From hyperplanes to hyperellipsoids: characterizing the inherent interpretability of linear and single-qubit mixed-state binary classification models

# From hyperplanes to hyperellipsoids: characterizing the inherent interpretability of linear and single-qubit mixed-state binary classification models
Source: [https://arxiv.org/html/2607.15433](https://arxiv.org/html/2607.15433)
\(July 9, 2026\)

###### Abstract

We characterize and compare the inherent interpretability offerings of a standard linear model with a single qubit mixed state model for the task of supervised binary classification\. A side by side comparison reveals that a single qubit mixed state model for binary classification is just the “ellipsoid version” of standard linear model classification\. More precisely, rather than learning a*hyperplane*to classify data, we learn a*hyperellipsoid*\. We discuss the consequences of the geometric inductive biases of both models, as well as how each model contains a different feature importance inductive bias\. This short characterization offers an accessible route to quantum machine learning \(ML\) ideas for readers who have zero background in quantum and are only familiar with linear classification in ML\. In support of ML pedagogy, we encourage instructors to utilize this piece to smoothly introduce quantum ML ideas into the undergraduate ML classroom\.

## 1Introduction

In a recent perspective piece\[gili2026inherent\], we argue for more research that characterizes machine learning \(ML\) models side by side for their inherent interpretability offerings\[Zschech2025Inherently\]\. The motivation is simple: the characterization reveals structural differences between models that imply distinct biases in their interaction with the same data\. Understanding these mathematical biases and their consequences enables ML researchers and practitioners to design methods in the real world with principled mathematical reasons that can be verified empirically\. These characterizations are also constructive from a pedagogical point of view\. When introduced to students, they encourage mechanistic reasoning\[mech\_reason\]with mathematical objects to understand model behavior and design models with desired behaviors\. In addition, they can help experts recognize connections between models within their own domain and those developed outside of it\.

In this brief characterization, we compare a standard linear model\[Bishop\-book\-2006,hastie2009elements,james2021introduction\]with a single qubit mixed state model\[nielsen2010quantum\]for the task of supervised binary classification\. Linear model classification is arguably the most widely used ML method across industry sectors – largely for its simplicity and inherent interpretability\[fan2008liblinear\]\. Often\-times these models are implemented as last layer classification heads for deep neural networks\[he2016deep,BERT\], or as classifiers for frozen embeddings from a foundation model, known as linear probing\[alain2016understanding,evci2022head2toe\]\. Occasionally, these models even perform on par with deep non\-linear ones\[schulz2020different\], or outperform them entirely for a specific problem\[arshad2023powersimplicitysimplelinear\]\. Linear classification is typically among the first parametric methods taught in an undergraduate introductory ML course\[james2021introduction\]\. On the opposite side of the spectrum, a single quantum bit \(qubit\) mixed\-state model is typically viewed as requiring specialized knowledge of quantum information science to understand and use\[nielsen2010quantum\]\. The model name contains the word “qubit” – which for many people immediately triggers a notion of inaccessibility\. In fact, students may only learn about this model in a graduate quantum information course – completely separate from ML methods\.

We show that if we look past the complex jargon, a single qubit mixed state model for binary classification is just the “ellipsoid version” of standard linear model classification\. More precisely, rather than learning a*hyperplane*to classify data, we learn a*hyperellipsoid*\. We characterize this difference in geometric inductive bias between the models, as well as the effect of the additional restriction imposed by the normalization constraint on the weights in the qubit case\. In the latter bias, we observe that linear models offer inherent interpretability through*absolute*feature importance, whereas single\-qubit mixed\-state models offer*relative*feature importance\.

We intentionally keep this characterization brief and accessible to readers with no background in quantum information, as we have an interest in supporting ML pedagogy\. Empirical investigations of model interpretability in real\-world data settings, building on the theory developed below, remain an open direction for future research\.

## 2Characterization of binary classification models

![Refer to caption](https://arxiv.org/html/2607.15433v1/Figure1.png)\(a\)Linear model
![Refer to caption](https://arxiv.org/html/2607.15433v1/Figure2.png)\(b\)Single qubit mixed state model

Figure 1:Toy classification example for two data features\. \(a\) The model can only learn the slope of a line to separate the data\. \(b\) The model can only learn the semi\-axis lengths of an ellipsoid to separate the data\.Let us consider a traditional supervised learning classification problem\. We have a training dataset𝒟=\{xi,yi\}i=1N\\mathcal\{D\}=\\\{x\_\{i\},y\_\{i\}\\\}\_\{i=1\}^\{N\}, wherexi∈ℝDx\_\{i\}\\in\\mathbb\{R\}^\{D\}andyi∈\{0,1\}y\_\{i\}\\in\\\{0,1\\\}\. Our goal is to train a parameterized model to classify the input dataxix\_\{i\}by its true binary labelyi=0y\_\{i\}=0oryi=1y\_\{i\}=1, and generalize to unseen data that is assumed to be sampled from the same underlying distribution as the training set\. Typically, we train such a model with a logistic loss function\. Below, we characterize the interpretability offerings of linear models and single qubit mixed state models for this task\.

### 2\.1Linear model

Let us consider a linear classification model\. For a single datapointxix\_\{i\}, a linear model is of the form:

fw​\(x\)=w1​x1\+w2​x2\+⋯\+wD​xD\+βf\_\{w\}\(\{x\}\)=w\_\{1\}x\_\{1\}\+w\_\{2\}x\_\{2\}\+\\dots\+w\_\{D\}x\_\{D\}\+\\beta\(1\)
Each trainable weightwd∈ℝw\_\{d\}\\in\\mathbb\{R\}influences a data featurexdx\_\{d\}directly, and the combined sum is the model output for the entire data inputxix\_\{i\}, plus a scalar bias offsetβ∈ℝ\\beta\\in\\mathbb\{R\}\. In this form, the parameterized weights are unbounded in training – although constraints can be added\. If raw data features are very large compared to others, we use amin−max\\min\-\\maxnormalization to map each datapointxi∈\[0,1\]x\_\{i\}\\in\[0,1\]\. This normalization makes the model output more sensitive to the learned feature weights than to the absolute scale of the input data\. We set the binary decision threshold to zero, and so the final decision function issign​\(fw​\(x\)\)\\text\{sign\}\(f\_\{w\}\(\{x\}\)\)\.

Absolute feature importance\.The sign \(positive or negative\) of each feature weightwdw\_\{d\}indicates whether that feature pushes the model output toward class0or class11\. The magnitude\|wd\|\|w\_\{d\}\|indicates*absolute importance*of the feature weight in its contribution to the classification output\. Thus, we understand what features have high or low impact on the model behavior\.

Learning a hyperplane\.Whenfw​\(x\)=0f\_\{w\}\(\{x\}\)=0, we have aD−D\-dimensional hyper\-plane, and thus we are learning this hyper\-plane that separates the data\. The weights determine the orientation and placement of the hyper\-plane\. Linear models have the constraint that they can*only*learn linear decision boundaries\. Hence, if the data cannot be linearly separated, a linear model is not geometrically expressive enough for perfect classification, no matter how much additional data is available\. See Fig\.[1\(a\)](https://arxiv.org/html/2607.15433#S2.F1.sf1)for a standard toy example with two data features\. Conversely, if we know that the data are linearly separable, why use anything more complicated?

### 2\.2Single qubit mixed state model

Let us now consider a single qubit mixed state model111As this is a*single*qubit model, it is clearly classically simulatable\. It is useful to classify this model as*quantum\-inspired*for the reason that we would not actually need to use quantum hardware resources to implement it\.\. In order to reveal the hyperellipsoid structure of the model, we need to introduce two ideas within quantum information science – a qubit state and a qubit measurement\[nielsen2010quantum\]\. However, we encourage the reader to not pay close attention to the physical meaning of these ideas \(e\.g\. the physical state of an atom and our observation of it via interaction\); rather, it is important that one pay attention to the mathematical objects that represent the physical ones \(i\.e\. a unit vector and a hermitian matrix\)\.

A*pure*single qubit state is a2−2\-dimensional column unit vector containing complex coefficientsα∈ℂ\\alpha\\in\\mathbb\{C\}andβ∈ℂ\\beta\\in\\mathbb\{C\}\. We denote this asvpure=\(αβ\)v\_\{\\text\{pure\}\}=\\begin\{pmatrix\}\\alpha\\\\ \\beta\\end\{pmatrix\}\. Given thatvpurev\_\{\\text\{pure\}\}is a unit vector, we know that the coefficients are constrained as\|α\|2\+\|β\|2=1\|\\alpha\|^\{2\}\+\|\\beta\|^\{2\}=1\. The coefficients indicate rotational coordinates of the vector on the unit sphere\.

A*mixed*single qubit state is a linear sum of pure model outer products, where each is multiplied by some probabilitypp\. We encode each data featureddin a datapointxix\_\{i\}into theα\\alphacoefficient of each pure state\. This technique is a form of amplitude encoding\[grover2002creating,Rebentrost\-PRL\-2014\]\. Since all of our data are real\-valued, from this point onward we restrictα\\alphaandβ\\betato be real\. Each probabilitypdp\_\{d\}is the trainable feature weight, analogous to the weightwwabove\. Asα2<1\\alpha^\{2\}<1, we use amin−max\\min\-\\maxnormalization to map all data featuresxd∈\[0,1\]x\_\{d\}\\in\[0,1\]\(same as in the linear model\)\. Under these conditions, we observe the following single qubit mixed state:

vmixed=∑d=1Dpd​vpure​\(xd\)​vpureT​\(xd\)=∑d=1Dpd​\(\|xd\|2xd​ββ​xd\|β\|2\)=∑d=1Dpd​\(\|xd\|2xd​ββ​xd1−\|xd\|2\)v\_\{\\text\{mixed\}\}=\\sum\_\{d=1\}^\{D\}p\_\{d\}\\ v\_\{\\text\{pure\}\}\(x\_\{d\}\)v\_\{\\text\{pure\}\}^\{T\}\(x\_\{d\}\)\\\\ =\\sum\_\{d=1\}^\{D\}p\_\{d\}\\begin\{pmatrix\}\|x\_\{d\}\|^\{2\}&x\_\{d\}\\beta\\\\ \\beta x\_\{d\}&\|\\beta\|^\{2\}\\\\ \\end\{pmatrix\}\\\\ =\\sum\_\{d=1\}^\{D\}p\_\{d\}\\begin\{pmatrix\}\|x\_\{d\}\|^\{2\}&x\_\{d\}\\beta\\\\ \\beta x\_\{d\}&1\-\|x\_\{d\}\|^\{2\}\\\\ \\end\{pmatrix\}\(2\)
Notice that the diagonals can be written in this form because\|α\|2\+\|β\|2=1\|\\alpha\|^\{2\}\+\|\\beta\|^\{2\}=1\. As such, quantum information imposes the general normalization constraintTr​\(vmixed\)=1\\text\{Tr\}\(v\_\{\\text\{mixed\}\}\)=1\.

So far, we have the mixed quantum statevmixed∈ℝ2×2v\_\{\\text\{mixed\}\}\\in\\mathbb\{R\}^\{2\\times 2\}, but not the full single qubit mixed state model\. The full model includes a measurement of this quantum state with respect to a chosen random variable \(e\.g\. system energies\)\. The expectation of measuring a particular random variable is the trace \(sum of the diagonal entries\) of the product of the statevmixedv\_\{\\text\{mixed\}\}and the possible random variable outcomes represented as the hermitian \(real\-eigenvalued\) matrix𝒪∈ℂ2×2\\mathcal\{O\}\\in\\mathbb\{C\}^\{2\\times 2\}\.

If we choose a standard binary computational measurementZ=\(100−1\)Z=\\begin\{pmatrix\}1&0\\\\ 0&\-1\\end\{pmatrix\}, the single qubit mixed state model is written as:

fp​\(x\)\\displaystyle f\_\{p\}\(x\)=Tr⁡\[vmixed​𝒪\]\\displaystyle=\\operatorname\{Tr\}\\\!\\left\[v\_\{\\text\{mixed\}\}\\mathcal\{O\}\\right\]\(3\)=Tr⁡\[∑d=1Dpd​\(\|xd\|2xd​ββ​xd1−\|xd\|2\)​\(100−1\)\]\\displaystyle=\\operatorname\{Tr\}\\\!\\left\[\\sum\_\{d=1\}^\{D\}p\_\{d\}\\begin\{pmatrix\}\|x\_\{d\}\|^\{2\}&x\_\{d\}\\beta\\\\ \\beta x\_\{d\}&1\-\|x\_\{d\}\|^\{2\}\\end\{pmatrix\}\\begin\{pmatrix\}1&0\\\\ 0&\-1\\end\{pmatrix\}\\right\]=Tr⁡\[∑d=1Dpd​\(\|xd\|2−xd​ββ​xd\|xd\|2−1\)\]\\displaystyle=\\operatorname\{Tr\}\\\!\\left\[\\sum\_\{d=1\}^\{D\}p\_\{d\}\\begin\{pmatrix\}\|x\_\{d\}\|^\{2\}&\-x\_\{d\}\\beta\\\\ \\beta x\_\{d\}&\|x\_\{d\}\|^\{2\}\-1\\end\{pmatrix\}\\right\]=2​\(p1​\|x1\|2\+p2​\|x2\|2\+⋯\+pD​\|xD\|2\)−1\\displaystyle=2\\left\(p\_\{1\}\|x\_\{1\}\|^\{2\}\+p\_\{2\}\|x\_\{2\}\|^\{2\}\+\\dots\+p\_\{D\}\|x\_\{D\}\|^\{2\}\\right\)\-1
Each trainable weightpdp\_\{d\}directly scales the squared contribution of the corresponding input featurexdx\_\{d\}, and the combined sum is the model output for the entire data inputxix\_\{i\}\. In this standard form, the parameterized weights*are bounded*in training – unlike the standard linear model\. The probabilities are bounded as0≤pd≤10\\leq p\_\{d\}\\leq 1\. We set the binary decision threshold to zero, and so the final decision function issign​\(fp​\(x\)\)\\text\{sign\}\(f\_\{p\}\(\{x\}\)\)\.

Relative feature importance\.All probabilistic weights are positive, and thus the sign of each weight provides no information regarding its contribution to the classification outcome\. However, the constraint∑d=1Dpd=1\\sum\_\{d=1\}^\{D\}p\_\{d\}=1adds an inherently interpretable bias of relative importance for each feature’s contribution to the model output\. Thus, the model encodes the bias that some features are more important than others – analogous to the bias of self\-attention in transformer architectures\.

Learning a hyperellipsoid\.Observe that the single qubit mixed state model takes a final form that is similar to the linear model in Eq\.[1](https://arxiv.org/html/2607.15433#S2.E1), except that rather than containing the structure of linear hyper\-plane, the model structure is that of a*hyperellipsoid*\. When the decision threshold is zero, we obtain the following hyperellipsoid boundary, centered at the origin:

x12a12\+x22a22\+⋯\+xD2aD2=1\\frac\{x\_\{1\}^\{2\}\}\{a\_\{1\}^\{2\}\}\+\\frac\{x\_\{2\}^\{2\}\}\{a\_\{2\}^\{2\}\}\+\\dots\+\\frac\{x\_\{D\}^\{2\}\}\{a\_\{D\}^\{2\}\}=1\(4\)The coefficientsada\_\{d\}are semi\-axis lengths – i\.e\. the distances from the origin to the ellipsoid’s boundary along each direction\. From our model in Eq\.[3](https://arxiv.org/html/2607.15433#S2.E3), we observe thatad=12​pda\_\{d\}=\\frac\{1\}\{\\sqrt\{2p\_\{d\}\}\}\. As a result, changing each probability is geometrically equivalent to stretching or shrinking the ellipsoid along a distinct feature direction\. See Fig\.[1](https://arxiv.org/html/2607.15433#S2.F1)for a comparison between both models with toy data whenD=2D=2\. When all of the data features are of equal relative importance \(pd=1Dp\_\{d\}=\\frac\{1\}\{D\}\), we obtain a*hyper\-sphere*around the origin with a radiusr=D2r=\\sqrt\{\\frac\{D\}\{2\}\}\. The hyperellipsoid is most compressed along a feature direction whenpd=1p\_\{d\}=1and stretches without bound aspd→0p\_\{d\}\\to 0\. Whenpd=0p\_\{d\}=0, the corresponding feature no longer contributes to the decision boundary, causing the hyperellipsoid to become unbounded in that direction and geometrically reducing it to a hypercylinder\. Thus, from a practical implementation standpoint, it is important to lower\-bound the probabilistic weights by a small positive value, such as10−910^\{\-9\}, to prevent the decision boundary from becoming unbounded along any feature direction\. Similar to the linear model, the single qubit mixed state model can only learn decision boundaries that are ellipsoidal\. Fig\.[1\(a\)](https://arxiv.org/html/2607.15433#S2.F1.sf1)presents a toy setting in which this ellipsoidal boundary can separate data that a linear classifier cannot\. While this provides greater curvature around the data, the model remains more restricted than one that could also translate the hyperellipsoid away from the origin\.

For the reader who is interesting in learning more about single qubit mixed state models for ML, we refer them to Refs\.\[sergioli2016pattern,leporini2022efficient,bertini2023quantum\]\. In these different Refs, the authors explore models where the mixed state probabilities are fixed, and individual datapoints are encoded into each pure state to be classified via distance metrics\. Alternatively, for readers interested in the more widely studied single\-qubit quantum classifier, we refer them to Ref\.\[perez2020reuploading\]\. This method classifies data by encoding each input into a single pure quantum state, training the encoding via unitary rotations, and then computing the expectation with respect to the random variableZZ\. Lastly, if one is interested in different ML problems that can be tackled with only a single qubit, we refer the reader to Ref\.\[cuellar2024onequbit\]\. To the best of our knowledge, the exact single qubit mixed model discussed here, particularly the feature of the probabilities themselves being trainable, has not been studied in other works\. At least, this model does not seem to be present in the mainstream quantum ML literature\. While we introduce the model theory to characterize its simple inductive bias and draw a connection to a widely known model, we encourage interested researchers and practitioners to investigate its empirical behavior on real\-world data\.

## 3Conclusion

By characterizing models through their inherent structure, and discussing their differences in a form of analysis that is closer to*qualitative*than quantitative, we are able to draw connections between objects that may once have seemed far apart\. For ML experts who have viewed quantum machine learning models as somewhat “fuzzy” objects, with little clarity about how they relate to familiar classical models, we hope this perspective makes quantum ideas feel closer and more concrete\. For ML instructors, we hope it provides a way to bring quantum ideas into the classroom that is only one step removed from linear classification\. For quantum information instructors, we hope it offers a natural route for introducing ML alongside qubits and mixed states, while providing an intuitive geometric perspective\. And lastly, for the quantum ML research community, we hope it serves as a reminder that understanding the inductive bias of a model matters alongside the amount of compute resources required to implement it\.

## References