The quantum Wasserstein distance (W-distance) is a fundamental metric for quantifying the distinguishability of quantum operations, with critical applications in quantum error correction. However, computing the W-distance remains computationally challenging for multiqubit systems due to exponential scaling. We present a machine learning framework that efficiently predicts the quantum W-distance by extracting physically meaningful features from quantum state pairs, including Pauli measurements, statistical moments, quantum fidelity, and entanglement measures. Our approach employs both classical neural networks and traditional machine learning models. On three-qubit systems, the best-performing Random Forest model achieves near-perfect accuracy ($R^2 = 0.9999$) with mean absolute errors on the order of $10^{-5}$. We further validate the framework's practical utility by successfully verifying two fundamental theoretical propositions in quantum information theory: the bound on measurement probability differences between unitary operations and the $W_1$ gate error rate bound. The results establish machine learning as a viable and scalable alternative to traditional numerical methods for W-distance computation, with particular promise for real-time quantum circuit assessment and error correction protocol design in NISQ devices.
The quantum Wasserstein distance (W-distance) is a fundamental metric for quantifying the distinguishability of quantum operations, with critical applications in quantum error correction. However, computing the W-distance remains computationally challenging for multiqubit systems due to exponential scaling. We present a machine learning framework that efficiently predicts the quantum W-distance by extracting physically meaningful features from quantum state pairs, including Pauli measurements, statistical moments, quantum fidelity, and entanglement measures. Our approach employs both classical neural networks and traditional machine learning models. On three-qubit systems, the best-performing Random Forest model achieves near-perfect accuracy (R2=0.9999) with mean absolute errors on the order of 10−5. We further validate the framework's practical utility by successfully verifying two fundamental theoretical propositions in quantum information theory: the bound on measurement probability differences between unitary operations and the W1 gate error rate bound. The results establish machine learning as a viable and scalable alternative to traditional numerical methods for W-distance computation, with particular promise for real-time quantum circuit assessment and error correction protocol design in NISQ devices.
Quantifying the effect of noise on unitary operations is an essential task in quantum information processing. We propose the quantum Wasserstein distance between unitary operations, which shows an explanation for quantum circuit complexity and characterizes the local distinguishability of multiqudit operations. We show analytical calculations of the distance between identity and widely used quantum gates including swap, cnot, and other controlled gates. As an application, we estimate the closeness between quantum gates in a circuit and show that the noisy operation simulates the ideal one well when they become close under the distance. Further, we introduce the W1 error rate by the distance and establish the relation between the W1 error rate and two practical cost measures of recovery operations in quantum error correction under typical noise scenarios. These applications allow the distance to quantify the effect of noise on unitary operations from the perspective of experiment resources, which cannot be achieved by existing distance.
This study investigates the application of machine learning techniques for predicting gate depth in quantum computational systems, focusing on the relationship between quantum circuit parameters and computational performance. Using a dataset of 50 observations encompassing qubit count, gate depth, entanglement rate, and logical error rate, we developed and compared Linear Regression (LR) and Random Forest Regression (RFR) models for gate depth prediction. The research addresses critical challenges in quantum fault tolerance and circuit optimization, which are essential for achieving scalable quantum computation. Our analysis revealed strong negative correlation (-0.65) between qubit count and logical error rate, while gate depth showed moderate positive correlation (0.62) with error rates. The Random Forest model demonstrated superior performance with R² values of 0.96068 on training data and 0.87262 on testing data, significantly outperforming Linear Regression (R²: 0.87552 training, 0.60257 testing). Error metrics further confirmed RFR's superiority, achieving RMSE of 2.33 (training) and 2.97 (testing) compared to LR's 4.15 and 5.25 respectively. The study highlights the importance of quantum error correction mechanisms and the inherent trade-offs between circuit complexity and computational reliability. Both models showed expected performance degradation from training to testing, with RFR maintaining better generalization capabilities. These findings contribute to the development of typed models for quantum mechanics in computational applications, offering insights into quantum circuit optimization and fault-tolerant design. The research demonstrates that ensemble methods like Random Forest provide more robust predictions for quantum system parameters, supporting the advancement of practical quantum computing implementations and algorithm development. KeyWords: Quantum Computation, Gate Depth Prediction, Random Forest Regression, Quantum Error Correction, Machine Learning, Fault Tolerance.
Distance metrics are central to machine learning, yet distances between ensembles of quantum states remain poorly understood due to fundamental quantum measurement constraints. We introduce a hierarchy of integral probability metrics, termed MMD-k, which generalizes the maximum mean discrepancy to quantum ensembles and exhibits a strict trade-off between discriminative power and statistical efficiency as the moment order k increases. For pure-state ensembles of size N, estimating MMD-k with arbitrary measurement schemes requires Θ(N1−1/k) samples for constant k. At the same time, we prove that any stable distance metric with full discriminative power admits an O(NlogN) upper bound and an Ω(N) instance-gap lower bound. For quantum Wasserstein distance, with sufficiently large fixed state dimension, we establish a nearly linear lower bound in the ensemble size at constant additive accuracy, together with an O(NlogN) upper bound. These results provide principled guidance for the design of loss functions in quantum machine learning, as we illustrate in training quantum denoising diffusion probabilistic models.
The pursuit of high-fidelity digital quantum simulation is fundamentally constrained by the inevitable presence of noise in current and near-term quantum processors. While quantum error correction ultimately promises fault-tolerance, its substantial qubit overhead renders it impractical for the noisy intermediate-scale quantum (NISQ) era. This research addresses the critical gap between hardware imperfections and simulation accuracy by designing systematic, resource-efficient machine learning (ML) error-mitigation protocols tailored specifically for digital quantum simulation. Unlike generic error-suppression methods, our approach exploits the unique structural properties of simulated quantum dynamics—including time-translation invariance, locality of interactions, and conserved quantities—to inform the training of lightweight neural networks and Gaussian process regressors. We develop a hybrid framework that combines randomized compiling with data-driven noise-inversion models, enabling real-time estimation and subtraction of both coherent and incoherent errors without exponential sampling overhead. Through extensive numerical benchmarks on spin-chain and Fermi-Hubbard models, we demonstrate that our ML protocols achieve up to an order-of-magnitude reduction in infidelity compared to standard probabilistic error cancellation and zero-noise extrapolation, while requiring fewer calibration circuits. Furthermore, we introduce an adaptive online learning scheme that continuously refines the mitigation model during the simulation runtime, automatically adjusting to time-dependent drift in hardware noise. Our results establish a practical pathway toward scalable, high-accuracy quantum simulations on existing devices, with implications for condensed matter physics, quantum chemistry, and the development of quantum advantage benchmarks.
A quantum error mitigation technique based on machine learning is proposed, which learns how to adjust the probabilities estimated by measurement in the computational basis. Neural networks in two different designs are trained with random quantum circuits consisting of a set of one- and two-qubit unitary gates whose measurement statistics in the ideal (noiseless) and real (noisy) cases are known. Once the neural networks are trained, they infer the amount of probability adjustment to be made on the measurement obtained from executing an unseen quantum circuit to reduce the error. The proposed schemes are tested with two-, three-, five-, and seven-qubit quantum circuits of depth up to 20 by computer simulations with realistic error models and experiments using the IBM quantum cloud platform. In all test cases, the proposed mitigation technique reduces the error effectively. Our method can be used to improve the accuracy of noisy intermediate-scale quantum (NISQ) algorithms without relying on extensive error characterization or quantum error correction.
Abstract Quantum key distribution (QKD) systems represent a cornerstone of secure quantum communication, relying on post-processing procedures to generate shared keys. Among these, error correction (EC) is a critical and computationally demanding step, where the estimation of the quantum bit error rate (QBER) plays a pivotal role in optimizing code rates for LDPC-based protocols. Traditional QBER estimation methods, grounded in physical models and statistics, often struggle with real-world variability and outliers, leading to suboptimal performance. In this study, we propose a novel machine learning framework for QBER forecasting, reframing the problem as a time-series prediction task. Leveraging real-world data from quantum transmitters, we evaluated a suite of models, including gradient boosting with autoregressive components, exponential smoothing hybrids, and deep neural networks, to predict QBER with greater precision. Through extensive experiments, our approach demonstrates significant improvements in error correction efficiency compared to traditional methods, particularly under challenging conditions like system calibration drifts and non-stationary signal behavior. The results highlight the potential of lightweight and interpretable machine learning models, such as gradient-boosted exponential smoothing, to outperform complex neural architectures in real-world scenarios. This research contributes to the advancement of quantum cryptography by enabling faster and more robust key generation, with reduced information disclosure during reconciliation. Our findings open new avenues for integrating machine learning into QKD systems for adaptive and scalable quantum communication.