Uncertainty Quantification for Quantum Fidelity Estimation

The gist The hybrid “Bayesian-informed” conformal prediction-based framework provides distribution-free, rigorous prediction intervals for quantum state fidelity estimation which can be used, with care, to approximate confidence intervals with respect to the quantum randomness of Born’s rule<ref:2610.10885#pg5>.

Introduction and Motivation

Quantum fidelity estimation is a fundamental task in quantum information and computing to benchmark the quality of quantum states — and hence of quantum processes — on noisy quantum hardware<ref:2610.10885#pg4>. Point estimates of fidelity by existing methods generally lack the rigorous uncertainty quantification required for reliable preparation and verification of these tasks<ref:2610.10885#pg4>. While Bayesian and bootstrap frameworks are widely used for uncertainty quantification, they are sensitive to model misspecification or lack formal coverage guarantees in non-asymptotic regimes<ref:2610.10885#pg4>. The paper introduces a hybrid “Bayesian-informed” conformal prediction-based framework to provide distribution-free, rigorous prediction intervals for quantum state fidelity estimation which can be used, with care, to approximate confidence intervals with respect to the quantum randomness of Born’s rule<ref:2610.10885#pg4>.

Methodology Overview

The proposed methodology combines Bayesian assumptions with split conformal prediction to generate prediction intervals, and then we discuss and use simulations to empirically validate using these intervals as confidence intervals for fidelity estimation<ref:2610.10885#pg4>. The framework requires a point estimator, and we consider both classical point estimators as well as data-driven neural net estimators<ref:2610.10885#pg4>.

The core of the approach involves defining an “instance” by a realization of a quantum state ρ ∈ Hd, and the corresponding input feature vector X ∈ X = RM is the set of empirical frequencies f = ⌉fkM k=1 collected across the Pauli POVMs<ref:2610.10885#pg4>. The corresponding target Y ∈ Y = [0, 1] is the true quantum state fidelity, calculated between the state ρ and the pure reference state σ ∈ Hd<ref:2610.10885#pg4>.

Bayesian-Informed Conformal Prediction

The framework aims to obtain a stateconditional coverage guarantee for a specific, fixed underlying state ρ: PXn+1 ∼ L(X ρ) Yn+1 ∼ δ⟨σ,ρ⟩ Yn+1 ∈ Cˆ n(Xn+1) ρ ≥ 1 − α<ref:2610.10885#pg4>. Since exact distribution-free conditional guarantees are mathematically impossible to satisfy for all individual states ρ in finite samples, the marginal conformal prediction framework guarantees validity on average over a joint datagenerating distribution P: P D1 i.i.d. ∼ P D2 i.i.d. ∼ P (Xn+1, Yn+1) i.i.d ∼ P Yn+1 ∈ Cˆ n(Xn+1) ≥ 1 − α<ref:2610.10885#pg4>. This joint distribution is given by P(ρ, X, Y) = π(ρ)P(X ρ)P(Y ρ, X)δ(Y − ⟨σ, ρ⟩)<ref:2610.10885#pg4>. The hybrid strategy wraps these Bayesian priors within the conformal framework<ref:2610.10885#pg4>.

Fidelity Estimation Models

The model Fˆ n1 can be any of the methods described in Sec. II B<ref:2610.10885#pg4>. The methods considered include:

1. Plugin estimators, which estimate ρˆ from the instance and then return Fˆ(X) = F(σ, ρˆ)<ref:2610.10885#pg4>.

2. PLS Projected least-squares (PLS) estimates ρˆ by first solving the linear least-squares problem minρ PM i=1 (⟨Wk, ρ⟩ − p˜k) where ˜pk = 2fk − 1<ref:2610.10885#pg4>.

3. MLE Maximum likelihood estimation (MLE) is similar to PLS but uses a convex non-quadratic objective function based on the likelihood, and explicitly enforces the physical constraints ρ = ρ†, ρ⪰ 0,tr(ρ) = 1 during the optimization<ref:2610.10885#pg4>.

4. Neural Net Architecture and Training involves employing a shallow feed-forward neural network, with a single hidden layer of 256 neurons, which maps samples X ∈ X = RM to estimates of fidelity Y ∈ Y = [0, 1] and is trained on the data in D1 using the squared loss function<ref:2610.10885#pg4>.

Variants of Conformal Prediction

The paper evaluates several variants of Algo. 1, including:

** Adaptive split conformal (local) NN, which uses residuals from the prediction of the first model as targets/labels to scale the original score to incorporate the variance in the data as S˜i = Si rˆ(Xi)<ref:2610.10885#pg4>. The Quantile Neural Net (Non-Conformal), which estimates lower and upper fidelity quantiles directly using a two-output neural network trained by minimizing the pinball loss<ref:2610.10885#pg4>. The Asymmetric CQR Neural Net, which applies conformal calibration to the lower and upper quantile estimates using D2 as described in §II D 2<ref:2610.10885#pg4>. The Y-conditioned SC method, which partitions the calibration dataset according to the observed variable X and independently applies the conformal prediction procedure to each partition<ref:2610.10885#pg4>. The Y-conditioned split conformal method is better at achieving approximate state-conditional validity<ref:2610.10885#pg4>.**

Validation and Efficiency

The results show that all conformalized methods successfully maintain the target coverage level of 1 −α = 0.9, independent of the number of shots N, as expected by the theory<ref:2610.10885#pg4>. Regarding efficiency, NN-based CP methods yield comparable interval lengths<ref:2610.10885#pg4>. The observed superiority of NN-based intervals can be attributed to the relationship between interval length and estimator variance, as the neural network implicitly adopts the distribution as an inductive bias, leading to more concentrated residuals and therefore, allowing the conformal framework to produce tighter bounds<ref:2610.10885#pg4>.

Robustness Analysis

The study evaluates robustness across various experimental challenges:

** Model Misspecification: When comparing Bayesian methods with standard CP using a 1-D line prior, the Bayesian approach increasingly undercovers in terms of coverage and does not attempt to adjust the interval length in order to compensate<ref:2610.10885#pg4>. In contrast, our CP method (using a NN prediction model) achieves target coverage for different levels of model misspecification with slight over-coverage for ε > 0<ref:2610.10885#pg4>. The strength of the conformal prediction framework is precisely its ability to bypass the need for explicit parametric modeling, leveraging empirical calibration data to provide rigorous safety guarantees even when the underlying physical processes are poorly understood<ref:2610.10885#pg4>. **

** Finite-sample effects: Bootstrap techniques exhibit significant undercoverage alongside excessively wide interval lengths at lower shot count N, but as N becomes practically large, the empirical coverage starts to converge to the target level while the corresponding interval length systematically reduces<ref:2610.10885#pg4>. **

** Incomplete Measurements: When subsampling Pauli POVMs, conformalbased methods roughly achieve the target coverage level of 90% independent of the POVM subsampling, but the interval lengths are much wider in the undersampled regime, reflecting the lower information<ref:2610.10885#pg4>. **

** Conditional Coverage: The Y-conditioned split conformal method demonstrates remarkable resilience in the mid-to-high fidelity regimes (η ≳ 0.5) when tested under a depolarizing noise model, successfully restoring and maintaining the nominal target coverage where the standard CP approach fails<ref:2610.10885#pg4>. **

The paper concludes that this framework provides experimentalists with a scalable, model-agnostic, and mathematically rigorous “safety wrapper” for quantum hardware validation<ref:2610.10885#pg4>. The core contribution was to address the gap between the marginal guarantees provided by standard conformal prediction and the state-conditional coverage demanded by experimentalists<ref:2610.10885#pg4>. The Y-conditioned split conformal method is better at achieving approximate state-conditional validity by conditioning the calibration architecture directly on the target fidelity space<ref:2610.10885#pg4>. This approach dynamically adapts to the local structure of the Hilbert space, ensuring robust coverage for specific physical state realizations<ref:2610.10885#pg4>.

1. The framework requires a point estimator, and we consider both classical point estimators as well as data-driven neural net estimators. This enables the AI system to be flexible; it can integrate any existing fidelity estimation method—from Plugin estimators like PLS or MLE to data-driven neural networks—into the rigorous uncertainty quantification pipeline.

2. The system can generate distribution-free, rigorous prediction intervals for quantum state fidelity estimation which can be used, with care, to approximate confidence intervals with respect to the quantum randomness of Born’s rule. This allows experimentalists to make more informed downstream decisions without having to trust the results blindly by providing mathematically guaranteed coverage.

3. The hybrid strategy wraps these Bayesian priors within the conformal framework allowing it to investigate and optimize the state-conditional adaptivity of our estimator. This means the AI system can be tuned, using physics knowledge as a prior (e.g., assuming noise is depolarizing), to maintain coverage even under out-of-distribution physical noise.

4. The implementation of a Y -conditioned split conformal prediction framework ensures that the system maintains nominal target coverage across the majority of the fidelity bins, preventing systematic over- or under-coverage for specific classes. This provides robust local reliability, specifically in scenarios where experimental states might fall into different fidelity regimes.

5. The system can utilize a neural network trained to map measurement statistics directly to fidelity, learning the mapping from statistical measurement outcomes to the state fidelity without needing an intermediate state reconstruction estimate. This makes it highly efficient for high-dimensional problems, as the architecture is shown to be highly robust across different system sizes.