LUQPI: Quantum Privileged Information Learning

• LUQPI is a quantum machine learning paradigm that uses a quantum device for one-time per-instance feature extraction, enabling exponential quantum-classical separations under cryptographic hardness assumptions.
• It augments classical training by incorporating quantum-generated features exclusively during training, boosting performance in low-data and hard-boundary regimes.
• Empirical studies in many-body physics demonstrate that LUQPI significantly improves classification accuracy over traditional SVM models by correcting boundary-region errors.

Learning Under Quantum Privileged Information (LUQPI) is a machine learning paradigm in which a quantum device is used exclusively as a feature extractor on individual, unlabeled data points during training. The quantum computer operates without access to labels or aggregate dataset information, is never used at deployment, and augments the classical training dataset with quantum-generated features. Despite this extremely limited quantum involvement, the LUQPI framework enables provable learning advantages—specifically, exponential quantum-classical separations for suitable problems under plausible computational hardness assumptions. LUQPI is formally positioned as an extension of the classical Learning Under Privileged Information (LUPI) concept to the quantum regime, facilitating practical hybrid quantum-classical workflows while maintaining classical deployment (Bokov et al., 29 Jan 2026).

1. Formal Definition and Theoretical Model

Let $\mathcal{X}_n$ denote the input domain, $\mathcal{Y}$ the label space, and $Q_n: \mathcal{X}_n \to \mathcal{F}_n$ a quantum feature extractor implementable in polynomial time on a quantum device. The extended example oracle, $\mathrm{EX}_{\mathrm{ext}}$, outputs triplets $(x, z = Q_n(x), y)$ for $x \sim D_n$ and $y = c(x)$.

A LUQPI learner is a classical polynomial-time algorithm $\mathcal{A}_{\mathrm{off}}$, which, given $m = \mathrm{poly}(n,1/\epsilon,\log(1/\delta))$ samples from $\mathrm{EX}_{\mathrm{ext}}$, outputs a hypothesis $\mathcal{Y}$0 such that, with probability at least $\mathcal{Y}$1,

$\mathcal{Y}$2

At test time, only $\mathcal{Y}$3 is accessible; neither $\mathcal{Y}$4 nor quantum devices are used. Each training example receives a single quantum feature extraction query, independent of labels and other examples.

2. Exponential Quantum-Classical Separations

The central separation is established using an order-$\mathcal{Y}$5 cyclic group $\mathcal{Y}$6 and a family of concept classes $\mathcal{Y}$7, where each concept

$\mathcal{Y}$8

with $\mathcal{Y}$9. The learning task is defined under the uniform distribution $Q_n: \mathcal{X}_n \to \mathcal{F}_n$0 over $Q_n: \mathcal{X}_n \to \mathcal{F}_n$1.

Assuming the circular Decisional Diffie-Hellman (DDH) problem is hard for classical (even nonuniform $Q_n: \mathcal{X}_n \to \mathcal{F}_n$2) algorithms, no classical polynomial-time learner can Probably Approximately Correct (PAC)-learn $Q_n: \mathcal{X}_n \to \mathcal{F}_n$3 with error less than $Q_n: \mathcal{X}_n \to \mathcal{F}_n$4. Any classical learner achieving lower error would break the circular DDH assumption by distinguishing structured from random tuples.

In contrast, the quantum LUQPI setting admits a quantum feature map $Q_n: \mathcal{X}_n \to \mathcal{F}_n$5, with $Q_n: \mathcal{X}_n \to \mathcal{F}_n$6, efficiently computable by Shor’s algorithm. Classical post-processing then leverages these quantum-derived features and the corresponding labels to recover $Q_n: \mathcal{X}_n \to \mathcal{F}_n$7 in polynomial time, yielding exponential separation between quantum-augmented and fully classical learning under standard group-based assumptions. These results persist even against nonuniform adversaries (Bokov et al., 29 Jan 2026).

3. Taxonomy of Learning Settings

The following table situates LUQPI within a hierarchy of classical and quantum learning models:

Paradigm	Privileged Features	Quantum Involvement at Deployment
PAC (classical)	None	None
LUPI (classical privileged)	Classical	None
Quantum online	Quantum (train+test)	Yes
Semi-supervised quantum privileged	Quantum (unlabeled)	None
LUQPI (quantum offline privileged)	Quantum (train only)	None

LUQPI uniquely combines offline quantum feature extraction—used exactly once per labeled training example, never during test or inference—with fully classical model training and deployment. This is the minimal quantum involvement sufficient to achieve exponential learning-theoretic separations under natural distributions and cryptographic assumptions.

4. Adapting Classical SVM+ to the LUQPI Setting

Support Vector Machine+ (SVM+) is the canonical algorithm for classical LUPI, extending the SVM framework to exploit privileged features $Q_n: \mathcal{X}_n \to \mathcal{F}_n$8 through modeling slack variables as functions of $Q_n: \mathcal{X}_n \to \mathcal{F}_n$9. The SVM+ primal formulation is: $\mathrm{EX}_{\mathrm{ext}}$0 Here, $\mathrm{EX}_{\mathrm{ext}}$1 and $\mathrm{EX}_{\mathrm{ext}}$2 denote mappings to respective kernel spaces for $\mathrm{EX}_{\mathrm{ext}}$3 and $\mathrm{EX}_{\mathrm{ext}}$4. At deployment, only $\mathrm{EX}_{\mathrm{ext}}$5 is evaluated; neither $\mathrm{EX}_{\mathrm{ext}}$6 nor $\mathrm{EX}_{\mathrm{ext}}$7 is needed. In LUQPI, one simply sets $\mathrm{EX}_{\mathrm{ext}}$8, applying SVM+ in the same manner—enabling incorporation of quantum-augmented features during training and purely classical inference.

5. Empirical Study: Quantum-Privileged Features in Many-Body Physics

A numerical investigation of LUQPI was conducted in the context of phase classification for a one-dimensional chain of $\mathrm{EX}_{\mathrm{ext}}$9 Rydberg atoms with Hamiltonian

$(x, z = Q_n(x), y)$0

where $(x, z = Q_n(x), y)$1. Ground states exhibit Disordered, $(x, z = Q_n(x), y)$2, and $(x, z = Q_n(x), y)$3 phases, characterized by order parameters

$(x, z = Q_n(x), y)$4

Class labels $(x, z = Q_n(x), y)$5 are assigned via thresholding $(x, z = Q_n(x), y)$6 and $(x, z = Q_n(x), y)$7 at $(x, z = Q_n(x), y)$8.

Privileged features $(x, z = Q_n(x), y)$9 are computed from ground states obtained by DMRG and play the role of "quantum" features (practically, these would come from QPU measurement). The dataset consists of $x \sim D_n$0 parameter points in the Hamiltonian phase diagram, with three train set selection regimes: uniform, light boundary ($x \sim D_n$1 near phase boundaries), and hard boundary ($x \sim D_n$2 near boundaries), over training sizes $x \sim D_n$3.

Baselines assessed include:

1. SVM (classical, $x \sim D_n$4 only)
2. SVM+ (classical, $x \sim D_n$5 with quantum-privileged $x \sim D_n$6 at train only)
3. Transformer-based "Quantum Feature Learner" [Wang et al.] leveraging predicted local POVM expectation values for quantum-inspired features at test time.

Performance is measured as test accuracy over a uniform parameter grid, averaged over 30 splits. Under challenging "hard boundary" sampling at $x \sim D_n$7, SVM+ achieves $x \sim D_n$8 versus SVM's $x \sim D_n$9, outperforming the Transformer approach for $y = c(x)$0. SVM+ corrects boundary-region errors that standard SVM fails on. These results demonstrate that modest quantum-privileged features provide material improvements for classical learners—particularly in low-data, high-difficulty regimes (Bokov et al., 29 Jan 2026).

6. Implications and Significance

LUQPI operationalizes a minimal hybrid quantum-classical workflow, requiring a quantum device solely for feature extraction on training data, without quantum involvement at test or deployment. This paradigm establishes exponential learning advantages over classical machines for select concept classes and distributions under cryptographic hardness assumptions, even when restricted to one quantum query per labeled example and against nonuniform adversaries. Empirical results in many-body physics show that LUQPI's practical gains are most pronounced in regimes of limited data and hard-to-separate classes. The LUQPI framework thereby delineates limits of classical learnability in the presence of quantum-computable privileged information and motivates further research into efficient, implementable hybrid algorithms for near-term quantum applications (Bokov et al., 29 Jan 2026).