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arXiv:2501.01434 · European Journal of Physics

Simulating Bell inequalities with Qibo

Isabella Masina, Giuseppe Lo Presti, Matteo Robbiati, Michele Grossi

Published Apr 15, 2025• 0 citations
arXivHTMLPDF

Abstract

Abstract We present educational material about Bell inequalities in the context of quantum computing. In particular, we provide software tools to simulate their violation, together with a guide for classroom discussion. The material is organized in three modules of increasing difficulty, and the relative implementation has been written in Qibo, an open-source software suite to simulate quantum circuits with the ability to interface with quantum hardware. The topic of inequalities allows us to not only introduce undergraduate or graduate students to crucial theoretical issues in quantum mechanics, such as entanglement, correlations, hidden variables and non-locality, but also to practically put hands on tools to implement a real simulation, where statistical aspects and noise coming from current quantum chips also come into play.

DOI
10.1088/1361-6404/adcd13

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Abstract We present educational material about Bell inequalities in the context of quantum computing. In particular, we provide software tools to simulate their violation, together with a guide for classroom discussion. The material is organized in three modules of increasing difficulty, and the relative implementation has been written in Qibo, an open-source software suite to simulate quantum circuits with the ability to interface with quantum hardware. The topic of inequalities allows us to not only introduce undergraduate or graduate students to crucial theoretical issues in quantum mechanics, such as entanglement, correlations, hidden variables and non-locality, but also to practically put hands on tools to implement a real simulation, where statistical aspects and noise coming from current quantum chips also come into play.
Abstract We present Qibo , a new open-source software for fast evaluation of quantum circuits and adiabatic evolution which takes full advantage of hardware accelerators. The growing interest in quantum computing and the recent developments of quantum hardware devices motivates the development of new advanced computational tools focused on performance and usage simplicity. In this work we introduce a new quantum simulation framework that enables developers to delegate all complicated aspects of hardware or platform implementation to the library so they can focus on the problem and quantum algorithms at hand. This software is designed from scratch with simulation performance, code simplicity and user friendly interface as target goals. It takes advantage of hardware acceleration such as multi-threading Central Processing Unit (CPU), single Graphics Processing Unit (GPU) and multi-GPU devices.
The evolution of quantum computing has created a pressing need to rethink how we introduce fundamental concepts in quantum mechanics and computation in education. Traditional classroom methods, based on classical teaching and theoretical ideas, often fail to effectively convey the unpredictable and strange nature of quantum systems. Simulating quantum environments offers a powerful strategy. It makes quantum principles more accessible, interactive, and engaging for learners at various educational levels. These simulated platforms, which include visual circuit builders and programmable virtual quantum computers, allow students to explore qubits, entanglement, and quantum gates without needing costly or experimental quantum hardware. This chapter examines the role of simulated quantum environments in reshaping STEM education, particularly in secondary and higher education. It starts by discussing the theoretical challenges of teaching quantum concepts, pointing out the limitations of classical analogies and rote calculations in promoting real understanding. Simulation tools help address these issues by providing hands-on, inquiry-based learning.
Abstract This laboratory work presents a practical introduction to the preparation and analysis of entangled quantum states (Bell pairs) using the Qiskit framework. The material is based on educational resources developed by the International Quantum Center at CERN (Switzerland) and is designed to provide hands-on experience with multi-qubit quantum circuits and entanglement generation. In Laboratory Work No. 3, participants construct and simulate a quantum circuit that prepares an entangled two-qubit state. As a representative example of multi-qubit operations in Qiskit, the circuit generates a Bell state by applying a Hadamard (H) gate to the first qubit q[0]q[0]q[0], followed by a controlled-NOT (CNOT) gate with q[0]q[0]q[0] as the control qubit and q[1]q[1]q[1] as the target qubit. This sequence creates a maximally entangled state, demonstrating the fundamental non-classical correlations characteristic of quantum systems. The laboratory exercise includes measurement operations applied to both qubits, illustrating how entanglement manifests in correlated classical outcomes. When the measurement operator is applied to the first qubit q[0]q[0]q[0], the corresponding classical bit records a definite value, and subsequent measurement of the second qubit q[1]q[1]q[1] yields a correlated result consistent with the structure of the Bell state. The probabilistic yet correlated outcomes provide direct evidence of quantum entanglement. The guide emphasizes the conceptual understanding of entanglement, circuit implementation in Qiskit, and the hybrid quantum-classical workflow underlying quantum computation. By bridging theoretical quantum mechanics with executable quantum circuits, this laboratory work enables students and researchers to experimentally explore one of the central phenomena of quantum information science using cloud-based quantum computing resources.
The predictions of quantum mechanics cannot be resolved with a completely classical view of the world. In particular, the statistics of space-like separated measurements on entangled quantum systems violate a Bell inequality. We put forward a computational perspective on a broad class of Bell tests that study correlators, or the statistics of joint measurement outcomes. We associate particular maps, or functions to particular theories. The violation of a Bell inequality then implies the ability to perform some functions, or computations that classical, or more generally, local hidden variable (LHV) theories cannot.
Written for a two-semester graduate course in Quantum Mechanics, this comprehensive text helps develop the tools and formalism of Quantum Mechanics and its applications to physical systems. It suits students who have taken some introductory Quantum Mechanics and Modern Physics courses at undergraduate level, but it is self-contained and does not assume any specific background knowledge beyond appropriate fluency in mathematics. The text takes a modern logical approach rather than a historical one and it covers standard material, such as the hydrogen atom and the harmonic oscillator, the WKB approximations and Bohr-Sommerfeld quantization. Important modern topics and examples are also described, including Berry phase, quantum information, complexity and chaos, decoherence and thermalization, nonstandard statistics, as well as more advanced material such as path integrals, scattering theory, multiparticles and Fock space. Readers will gain a broad overview of Quantum Mechanics, as solid preparation for further study or research.
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.