 4 internal anchors

[1]

Anonymous. 2021. The bitter truth about gate-based quantum algorithms in the nisq era. Quantum In- formation Processing (IOP) . Discusses how lim- ited qubit connectivity forces additional gates and increases circuit depth

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Martin Arjovsky, Amar Shah, and Yoshua Bengio

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Bennett, Richard Cleve, David P

Adriano Barenco, Charles H. Bennett, Richard Cleve, David P. DiVincenzo, Norman Margolus, Peter Shor, Tycho Sleator, John A. Smolin, and Harald Wein- furter. 1995. Elementary gates for quantum computa- tion. Physical Review A, 52(5):3457

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Olivier Coulaud, Luc Giraud, and Martina Iannacito. 2022. On some orthogonalization schemes in tensor train format. arXiv preprint arXiv:2211.08770. Available at https://arxiv. org/abs/2211.08770

work page internal anchor Pith review Pith/arXiv arXiv 2022
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work page 1992
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Houssein, Zainab Abohashima, Mohamed Elhoseny, and Waleed M

Essam H. Houssein, Zainab Abohashima, Mohamed Elhoseny, and Waleed M. Mohamed. 2022. Machine learning in the quantum realm: The state-of-the-art, challenges, and future vision. Expert Systems with Applications, 194:116512

work page 2022
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work page internal anchor Pith review Pith/arXiv arXiv 2016
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Nielsen and Isaac L

Michael A. Nielsen and Isaac L. Chuang. 2002. Quantum Computation and Quantum Information . Cambridge University Press

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work page 2025
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Rohit Sarma Sarkar and Bibhas Adhikari. 2024. A quantum neural network framework for scalable quantum circuit approximation of unitary matrices. Accessible at arXiv:2405.00012

work page internal anchor Pith review Pith/arXiv arXiv 2024
[12]

Shende, Stephen S

Vivek V . Shende, Stephen S. Bullock, and Igor L. Markov. 2006. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 25(6):1000–1010

work page 2006
[13]

Peter W. Shor. 1994. Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. In Proceedings of the 35th Annual Symposium on F oundations of Computer Sci- ence, pages 124–134. IEEE

work page 1994
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Hemant D. Tagare. 2011. Notes on optimization on stiefel manifolds. Technical report, Yale University

work page 2011
[15]

Kwok Ho Wan, Oscar Dahlsten, Hlér Kristjánsson, Robert Gardner, and M. S. Kim. 2016. Quantum generalisation of feedforward neural networks. arXiv preprint arXiv:1612.01045

work page internal anchor Pith review Pith/arXiv arXiv 2016
[16]

Hershey, Jonathan Le Roux, and Les Atlas

Scott Wisdom, Thomas Powers, John R. Hershey, Jonathan Le Roux, and Les Atlas. 2016. Full- capacity unitary recurrent neural networks. In Ad- vances in Neural Information Processing Systems , pages 4880–4888

work page 2016
[17]

Zomorodi, H

M. Zomorodi, H. Amini, M. Abbaszadeh, J. Sohrabi, V . Salari, and P. Plawiak. 2024. Optimal quantum circuit design via unitary neural networks. arXiv preprint arXiv:2408.13211. Appendix Optimization in Stiefel Manifold We consider the Stiefel manifold of all N × N complex-valued matrices whose columns are N orthonormal vectors in CN. It is defined as VN (...

Pith/arXiv arXiv 2024
[2016]

In Proceedings of the 33rd International Conference on Machine Learning (ICML) , pages 1120–1128

Unitary evolution recurrent neural networks. In Proceedings of the 33rd International Conference on Machine Learning (ICML) , pages 1120–1128

work page
[2024]

Preprint (arXiv)

Quantum circuit synthesis and compilation optimization: Overview and prospects. Preprint (arXiv). Provides a survey on translating algorithms into optimized gate sequences under NISQ con- straints

work page

This paper was first reviewed by deepseek-v4-flash on August 5, 2026.

[1] [1]

Anonymous. 2021. The bitter truth about gate-based quantum algorithms in the nisq era. Quantum In- formation Processing (IOP) . Discusses how lim- ited qubit connectivity forces additional gates and increases circuit depth

work page 2021

[2] [2]

Martin Arjovsky, Amar Shah, and Yoshua Bengio

work page

[3] [3]

Bennett, Richard Cleve, David P

Adriano Barenco, Charles H. Bennett, Richard Cleve, David P. DiVincenzo, Norman Margolus, Peter Shor, Tycho Sleator, John A. Smolin, and Harald Wein- furter. 1995. Elementary gates for quantum computa- tion. Physical Review A, 52(5):3457

work page 1995

[4] [4]

Olivier Coulaud, Luc Giraud, and Martina Iannacito. 2022. On some orthogonalization schemes in tensor train format. arXiv preprint arXiv:2211.08770. Available at https://arxiv. org/abs/2211.08770

work page internal anchor Pith review Pith/arXiv arXiv 2022

[5] [5]

David Deutsch and Richard Jozsa. 1992. Rapid so- lution of problems by quantum computation. Pro- ceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences, 439(1907):553– 558

work page 1992

[6] [6]

Yan Ge, Wu Wenjie, Chen Yuheng, and 1 others

work page

[7] [7]

Houssein, Zainab Abohashima, Mohamed Elhoseny, and Waleed M

Essam H. Houssein, Zainab Abohashima, Mohamed Elhoseny, and Waleed M. Mohamed. 2022. Machine learning in the quantum realm: The state-of-the-art, challenges, and future vision. Expert Systems with Applications, 194:116512

work page 2022

[8] [8]

Zakaria Mhammedi, Andrew Hellicar, Ashfaqur Rahman, and James Bailey. 2016. Efficient or- thogonal parametrisation of recurrent neural net- works using householder reflections. arXiv preprint arXiv:1612.00188

work page internal anchor Pith review Pith/arXiv arXiv 2016

[9] [9]

Nielsen and Isaac L

Michael A. Nielsen and Isaac L. Chuang. 2002. Quantum Computation and Quantum Information . Cambridge University Press

work page 2002

[10] [10]

Edward Parker. 2025. Unitary time evolution in quantum mechanics is a stronger physical postulate than linear time evolution. F oundations of Physics, 55:1–13

work page 2025

[11] [11]

Rohit Sarma Sarkar and Bibhas Adhikari. 2024. A quantum neural network framework for scalable quantum circuit approximation of unitary matrices. Accessible at arXiv:2405.00012

work page internal anchor Pith review Pith/arXiv arXiv 2024

[12] [12]

Shende, Stephen S

Vivek V . Shende, Stephen S. Bullock, and Igor L. Markov. 2006. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 25(6):1000–1010

work page 2006

[13] [13]

Peter W. Shor. 1994. Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. In Proceedings of the 35th Annual Symposium on F oundations of Computer Sci- ence, pages 124–134. IEEE

work page 1994

[14] [14]

Hemant D. Tagare. 2011. Notes on optimization on stiefel manifolds. Technical report, Yale University

work page 2011

[15] [15]

Kwok Ho Wan, Oscar Dahlsten, Hlér Kristjánsson, Robert Gardner, and M. S. Kim. 2016. Quantum generalisation of feedforward neural networks. arXiv preprint arXiv:1612.01045

work page internal anchor Pith review Pith/arXiv arXiv 2016

[16] [16]

Hershey, Jonathan Le Roux, and Les Atlas

Scott Wisdom, Thomas Powers, John R. Hershey, Jonathan Le Roux, and Les Atlas. 2016. Full- capacity unitary recurrent neural networks. In Ad- vances in Neural Information Processing Systems , pages 4880–4888

work page 2016

[17] [17]

Zomorodi, H

M. Zomorodi, H. Amini, M. Abbaszadeh, J. Sohrabi, V . Salari, and P. Plawiak. 2024. Optimal quantum circuit design via unitary neural networks. arXiv preprint arXiv:2408.13211. Appendix Optimization in Stiefel Manifold We consider the Stiefel manifold of all N × N complex-valued matrices whose columns are N orthonormal vectors in CN. It is defined as VN (...

Pith/arXiv arXiv 2024

[18] [2016]

In Proceedings of the 33rd International Conference on Machine Learning (ICML) , pages 1120–1128

Unitary evolution recurrent neural networks. In Proceedings of the 33rd International Conference on Machine Learning (ICML) , pages 1120–1128

work page

[19] [2024]

Preprint (arXiv)

Quantum circuit synthesis and compilation optimization: Overview and prospects. Preprint (arXiv). Provides a survey on translating algorithms into optimized gate sequences under NISQ con- straints

work page

Quantum Circuit Design using Complex valued Neural Network in Stiefel Manifold

The reading

Core claim §

What carries the argument

Load-bearing premise §

What would settle it §

If this is right

Where Pith is reading between the lines

discussion (0)

Core claim

Load-bearing premise

What would settle it