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Adaptive Conformal Prediction for Quantum Machine Learning

This paper introduces Adaptive Quantum Conformal Prediction (AQCP), an algorithm that ensures reliable uncertainty quantification for quantum machine learning by maintaining target coverage guarantees despite the time-varying noise inherent in quantum hardware, outperforming standard quantum conformal prediction in stability and validity.

Original authors: Douglas Spencer, Samual Nicholls, Michele Caprio

Published 2026-05-18

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CC BY 4.0

Original authors: Douglas Spencer, Samual Nicholls, Michele Caprio

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to teach a very powerful, but slightly "drunk," robot to predict the weather. This robot is a Quantum Machine Learning (QML) model. It's incredibly fast and smart, but it lives on a quantum computer, which is currently a bit unstable. The hardware is like a ship in a stormy sea; the waves (noise) change constantly, sometimes gently, sometimes violently.

The problem is: How do you know when the robot is confident enough to trust its prediction?

In the classical world, we use a safety net called Conformal Prediction. Think of this as a "confidence belt." Instead of the robot saying, "It will rain tomorrow," it says, "It will rain between 2 PM and 4 PM." The goal is to make sure that "belt" is wide enough to catch the true answer 90% of the time (if you set it to 90%).

The Problem: The Drunk Robot's Unstable Belt

The paper points out a major flaw in how we currently use this safety net for quantum computers.

Standard safety nets assume the robot's "drunkenness" (noise) stays the same all day. But on real quantum computers, the noise changes. The hardware recalibrates every hour, temperature shifts, and cosmic rays can hit the machine.

The authors explain that because the noise changes over time, the robot's "confidence belt" becomes unreliable. It's like trying to use a ruler that stretches and shrinks randomly. If you calibrate the ruler in the morning, it might be too short by the afternoon. The standard math says, "We guarantee this belt catches the truth 90% of the time," but because the ruler is changing size, that guarantee breaks. The robot might think it's 90% sure, but it's actually only 70% sure, or maybe 99% sure.

The Solution: The "Self-Correcting" Belt

To fix this, the authors introduce a new method called Adaptive Quantum Conformal Prediction (AQCP).

Think of AQCP as a smart, self-correcting belt.

Old Method (QCP): You measure the ruler once in the morning and stick with it all day. If the ruler shrinks later, your measurements are wrong.
New Method (AQCP): You check the ruler every single time you make a prediction.
- If the robot was wrong (the true answer fell outside the belt), the system says, "Oops, my belt was too tight!" and immediately widens it for the next prediction.
- If the robot was right (the answer was inside), it might slightly tighten the belt to be more efficient.

This happens in real-time. The system constantly recalibrates itself based on whether it just got a "hit" or a "miss." This allows it to handle the "drunk" hardware, even if the noise is chaotic and changing every second.

How They Tested It

The researchers tested this on a real quantum computer made by IBM (called ibm_sherbrooke). They gave the robot a task: predict a number based on a sine wave pattern (a wavy line).

The Test: They ran thousands of predictions.
The Result: The old method (QCP) kept drifting away from the target. Sometimes it was way too wide (wasting resources), and sometimes it was too narrow (missing the truth).
The Winner: The new method (AQCP) stayed right on target. It kept the "catch rate" steady at the desired 90%, no matter how much the hardware noise jittered.

They also tested different ways to measure "how wrong" the robot was (called score functions). They found that some methods were better at keeping the prediction belt as small as possible while still catching the truth, but all of them worked much better with the self-correcting AQCP system than without it.

The Catch

There is one small limitation. Because this system learns by checking its mistakes one by one, it has to work sequentially.

Analogy: Imagine a teacher grading a stack of exams. The old method grades all 100 exams at once based on a rule set at the start. The new method grades one exam, checks the answer, adjusts the rule, and then grades the next one.
Consequence: You can't use this for "batch" processing (grading 1,000 exams all at once without seeing the answers). You have to wait for the result of one prediction before making the next.

Summary

In simple terms: Quantum computers are noisy and unpredictable. The old way of measuring their confidence didn't account for this changing noise, leading to unreliable predictions. The authors built a new "self-correcting" system that constantly adjusts its confidence levels in real-time. This ensures that even on a shaky, noisy quantum machine, the predictions remain trustworthy and accurate.

Technical Summary: Adaptive Conformal Prediction for Quantum Machine Learning

Problem Statement

Quantum Machine Learning (QML) aims to leverage quantum processors to enhance classical algorithms, yet the deployment of these models in high-stakes domains (e.g., healthcare, defense) requires rigorous uncertainty quantification. While Conformal Prediction (CP) offers distribution-free, finite-sample guarantees for classical models, its application to QML faces a critical hurdle: non-stationary hardware noise.

Standard Conformal Prediction, including the recently proposed Quantum Conformal Prediction (QCP) framework (Park & Simeone, 2023), relies on the assumption that calibration and test data are exchangeable. This assumption implies that the underlying noise processes within the quantum hardware remain stationary over time. However, current Noisy Intermediate-Scale Quantum (NISQ) devices exhibit time-varying noise due to factors such as temperature fluctuations, control equipment oscillations, and cosmic ray events. The authors demonstrate that when noise is non-stationary, the conformity scores derived from quantum measurements are no longer exchangeable, even if the input data distribution is stationary. Consequently, the theoretical coverage guarantees of standard QCP (i.e., that the true outcome lies within the prediction set with probability $1-\alpha$ ) are invalidated.

Methodology

To address the breakdown of exchangeability caused by time-dependent noise, the authors propose Adaptive Quantum Conformal Prediction (AQCP). This approach adapts the Adaptive Conformal Inference (ACI) framework (Gibbs & Candès, 2021) to the quantum setting.

Core Algorithm

AQCP operates in an online testing setting where the miscoverage level is dynamically adjusted based on observed empirical coverage. The algorithm proceeds as follows:

Initialization: A calibration dataset $D_{cal}$ is used to compute initial conformity scores.
Sequential Processing: For each test point $i$ , the algorithm constructs a prediction set $C_i(X_{n+i})$ using a current miscoverage level $\alpha_i$ .
Feedback Loop: After the true label $Y_{n+i}$ is revealed, the algorithm updates the miscoverage level:
$\alpha_{i+1} = \alpha_i + \gamma (\alpha - \text{err}_i)$
where $\gamma$ is a step-size hyperparameter and $\text{err}_i$ is 1 if the true label is outside the prediction set, and 0 otherwise.
Score Functions: Since quantum models produce stochastic samples (shots) rather than deterministic outputs, AQCP utilizes sample-based score functions. The paper evaluates four specific functions:
- Euclidean Distance: Distance between the candidate $y$ and the mean of the shot samples.
- k-Nearest Neighbour (k-NN): Distance to the $k$ -th nearest neighbor among the samples.
- Kernel Density Estimation (KDE): Negative log-density estimate at $y$ .
- High-Density Region (HDR): Estimated probability mass of the region where density is at least as high as at $y$ .

Theoretical Guarantees

Unlike standard CP which offers finite-sample guarantees under exchangeability, AQCP provides asymptotic average coverage guarantees under arbitrary noise conditions. The authors prove that as the number of test points $N \to \infty$ , the average miscoverage rate converges almost surely to the target $\alpha$ . A finite-sample bound is also derived under specific assumptions regarding hidden Markov chains governing the noise, though the authors note these assumptions may not hold exactly in practice.

Key Contributions

The paper makes three primary contributions:

Formalisation of Time Dependence: The authors provide a theoretical framework demonstrating how non-stationary noise in quantum processors invalidates the exchangeability of conformity scores, thereby undermining standard conformal guarantees.
Adaptive Algorithm (AQCP): They introduce AQCP, an algorithm that utilizes online recalibration to handle non-stationary noise. It is designed to work with score functions operating on samples from implicit probability distributions (quantum shots).
Comprehensive Hardware Evaluation: The study conducts extensive experiments on the IBM Quantum Processor (ibm_sherbrooke) and a classical simulator. It analyzes the validity and efficiency of AQCP, comparing various score functions and demonstrating that AQCP maintains target coverage levels with greater stability than standard QCP.

Experimental Results

The experiments were conducted on a univariate multimodal regression task, replicating the setup of Park & Simeone (2023).

Coverage Stability: Using data from the IBM quantum processor, the authors compared AQCP (with step size $\gamma = 0.03$ $γ = 0.03$ ) against "online QCP" (equivalent to AQCP with $\gamma = 0$ $γ = 0$ , i.e., no feedback adjustment).
- Online QCP exhibited significant deviations from the target coverage (e.g., over-coverage followed by under-coverage), failing to maintain the nominal 90% level consistently.
- AQCP demonstrated robust stability, consistently maintaining average coverage close to the 90% target across all tested score functions (k-NN, Euclidean, KDE, HDR).
Efficiency (Set Size): Using a classical simulator (FakeQuitoV2) to isolate the effects of shot numbers and score functions:
- KDE and HDR score functions produced the smallest average prediction set sizes, particularly as the number of shots ( $M$ ) increased.
- Euclidean and 1-NN score functions resulted in significantly larger prediction sets (1.5–2 times larger at $M=1000$ ) compared to HDR, despite achieving the same coverage.
- All methods achieved the target coverage, but the choice of score function significantly impacted the efficiency (size) of the prediction sets.

Significance and Claims

The paper claims that AQCP provides a reliable representation of uncertainty for QML models under arbitrary hardware noise conditions, a critical requirement for the trustworthy deployment of quantum algorithms.

The authors are modest in their claims, acknowledging that:

The theoretical guarantees are asymptotic rather than finite-sample (except under specific, potentially restrictive assumptions).
The empirical evaluation of robustness is based on data collected over a single day, meaning it does not fully characterize long-term drift or structured perturbations.
The method is inherently sequential, requiring the true response to be revealed before the next prediction, which limits its direct applicability to batch prediction settings.

Despite these limitations, the work establishes that adaptive recalibration is necessary to maintain validity in the presence of non-stationary quantum noise, offering a practical pathway for uncertainty quantification in current NISQ-era applications.