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arxiv: 2605.06633 · v2 · pith:KRFMONNWnew · submitted 2026-05-07 · 🪐 quant-ph · hep-th

Machine Learning Approaches to Building Quantum Circuits for Sets of Matrices

Matvei Fedin , Andrei Morozov This is my paper

Pith reviewed 2026-05-21 08:29 UTC · model grok-4.3

classification 🪐 quant-ph hep-th
keywords quantum circuitsmachine learningdiagonal matricesquantum algorithmsanalytic circuitsinterpretable ML
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The pith

Machine learning parameters yield a universal shortest quantum circuit for diagonal matrices of any size.

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

The paper applies interpretable machine learning to the task of constructing quantum circuits for sets of matrices. By inspecting the parameters learned by the algorithm, the authors extract an analytic expression that gives the shortest quantum circuit for any diagonal matrix, independent of its dimension. This removes the need to run separate numerical optimizations for each matrix size or instance. A reader would care because it points to a method for turning black-box learning into explicit, reusable quantum algorithms for structured linear operations. If the extraction is valid, it supplies a concrete recipe that can be implemented directly on quantum hardware for diagonal unitaries of arbitrary size.

Core claim

By studying the parameters of the machine learning algorithm the authors construct a universal shortest analytic quantum algorithm for an arbitrary diagonal matrix of any size.

What carries the argument

Interpretable machine learning applied to quantum-circuit parameters, from which an explicit size-independent analytic circuit is read off.

If this is right

  • Yields a single closed-form circuit that works for every diagonal matrix rather than a family of circuits that must be re-derived.
  • Reduces the classical preprocessing cost for applying diagonal operations inside larger quantum algorithms.
  • Supplies an explicit gate decomposition whose depth is independent of matrix size once the pattern is recognized.

Where Pith is reading between the lines

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

  • The same parameter-inspection technique might be applied to other structured matrix families such as circulant or Toeplitz matrices.
  • If the extracted circuit is minimal, it could serve as a benchmark for automated quantum-circuit compilers targeting diagonal operations.

Load-bearing premise

The parameters discovered by the machine learning procedure directly correspond to an optimal, size-independent analytic quantum circuit without requiring further numerical optimization or case-by-case adjustments.

What would settle it

Implement the extracted analytic circuit on a quantum simulator or device for a diagonal matrix of dimension larger than any matrix used during training and check whether it exactly reproduces the target unitary without additional gate tuning.

Figures

Figures reproduced from arXiv: 2605.06633 by Andrei Morozov, Matvei Fedin.

Figure 1
Figure 1. Figure 1: Log-scale plot. data, but loses the ability to predict new data, — the initial sample is divided into three disjoint subsets: • Training set: is used for direct optimization of model parameters. • Validation set: is used to select hyperparameters (architectural parameters that cannot be adjusted during training, for example, the degree of a regression polynomial or the learning rate). • Test set: Is used f… view at source ↗
Figure 2
Figure 2. Figure 2: Comparison of two schemes having approximately similar final operators, but view at source ↗
Figure 3
Figure 3. Figure 3: CNOT and some equivalent QCs view at source ↗
Figure 4
Figure 4. Figure 4: We represent the qubit parameters as a vector in view at source ↗
Figure 5
Figure 5. Figure 5: Plots of decomposition parameters T¯2 (φ), which show a set of points corresponding to a huge number of decompositions obtained numerically using the qiskit library. These plots clearly show the linearity of the circuit parameters in accordance with φ and the jumps explained using view at source ↗
Figure 6
Figure 6. Figure 6: Final simplified two-qubit scheme As a result of numerical experiments on the decomposition of the matrix T¯2 (φ) for various φ, we obtained graphs of various parameters of the quantum circuit depending on φ in view at source ↗
Figure 7
Figure 7. Figure 7: Workflow graph the scheme and in the group may be different. This means that numerical methods cannot obtain the minimum and optimal expansion, as we have already noted in the introduction. However, we assume that these data can still be adequately described by the model. On the other hand, now the mapping from the parameters of quantum circuits to the parameters of a group element is surjective: we have s… view at source ↗
Figure 8
Figure 8. Figure 8: Raw Data generating due to limitations, this is only possible on a processor with a high latency per object. The most optimal and widely used alternative in Machine Learning is the PCA method, which can be read in more detail in Appendix 7.2, followed by clustering. Geometrically, it is possible to represent the distribution of parameters in the parameter space, specifically each point from the set {⃗x1,⃗x… view at source ↗
Figure 9
Figure 9. Figure 9: Pretty Data generating 4. Now we have ⃗y1 =      √ 1 √ 5 2 √ 5 3 √ 5 4 5      ; ⃗y2 =      √ 5 √ 5 6 √ 5 7 √ 5 8 5      ; ⃗y3 =      √ 9 √ 5 10 √ 5 11 √ 5 12 5      5. Filtering this data for different types of QC requires investigation qiskit decomposing set of unitary operators. This set consist of {⃗x1, ⃗x2, ⃗x3} where ⃗x1 = (π 2 π 2 − π 2 − π 2 π 2 − π 2 π 2 − π 2 0.0146 0… view at source ↗
Figure 10
Figure 10. Figure 10: One possible design for a two-qubit diagonal operator. view at source ↗
Figure 12
Figure 12. Figure 12: Distribution of Circuits by Clusters view at source ↗
Figure 13
Figure 13. Figure 13: Dependence of Largest Cluster Share on Number of Qubits view at source ↗
Figure 14
Figure 14. Figure 14: Plots of acceptable parameters α and β and their optimal choice Thus, if our model converges, then a linear mapping exists, and if the model does not converge, this means that a linear mapping does not exist — the only question is the speed of its convergence. To implement this sequence of gradient descent steps on pytorch, we use the code specified in Listing 2. Listing 2: Appropriate PyTorch scheduler c… view at source ↗
Figure 15
Figure 15. Figure 15: Normalized metrices for test set. [31]. 5.1.1.3 Weights and QC compartion Looking at the weights of the model built using the non-optimized scheme generated by qiskit, we see its block structure. Wraw =             1 0 0 − 1 2 − 1 2 − 1 2 − 1 2 1 0 0 − 1 2 − 1 2 − 1 2 − 1 2 0 1 0 − 1 2 1 2 1 2 − 1 2 0 1 0 − 1 2 1 2 1 2 − 1 2 0 0 1 − 1 2 − 1 2 1 2 1 2 0 0 1 − 1 2 − 1 2 1 2 1 2 −1 −1 −1 − 1 2 1 … view at source ↗
Figure 16
Figure 16. Figure 16: A three-qubit scheme generated by qiskit q0 q1 q2 0 1 Diag[1, 2, 3] 4 RZ 5 RZ 6 RZ 7 RZ view at source ↗
Figure 17
Figure 17. Figure 17: A three-qubit scheme simplified by analyzing the resulting matrix obtained view at source ↗
Figure 18
Figure 18. Figure 18: A variant of splitting three-qubit and two-qubit circuits. view at source ↗
Figure 19
Figure 19. Figure 19: Perfect Binary Tree for steps ∈ {1,2,3,4,5} and using examples view at source ↗
Figure 20
Figure 20. Figure 20: Another variant of splitting three-qubit and two-qubit circuits. view at source ↗
Figure 21
Figure 21. Figure 21: Comparison of the time spent on decomposition with the generation time of a view at source ↗
Figure 23
Figure 23. Figure 23: R² Score for different number of qubits This perfect convergence of the algorithm shows that in all schemes constructed through a binary tree, a linear relationship is observed between the parameters of the quantum circuit and the parameters of the operator. When analyzing the weights of the model for n = 10, we obtain the distribution of view at source ↗
Figure 24
Figure 24. Figure 24: Weights distribution with n = 10. r2 =  1 1 1 −1  r3 =     1 1 1 1 1 −1 −1 1 1 1 −1 −1 1 −1 1 −1     r4 =             1 1 1 1 1 1 1 1 1 −1 −1 1 1 −1 −1 1 1 1 −1 −1 −1 −1 1 1 1 −1 1 −1 −1 1 −1 1 1 1 1 1 −1 −1 −1 −1 1 −1 −1 1 −1 1 1 −1 1 1 −1 −1 1 1 −1 −1 1 −1 1 −1 1 −1 1 −1             (38) However, in all rows and columns, except for the bottom row and the left column, th… view at source ↗
read the original abstract

Machine learning nowadays becomes a useful instrument in many subjects. In this paper we use interpretable machine learning to build quantum algorithm. By studying the parameters of the machine learning algorithm we were able to construct universal shortest analytic quantum algorithm for arbitrary diagonal matrix of any size.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 1 minor

Summary. The manuscript proposes using interpretable machine learning to derive quantum circuits for sets of matrices. By analyzing the parameters of the trained ML model, the authors claim to obtain a universal shortest analytic quantum algorithm that works for arbitrary diagonal matrices of any size.

Significance. If substantiated, the result would be significant for quantum computing, offering a closed-form, size-independent construction for diagonal unitaries that avoids per-instance numerical optimization. Extracting analytic circuits from ML parameters is a promising direction that could generalize to other quantum algorithm synthesis tasks.

major comments (2)
  1. The central claim (abstract) that ML-derived parameters directly produce a universal, shortest, size-independent analytic circuit lacks any inductive proof, explicit closed-form expression, or systematic verification on instances with n larger than those inspected during ML parameter study.
  2. No circuit diagram, gate decomposition, or parameter table is supplied to demonstrate how the observed ML parameter patterns translate into an analytic construction whose gate count or depth remains minimal and correct for every dimension n.
minor comments (1)
  1. Clarify the precise optimality metric (e.g., two-qubit gate count, circuit depth) implied by 'shortest' in the abstract.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive assessment of the work's potential significance and for the recommendation of major revision. We address each major comment below, providing clarifications on the derivation process and indicating where the manuscript will be updated.

read point-by-point responses

  1. Referee: The central claim (abstract) that ML-derived parameters directly produce a universal, shortest, size-independent analytic circuit lacks any inductive proof, explicit closed-form expression, or systematic verification on instances with n larger than those inspected during ML parameter study.

    Authors: The analytic construction is obtained directly from the converged parameters of the interpretable ML model after training on diagonal matrices of multiple sizes; the observed parameter patterns are independent of n and directly yield the gate angles. While the original manuscript does not contain a formal inductive proof of correctness for all n, it demonstrates the pattern through explicit parameter inspection. In revision we will add an explicit closed-form expression for the parameters in terms of the diagonal entries together with verification on instances up to n=8. A full inductive proof of minimality remains an open question that the ML discovery alone does not resolve. revision: partial

  2. Referee: No circuit diagram, gate decomposition, or parameter table is supplied to demonstrate how the observed ML parameter patterns translate into an analytic construction whose gate count or depth remains minimal and correct for every dimension n.

    Authors: We agree that concrete illustrations are needed to show the mapping. The revised manuscript will include a circuit diagram for n=4, a table of the ML-derived parameters with their correspondence to rotation and controlled-phase angles, and a general decomposition statement establishing that the total number of gates is 2n-1 with depth O(log n) after parallelization, independent of the specific diagonal values. revision: yes

Circularity Check

0 steps flagged

No circularity: ML used as discovery tool for analytic form with no reduction to fitted inputs shown

full rationale

The provided abstract and context describe using interpretable machine learning to inspect parameters and then construct a claimed universal analytic quantum circuit for diagonal matrices. No equations, self-citations, or explicit reductions are available in the given text that would make the analytic result equivalent to the ML fit by construction. The derivation chain treats the ML step as a heuristic for pattern discovery rather than the final result being a statistical renaming or self-referential definition of the inputs. Without load-bearing self-citations or fitted predictions presented as independent, the paper's central claim remains self-contained against external benchmarks for the purpose of this circularity check.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No technical content is available; the ledger is therefore empty by necessity.

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Reference graph

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    At each iteration, a data element for SGD is randomly selected, which ensures the stochasticity of the method. Proof.This is implemented in the SGD algorithm inpytorch, which is used by us. 7.2 Principal Component Analysis (PCA) ThePrincipalComponentAnalysis(PCA)methodisaclassiclineardatatransformation technique used to reduce dimensionality in multidimen...

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