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arxiv: 2604.15666 · v1 · submitted 2026-04-17 · 🪐 quant-ph

Explainable quantum regression algorithm with encoded data structure

C.-C. Joseph Wang , F. Perkkola , I. Salmenper\"a , A. Meijer-van de Griend , J. K. Nurminen This is my paper

Pith reviewed 2026-05-10 09:50 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum regressioninterpretable machine learningvariational quantum algorithmshybrid quantum-classicaldata encodingquantum machine learningnoisy intermediate-scale quantum
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The pith

Quantum regression becomes interpretable when the quantum state encodes the data table exactly and variational parameters equal the regression coefficients.

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

The paper builds a quantum regression algorithm in which the quantum state holds the classical data table without loss of information. Variational parameters are constructed to match the regression coefficients directly as real numbers, so users can read off the model weights instead of treating the circuit as a black box. This design keeps the optimization cost low because the ansatz has exactly the needed expressiveness. The encoded structure is further used to cut the number of gates required for the regression map, and nonlinear cases are handled by classical preprocessing of features. Model quality is checked by measuring a cost function that is synchronous with mean squared error, and the authors derive the readout errors and sample complexity for both one-hot and compact encodings on noisy hardware.

Core claim

We construct the first interpretable quantum regression algorithm, in which the quantum state exactly encodes the classical data table and the variational parameters correspond directly to the regression coefficients, which are real numbers by construction, providing a high degree of model interpretability and minimal cost to optimize due to the right expressiveness. We also exploit the encoded data structure to reduce the gate complexity of computing the regression map, and extend the method to nonlinear regression via classical preprocessing of independent encoded column vectors. By design, the model performance is determined by the cost function measurement results synchronous to the mean

What carries the argument

The exact encoding of the classical data table into the quantum state, which forces the variational parameters to be identical to the real-valued regression coefficients.

Load-bearing premise

The quantum state must encode the full classical data table without any information loss and the variational parameters must map directly onto the regression coefficients even after optimization on noisy hardware.

What would settle it

Run the algorithm on a small synthetic dataset whose classical linear regression coefficients are known exactly and check whether the final variational parameters recovered from the circuit match those coefficients within the error budget set by the cost-function variance.

read the original abstract

Hybrid variational quantum algorithms are promising for solving practical problems, such as combinatorial optimization, quantum chemistry simulation, quantum machine learning, and quantum error correction on noisy quantum computers. However, variational quantum algorithms (derived from randomized hardware-efficient ansatz or adaptive ansatz) become a black box, not trustworthy for model interpretation, and not to mention for application deployment in informing critical decisions. In this paper, we construct the first interpretable quantum regression algorithm, in which the quantum state exactly encodes the classical data table and the variational parameters correspond directly to the regression coefficients, which are real numbers by construction, providing a high degree of model interpretability and minimal cost to optimize due to the right expressiveness. We also exploit the encoded data structure to reduce the gate complexity of computing the regression map. To reduce circuit depth in nonlinear regression, our algorithm can be extended by directly constructing nonlinear features via classical preprocessing, such as independent encoded column vectors. By design, the model performance is determined by the cost function measurement results $\mathcal{C}$ synchronous to the mean squared errors (MSE) for the regression models. We derived the read-out errors induced by one-hot encoding and compact encoding; the required physical qubit resources are exponentially compressed for the compact encoding to be favorable for noisy quantum devices. We also derive the cost function dependent sample complexity $ \in \mathcal{O}\left(\sigma^{2}(\mathcal{C}) \ln (1/\alpha)/\epsilon^{2}\right)$ under the error budget $\epsilon$ and confidence tolerance $\alpha$.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 1 minor

Summary. The paper claims to introduce the first interpretable quantum regression algorithm in which a quantum state exactly encodes the classical data table (with no information loss), variational parameters map directly to real-valued regression coefficients, the measured cost C is synchronous with classical MSE, read-out errors are derived for one-hot and compact encodings (with the latter exponentially reducing qubit count), and a sample complexity bound O(σ²(C) ln(1/α)/ε²) is obtained under an error budget.

Significance. If the exact-encoding claim and direct coefficient interpretability could be established without contradiction from read-out errors, the work would offer a notable contribution to explainable quantum machine learning by enabling trustworthy regression models with reduced optimization cost and hardware resources on NISQ devices.

major comments (2)
  1. [Abstract] Abstract: the central claim that 'the quantum state exactly encodes the classical data table' without information loss (enabling direct correspondence of variational parameters to regression coefficients and C synchronous to MSE) is contradicted by the derivation of nonzero read-out errors for the compact encoding, which the paper recommends as favorable for noisy hardware. Any such error propagates into the estimated coefficients and cost, undermining the interpretability and 'right expressiveness' arguments.
  2. [Abstract] Abstract (sample complexity derivation): the bound O(σ²(C) ln(1/α)/ε²) is expressed in terms of the variance of the cost function C, yet C is obtained only after fitting the variational parameters to the data. This creates a circular dependence in which the complexity bound incorporates post-optimization quantities by construction, rather than providing an a priori guarantee independent of the fitted model.
minor comments (1)
  1. [Abstract] The abstract states that the algorithm 'can be extended by directly constructing nonlinear features via classical preprocessing' but provides no explicit circuit construction, gate-count analysis, or verification that this preserves the exact-encoding property.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and insightful comments on our manuscript. We address each major comment point by point below, offering clarifications on the theoretical claims versus practical considerations and proposing revisions to improve clarity.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that 'the quantum state exactly encodes the classical data table' without information loss (enabling direct correspondence of variational parameters to regression coefficients and C synchronous to MSE) is contradicted by the derivation of nonzero read-out errors for the compact encoding, which the paper recommends as favorable for noisy hardware. Any such error propagates into the estimated coefficients and cost, undermining the interpretability and 'right expressiveness' arguments.

    Authors: We appreciate the referee pointing out this potential ambiguity. The claim of exact encoding without information loss refers specifically to the ideal quantum state preparation, where the classical data table is mapped into the quantum state via the chosen encoding (one-hot or compact) such that all information is preserved in the amplitudes or basis states. In this ideal setting, the variational parameters correspond directly to the real-valued regression coefficients, and the measured cost C aligns with the classical MSE. The nonzero read-out errors we derive and bound are the statistical sampling errors (from finite shots) and hardware-induced noise effects that arise when measuring the cost observable on NISQ devices. These errors affect the estimated values of C and the coefficients but do not invalidate the exact encoding of the data or the direct interpretability of the parameters in the model definition. The compact encoding is recommended for its exponential qubit reduction, with explicit error bounds provided to quantify the impact. To resolve the concern, we will revise the abstract to explicitly distinguish the ideal exact encoding from practical measurement errors and add a brief clarification on how interpretability is preserved in the theoretical model while error propagation is analyzed separately. revision: yes

  2. Referee: [Abstract] Abstract (sample complexity derivation): the bound O(σ²(C) ln(1/α)/ε²) is expressed in terms of the variance of the cost function C, yet C is obtained only after fitting the variational parameters to the data. This creates a circular dependence in which the complexity bound incorporates post-optimization quantities by construction, rather than providing an a priori guarantee independent of the fitted model.

    Authors: The sample complexity bound is a standard application of concentration inequalities (e.g., Chebyshev) for estimating the expectation value of the cost observable to precision ε with probability 1-α. In this context, σ²(C) is the variance of the measurement outcomes for the fixed observable corresponding to a given set of variational parameters; it is not the optimized cost value itself but a property of the quantum state and measurement for that model. The optimization step determines the parameters that minimize C, after which the bound is used to compute the number of shots needed to estimate the achieved C accurately. This is not circular, as the bound provides a guarantee on estimation resources for any fixed model and is commonly used in quantum algorithms literature (e.g., for VQE or QML expectation estimation). If an a priori bound on σ²(C) is available, the sample count can be determined before optimization. We will revise the manuscript to clarify this distinction, emphasize that the bound applies post-parameter fitting for estimation accuracy, and note options for bounding the variance independently. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained by explicit construction and standard bounds

full rationale

The paper's core contribution is an explicit algorithmic construction in which the quantum state is defined to encode the data table and variational parameters are set to match regression coefficients by design. This is not a derivation that reduces to its own outputs. The sample-complexity bound O(σ²(C) ln(1/α)/ε²) is a standard concentration inequality whose dependence on the variance of the measured cost C is the usual statistical relation between estimator variance and number of shots; it does not presuppose the fitted value of C itself. Read-out error derivations for the two encodings are separate analytic calculations performed after the encoding is specified, not inputs that are renamed as predictions. No self-citations, uniqueness theorems, or ansatzes imported from prior work appear in the provided text. The central interpretability claim therefore rests on the stated encoding and parameterization rather than on any tautological reduction.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 1 invented entities

The central claim rests on standard quantum mechanics for state preparation and measurement plus the domain assumption that the chosen encoding preserves data exactly and that cost function measurements align with classical MSE without additional artifacts from noise or fitting.

free parameters (1)
  • variational parameters as regression coefficients
    These parameters are optimized during training and directly define the model; their values are fitted to data.
axioms (2)
  • domain assumption Quantum states can exactly encode classical data tables via one-hot or compact encoding
    Invoked to support direct interpretability and gate reduction.
  • domain assumption Cost function measurement results C are synchronous to mean squared error
    Stated as the basis for performance evaluation and sample complexity.
invented entities (1)
  • encoded data structure no independent evidence
    purpose: Allows direct mapping of variational parameters to regression coefficients and reduces gate complexity
    New structuring of the quantum state for the regression task; no independent evidence outside the algorithm design.

pith-pipeline@v0.9.0 · 5596 in / 1603 out tokens · 32731 ms · 2026-05-10T09:50:02.795705+00:00 · methodology

discussion (0)

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