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arXiv:2504.18429 · IEEE International Conference on Quantum Computing and Engineering (QCE)

CHSH Violations Using Dynamic Circuits

Jean-Baptiste Waring, Christophe Père, Sébastien Le Beux

Published Aug 30, 2025• 0 citations
arXivHTMLPDF

Abstract

Scalable quantum computing relies on high-quality, long-range entanglement, a challenge on noisy, near-term devices. The need for practical insights for near-term algorithm design calls for trade-offs exploration in implementing dynamic circuits on current hardware. In this work, we experimentally compare three CNOT implementations for generating Bell states across varying qubit separations on a 127 -qubit IBM Quantum Eagle processor (ibm_quebec): a unitary (SWAP-based) approach, a dynamic approach with mid-circuit measurements and classical feedforward, and a post-processed approach. We use Clauser-Horne-Shimony-Holt (CHSH) inequality violations to quantify entanglement quality. We observe that, beyond 10 qubits, dynamic circuits lead to higher$\vert S \vert$values than the unitary approach, demonstrating improved distance-dependent entanglement preservation. The post-processed approach yields the highest CHSH values, reaching$\vert S \vert >2$up to 13 qubits. Our results underscore the critical need for faster classical feedforward and higher readout fidelity.

DOI
10.1109/qce65121.2025.00263

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Dynamic circuits, which augment unitary operations with mid-circuit measurements and classical feedforward, can generate long-range entanglement in constant depth, enabling low-depth primitives ranging from nontrivial state preparation to many-qubit entangling gates. Escaping the light-cone constraints of unitary circuits, however, comes at a cost: these primitives typically require a number of mid-circuit measurements that scales with system size and that, together with feedforward latency, can introduce errors that degrade the long-range entanglement they rely on. Here, we alleviate this tension by showing that many such primitives, when cast into a common framework, admit an error-detection scheme that trades infidelity for postselection overhead with no additional ancillas. Our framework thus unifies and upgrades a broad class of primitives including fan-out gates, multi-qubit Pauli rotations, the preparation of W and higher-weight Dicke states, and certain non-normal matrix product states. We also introduce a reduced-depth, error-detected implementation of the Hadamard test, extending the use cases of dynamic circuits to a key algorithmic primitive. Finally, we establish the practical utility of our scheme through experiments on a superconducting quantum processor. We demonstrate the error-detected preparation of a long-range entangled Bell pair spanning a 100-qubit chain with fidelity F=0.59±0.02F=0.59\pm0.02F=0.59±0.02, surpassing the entanglement-certification threshold F>0.5F>0.5F>0.5 that the baseline dynamic-circuit implementation fails to reach (0.39±0.010.39\pm0.010.39±0.01). Separately, we demonstrate constant-depth preparation of W states of up to 20 qubits by consuming GHZ states of up to 40 qubits, finding absolute fidelity improvements of ΔF≈0.2ΔF\approx 0.2ΔF≈0.2 across the largest sizes studied. Altogether, these results bring low-depth dynamic-circuit primitives within practical reach on present-day hardware.
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|S| \le 2 + 6δ_{\mathrm{ens}}∣S∣≤2+6δens​, where δensδ_{\mathrm{ens}}δens​ quantifies preparation nonstationarity. Because δensδ_{\mathrm{ens}}δens​ is not directly observable, we develop an operational witness δopδ_{\mathrm{op}}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.
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.
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'.
Reliable verification of entanglement is a central requirement for quantum networks. This paper presents a practical verification approach based on violations of the Clauser-Horne-Shimony-Holt (CHSH) inequality. We derive tight mathematical bounds that relate the CHSH value to entanglement fidelity and introduce a statistical framework that optimizes resource usage while ensuring reliable certification. Our main contributions are: (i) fidelity bounds derived directly from the CHSH measure, which also enable nonlocality certification at sufficiently high fidelities; (ii) a sample-complexity analysis that quantifies the number of measurements required to achieve desired confidence levels for the CHSH measure and the entanglement fidelity; and (iii) verification protocols, some with rigorous mathematical guarantees and others with numerical evaluation. Using NetSquid, we develop a simulation framework that models diverse network conditions and enables systematic exploration of trade-offs in CHSH-based verification. This framework highlights the interplay between accuracy, efficiency, and operational parameters, providing concrete guidelines for deploying entanglement verification in resource-constrained quantum networks.
Abstract Quantum teleportation transfers an unknown qubit state from a sender to a receiver using a shared entangled pair and a classically conditioned correction without moving the physical carrier. On the dynamic-circuit hardware, the complete protocol (which includes mid-circuit measurement and conditional feedforward) now runs on a single circuit. We use this to benchmark three-qubit teleportation on a 156-qubit IBM Heron r2 processor (ibm_fez). A single-qubit RY(θ) input was teleported at six angles along a meridian of the Bloch sphere and reconstructed by state tomography alongside two ablation controls; one without entanglement and the other without feedforward. The full protocol returns a mean fidelity of about 0.945, nearly constant across the angles and above the 2/3 full-Bloch-sphere reference at all six sampled angles. Removing feedforward lowered the fidelity to about 0.51, close to a maximally mixed state. On the other hand, removing entanglement yields (1 + cos²θ)/2 with a mean near 0.75, transferring the population but losing the coherence. Together, these results establish that entanglement and feedforward are both necessary. The calibration-limited fidelity serves as a reference for this class of hardware. A separate run returned a Bell-state fidelity value of 0.957. A branch-resolved analysis confirmed that the four measurement branches are Pauli rotations of the target realigned by feedforward. The reported fidelity depends only on the rotation angle and the device noise.