Bell or Clauser-Horne-Shimony-Holt (CHSH) tests on superconducting quantum processors are commonly interpreted under the assumption that repeated circuit executions sample a single, stationary preparation ensemble. Here we show that this assumption can be violated on contemporary hardware, with direct implications for the interpretation of observed Bell violations. We introduce an ensemble-divergence framework in which slow temporal drift of the preparation process induces context-dependent effective ensembles, even when measurement independence and locality are preserved. This leads to a relaxed Bell bound $|S| \le 2 + 6δ_{\mathrm{ens}}$, where $δ_{\mathrm{ens}}$ quantifies preparation nonstationarity. Because $δ_{\mathrm{ens}}$ is not directly observable, we develop an operational witness $δ_{\mathrm{op}}$ based on bin-resolved outcome statistics for fixed measurement channels. Using Pauli-axis measurements on IBM superconducting processors, we observe statistically significant operational drift that persists after full two-qubit readout mitigation, ruling out measurement artifacts. In contrast, drift extracted from CHSH-optimal measurements is eliminated by mitigation, demonstrating that such settings are unsuitable for diagnosing preparation nonstationarity. We further show that the observed Bell violations imply only modest ensemble divergences, comparable in scale to those required in Hall-type measurement-dependence models, but arising here solely from preparation drift combined with experimental scheduling. Our results identify a preparation-dependent loophole relevant to Bell tests on noisy intermediate-scale quantum devices and highlight the necessity of drift-aware protocols for reliable quantum certification.
Bell or Clauser-Horne-Shimony-Holt (CHSH) tests on superconducting quantum processors are commonly interpreted under the assumption that repeated circuit executions sample a single, stationary preparation ensemble. Here we show that this assumption can be violated on contemporary hardware, with direct implications for the interpretation of observed Bell violations. We introduce an ensemble-divergence framework in which slow temporal drift of the preparation process induces context-dependent effective ensembles, even when measurement independence and locality are preserved. This leads to a relaxed Bell bound ∣S∣≤2+6δens, where δens quantifies preparation nonstationarity. Because δens is not directly observable, we develop an operational witness δop based on bin-resolved outcome statistics for fixed measurement channels. Using Pauli-axis measurements on IBM superconducting processors, we observe statistically significant operational drift that persists after full two-qubit readout mitigation, ruling out measurement artifacts. In contrast, drift extracted from CHSH-optimal measurements is eliminated by mitigation, demonstrating that such settings are unsuitable for diagnosing preparation nonstationarity. We further show that the observed Bell violations imply only modest ensemble divergences, comparable in scale to those required in Hall-type measurement-dependence models, but arising here solely from preparation drift combined with experimental scheduling. Our results identify a preparation-dependent loophole relevant to Bell tests on noisy intermediate-scale quantum devices and highlight the necessity of drift-aware protocols for reliable quantum certification.
Self-testing protocols enable the certification of quantum systems in a device-independent manner, i.e., without knowledge of the inner workings of the quantum devices under test. Here, we demonstrate this high standard for characterization routines with superconducting circuits, a prime platform for building large-scale quantum computing systems. We first develop the missing theory allowing for the self-testing of Pauli measurements. We then self-test Bell pair generation and measurements at the same time, performing a complete self-test in a system composed of two entangled superconducting circuits operated at a separation of 30 m. In an experiment based on 17 million trials, we measure an average CHSH (Clauser-Horne-Shimony-Holt) S value of 2.236. Without relying on additional assumptions on the experimental setup, we certify an average Bell state fidelity of at least 58.9% and an average measurement fidelity of at least 89.5% in a device-independent manner, both with 99% confidence. This enables applications in the field of distributed quantum computing and communication with superconducting circuits, such as delegated quantum computing.
Abstract This laboratory work presents a practical implementation of the CHSH game using the Qiskit framework and IBM quantum hardware. The material is based on educational resources developed by the International Quantum Center at CERN (Switzerland) and aims to provide an experimental demonstration of quantum nonlocality and Bell inequality violation. In Laboratory Work No. 4, participants implement a simplified version of the Clauser–Horne–Shimony–Holt (CHSH) game to test whether IBM quantum processors exhibit genuinely quantum behavior. The CHSH game is a fundamental protocol in quantum information theory that distinguishes classical correlations from quantum correlations through statistical violation of Bell inequalities. Using Qiskit, a two-qubit entangled state is prepared and measurement settings corresponding to different basis choices are applied. By executing the circuit multiple times on a simulator and real quantum hardware, participants collect measurement statistics and compute the CHSH correlation parameter. If the experimentally obtained value exceeds the classical bound, the results provide empirical evidence of quantum entanglement and nonlocal correlations. This laboratory exercise demonstrates how foundational concepts of quantum mechanics—entanglement, superposition, and measurement basis dependence—can be directly tested using cloud-accessible quantum processors. The experiment not only reinforces theoretical understanding of Bell inequalities but also introduces practical skills in circuit construction, parameterized measurements, statistical analysis, and noise considerations in real quantum devices.
Quantum computing holds the promise of transforming computational paradigms across science and engineering. But realizing potential on near term hardware demands on understanding of the noise processes that corrupt quantum circuit outputs. On superconducting processors, fragile qubits are continuously subjected to environmental disturbances including T1 relaxation, charge noise, and magnetic flux drift that cause measurement statistics to evolve in a nonstationary manner across repeated circuit executions, undermining the reliability of any result aggregated over a long experimental session. In this work, we treat the sequence of bitstring count outcomes from repeated executions of a fixed quantum circuit as a time series and conduct a systematic statistical investigation of its structure. The first contribution is constructing a predictive model of circuit output, demonstrating that future measurement outcomes can be forecast by exploiting temporal correlations in the execution record. We then extend the analysis to two circuits sharing the same physical qubit, examining cross-series dependencies and evaluating whether the drift behavior of one circuit can inform predictions about the other. The temporal structure of the signals is characterized in depth. The other contribution of this dissertation is to perform Wavelet-based multifractal analysis to reveal that quantum circuit outputs exhibit genuine multifractality. This is a hallmark of long range correlated, scale free processes rather than the simple Gaussian white noise assumed by standard error models. Complementary frequency-domain analysis identifies a residual 1/f noise component that persists after linear detrending, consistent with charge and flux noise mechanisms well documented in superconducting qubit literature. Motivated by these findings, The other contribution is to design and evaluate a suite of drift-mitigation filters including piecewise Huber regression, rolling median, smoothing spline, and Savitzky–Golay approaches under a rigorous chronological train/test protocol that mimics real operational deployment. The smoothing spline and median filter achieve near-zero corrected bias, while the rolling median emerges as the preferred solution for online, shot-by-shot correction due to its causal update structure and maximal robustness to impulsive hardware events. Taken together, our results characterize quantum circuit outputs as non-stationary, multifractal processes exhibiting long memory, seasonality, and inter-circuit causality, and establish a principled filtering framework for mitigating temporal bias drift in NISQ era quantum hardware.
Bell tests, as primitive tools to detect nonlocality in bipartite systems, rely on an assumption, i.e., measurement independence. Since ensuring measurement independence in a practical Bell test is difficult, it is crucial to explore the effects of relaxing this assumption. Recently, in the simplest Clauser-Horne-Shimony-Holt Bell (CHSH-Bell) test, Koh et al. [Phys. Rev. Lett. 109, 160404 (2012)] built the relation among measurement dependence, guessing probability, and the maximum value of CHSH-Bell correlation function that an adversary (Eve) can fake. As well, Pope et al. [Phys. Rev. A 88, 032110 (2013)] and Yuan et al. [Phys. Rev. A. 91, 032111 (2015)] settled the same problem in the multiple-run scenario with the general input distribution and the factorizable one, respectively. However, pertinent results in the generalized CHSH-Bell test are still missing. Here, we study this problem and establish the relation among measurement dependence, guessing probability, and the maximum value of the generalized CHSH-Bell correlation function that Eve can fake. Furthermore, we also consider the multiple-run scenario and show the relations in both input distributions. Interestingly, compared with the simplest CHSH-Bell test, we find that it is more difficult for Eve to fake a violation in the generalized CHSH-Bell test in some special cases. We expect that our conclusions will serve as a reference for quantum information processing tasks such as quantum key distribution, randomness expansion, and other tasks in the device-independent framework.
We experimentally study error mitigation and finite-shot sampling on superconducting quantum hardware under a constrained execution budget. The study combines calibration-aware qubit selection, circuit-depth scaling, zero-noise extrapolation, dynamical decoupling, readout-error mitigation, and repeated-shot estimation on an IBM Quantum processor. Experiments are organized across ideal simulation, noise-model simulation, and physical-device execution to separate sampling uncertainty from device-induced error. We investigate how mitigation performance changes with circuit depth, effective noise scale, qubit connectivity, and measurement budget, and quantify accuracy using expectation-value error, mean-squared error, statistical uncertainty, and mitigation gain. A fixed hardware-execution budget is used to evaluate shot allocation strategies and repeated measurements without relying on unlimited sampling. The resulting analysis provides a hardware-aware characterization of when mitigation improves expectation-value estimation and when finite-shot fluctuations offset the benefit of additional mitigation overhead. The implementation uses contemporary Qiskit and IBM Quantum Runtime workflows and is designed to provide reproducible experimental evidence for error-mitigation studies on current quantum processors.
This work provides quantitative tests of the extent of violation of two inequalities applicable to qubits coupled into Bell states, using IBM's publicly accessible quantum computers. Violations of the inequalities are well established. Our purpose is not to test the inequalities, but rather to determine how well quantum mechanical predictions can be reproduced on quantum computers, given their current fault rates. We present results for the spin projections of two entangled qubits, along three axes A, B, and C, with a fixed angle θ between A and B and a range of angles θ' between B and C. For any classical object that can be characterized by three observables with two possible values, inequalities govern relationships among the probabilities of outcomes for the observables, taken pairwise. From set theory, these inequalities must be satisfied by all such classical objects; but quantum systems may violate the inequalities. We have detected clear-cut violations of one inequality in runs on IBM's publicly accessible quantum computers. The Clauser-Horne-Shimony-Holt (CHSH) inequality governs a linear combination S of expectation values of products of spin projections, taken pairwise. Finding S > 2 rules out local, hidden variable theories for entangled quantum systems. We obtained values of S greater than 2 in our runs prior to error mitigation. To reduce the quantitative errors, we used a modification of the error-mitigation procedure in the IBM documentation. We prepared a pair of qubits in the state |00〉, found the probabilities to observe the states |00〉, |01〉, |10〉, and |11〉 in multiple runs, and used that information to construct the first column of an error matrix M. We repeated this procedure for states prepared as |01〉, |10〉, and |11〉 to construct the full matrix M, whose inverse is the filtering matrix. After applying filtering matrices to our averaged outcomes, we have found good quantitative agreement between the quantum computer output and the quantum mechanical predictions for the extent of violation of both inequalities as functions of θ'.