Adaptive Quantum Conformal Prediction

• Adaptive Quantum Conformal Prediction (AQCP) is a method that adjusts prediction intervals in quantum machine learning to counter non-stationary hardware noise.
• It uses an adaptive feedback mechanism to update the miscoverage parameter in real time, ensuring asymptotic coverage and robust performance across various score functions.
• Empirical tests on IBM superconducting processors show AQCP stabilizes coverage near nominal levels, outperforming static quantum conformal methods.

Adaptive Quantum Conformal Prediction (AQCP) is an uncertainty quantification algorithm designed for quantum machine learning scenarios where time-varying hardware noise precludes classical conformal guarantees. By incorporating adaptive feedback mechanisms from conformal inference, AQCP maintains asymptotic coverage validity even under arbitrary non-stationary noise, as empirically documented on IBM’s superconducting quantum processors (Spencer et al., 23 Nov 2025).

1. Quantum Prediction and Classical Conformal Guarantees

In quantum machine learning, prediction relates typically to inferring a real-valued target $Y_{n+1}$ from classical or quantum features $X_{n+1}$, given i.i.d. training pairs $(X_1, Y_1), \ldots, (X_n, Y_n) \sim P_{X,Y}$. A parametrised quantum circuit (PQC) $U(\theta)$ models $Y$ as a distribution generated by applying $U(\theta)$ to measurement samples at times $T = \{ t_1, \ldots, t_M \}$, yielding a multiset $$\mathcal A_{x,T} = \{ \hat y_1, \ldots, \hat y_M \}$$ which is mapped to targets via a task mapping $f: \{0,1\}^Q \to \mathcal Y$.

Conformal prediction algorithms operate by calibrating a statistical score function $S : (x, y; \mathcal A_{x,T}) \to \mathbb R$ on a set of calibration scores $s_i = S(X_i, Y_i; \mathcal A_{X_i, T_i})$. Assuming exchangeability in $(X_i, Y_i)$ and permutation invariance in $S$, the standard conformal approach produces a coverage guarantee:

$P[Y_{n+1} \in C(X_{n+1})] \geq 1-\alpha,$

where the prediction set is $C(x) = \{ y : S(x, y) \leq \lambda \}$ and $\lambda$ is chosen as the $\lceil (n+1)(1-\alpha) \rceil/n$ quantile of $\{s_1, \ldots, s_n, \infty \}$.

2. Breakdown of Exchangeability under Quantum Hardware Noise

On practical noisy intermediate-scale quantum (NISQ) devices, hardware noise is not static. Each gate and measurement is accompanied by a time-indexed CPTP map $\mathcal E_t$, so the observed noisy state for input $x$ at time $t$ is:

$$\rho_{\text{noisy}}(x, t) = \mathcal E_{t, N_G} \circ \cdots \circ \mathcal E_{t, 1} (|0\rangle\langle 0|).$$

The measurement outcomes $\mathcal A_{x, T}$ thus depend on time $T$, causing the augmented tuple $Z_i = (X_i, Y_i; \mathcal A_{X_i, T_i})$—and consequently the conformity scores $S_i = S(Z_i)$—to lose exchangeability. The implication is that the rank of $S_{n+1}$ among $\{S_1, \ldots, S_{n+1}\}$ is no longer uniformly random, therefore voiding the classical marginal coverage proof for conformal prediction sets.

3. Algorithmic Structure of Adaptive Quantum Conformal Prediction

AQCP addresses this lapse of exchangeability by reacting to observed coverage errors with real-time adaptation of the miscoverage parameter $\alpha$. The workflow consists of calibration and online adaptation steps. The procedure—expressed as pseudocode in the source paper—operates as follows:

Algorithm AQCP(α₁, γ, Ŝ, D_cal, D_test, M) Input: α₁ ∈ [0,1] # initial miscoverage γ > 0 # adaptation step size Ŝ # score function (x,y;𝒜) D_cal = {(x_i,y_i)}_{i=1}^n # calibration data D_test = {(x_i,y_i)}_{i=n+1}^{n+n′} # test stream M # number of shots per query Output: {C_i(x_i)}_{i=n+1}^{n+n′} S ← ∅ for (x_i, y_i) in D_cal do T_i ← take M shots on PQC at input x_i s_i ← Ŝ(x_i, y_i; 𝒜_{x_i,T_i}) S ← S ∪ {s_i} end for α ← α₁ for i = n+1 to n+n′ do λ ← quantile_{1−α}(S ∪ {+∞}) T_i ← take M shots on PQC at x_i C_i(x_i) ← { y∈𝒴 : Ŝ(x_i, y; 𝒜_{x_i,T_i}) ≤ λ } err_i ← 1{ y_i ∉ C_i(x_i) } α ← α + γ (α₁ − err_i) s_i ← Ŝ(x_i, y_i; 𝒜_{x_i,T_i}) S ← S ∪ {s_i} end for return {C_i(x_i)}

The core update mechanism is expressed as:

$$\alpha_{i+1} = \alpha_i + \gamma(\alpha_1 - \text{err}_i),$$

where $\text{err}_i$ records miscoverage at step $i$ ($1$ if $y_i \notin C_i(x_i)$, $0$ else). The adaptation step-size $\gamma$ determines the balance between responsiveness and stability.

4. Theoretical Asymptotic Coverage Guarantees

The central theoretical result of AQCP is its asymptotic average coverage property, which guarantees that, without requiring exchangeability, the frequency of coverage errors converges to the nominal level $\alpha_1$:

$$\left| \frac{1}{N}\sum_{i=1}^N \text{err}_i - \alpha_1 \right| \leq \frac{\max\{\alpha_1, 1-\alpha_1\} + \gamma}{N \gamma}.$$

Thus,

$$\lim_{N \to \infty} \frac{1}{N} \sum_{i=1}^N \text{err}_i = \alpha_1 \quad \text{a.s.}$$

Under mild mixing conditions (scores governed by a hidden Markov model with spectral gap $1-\eta$), the finite-sample tail bound is:

$$\mathbb P \left( \left| \frac{1}{N} \sum_{i=1}^N \text{err}_i - \alpha_1 \right| \geq \varepsilon \right) \leq 2e^{-N\varepsilon^2/8} + 2\exp \left(-\frac{N(1-\eta)\varepsilon^2}{8(1+\eta)\sigma_B^2 + 20B\varepsilon}\right),$$

with $$B = \sup_b \left| \mathbb E[\text{err}_i | B_i = b] - \alpha_1 \right|$$ and $$\sigma_B^2 = \mathbb E ( \mathbb E[\text{err}_i | B_i] - \alpha_1 )^2$$, where $B_i$ denotes the hidden Markov state. The proof rests on a decomposition into martingale difference (Hoeffding) and Markov drift (Bernstein).

5. Empirical Analysis on IBM NISQ Hardware

Empirical validation uses multimodal regression as in Park & Simeone (2023): $X \sim \text{Uniform}(-10, 10)$, $$Y|X = \frac{1}{2} N(-\mu(X), 0.05^2) + \frac{1}{2} N(+\mu(X), 0.05^2)$$, with $\mu(x) = \frac{1}{2}\sin(4x/5) + x/20$. The quantum model is a 5-qubit hardware-efficient ansatz ($L=5$ layers), trained using a classical NN encoder with output mapped to PQC rotation angles. Measurement outcomes $\{ 0,1 \}^5$ map to $[-1.5, 1.5]$.

Multiple score functions are benchmarked: $k$-NN ($k = \lceil \sqrt{M} \rceil$), Euclidean distance, kernel density estimation (Gaussian kernel, Silverman bandwidth), HDR mass. Calibration uses $n=100$ points on IBM’s Sherbrooke processor; test stream contains $n'=9900$ queries, $M=100$ shots/query, with rolling-window quantiles estimated over the last $500$ scores.

The key findings are:

• With the static quantum conformal method (QCP, $\gamma=0$), coverage fluctuates around $90\%$, sometimes dipping near $80\%$ or exceeding $95\%$.
• Under AQCP ($\gamma=0.03$), moving average coverage remains tightly concentrated near the $90\%$ nominal level.
• Coverage stability improvement is consistent across all tested score functions (k-NN, Euclidean, KDE, HDR).
• On noiseless simulators, AQCP achieves target coverage with HDR/KDE scores yielding the smallest prediction sets, Euclid/k-NN generating larger sets, especially when $M$ is increased.

Score Function Performance Table

Score Function	Coverage Stability	Average Set Size
k-NN	High (AQCP)	Larger
Euclidean	High (AQCP)	Larger
KDE	High (AQCP)	Smaller
HDR	High (AQCP)	Smallest

6. Practical Considerations and Limitations

The adaptation step-size $\gamma$ is critical: small $\gamma$ yields slow adaptation, while large $\gamma$ induces high-frequency oscillations. Automated or strongly-adaptive $\gamma$ selection strategies constitute promising future work. AQCP’s requirement for online labels at each test point limits applicability to settings with frequent supervision; semi-supervised or unsupervised extensions are needed for broader utility.

Only asymptotic or HMM-based tail bounds are available for finite samples; no distribution-free finite-sample guarantee exists under arbitrary noise. Research into beyond-exchangeability frameworks—such as conformal reweighting or data-drift detection—may provide stronger bounds. Sample-based density score functions (KDE, HDR) offer optimal set sizes but at greater computational cost relative to proximity-based alternatives. Development of more efficient or hardware-aware nonconformity metrics is an ongoing concern. Potential enhancement could involve robust modeling or estimation of time-varying noise, enabling hybrid conformal-reweighting and drift-detection approaches for tighter coverage guarantees.

In summary, AQCP implements an asymptotically valid uncertainty quantification protocol for quantum machine learning on noisy hardware, restoring target coverage and coverage stability where classic quantum conformal methods fail due to non-stationary device error profiles (Spencer et al., 23 Nov 2025).