Description This laboratory work presents a practical and theoretical study of Pauli matrices (X, Y, Z) and their fundamental role in quantum teleportation implemented on IBM Quantum hardware. The work explores the mathematical structure of single-qubit operations in a two-dimensional Hilbert space and their geometric interpretation as rotations on the Bloch sphere. A Bell state is prepared using Hadamard and CNOT gates, forming the entangled resource required for teleportation. The protocol is analyzed in matrix form, demonstrating how measurement outcomes correspond to Pauli correction operators (I, X, Z, XZ). The study emphasizes that quantum teleportation represents controlled amplitude and phase correction based on classical measurement results rather than physical transfer of matter. Special attention is given to the applied implementation on IBM Quantum, where Pauli operators are realized through physical microwave pulses corresponding to exponentials of SU(2) generators. While the work operates at the level of quantum circuits, it highlights the underlying algebraic structure of the Lie algebra su(2), which governs qubit rotations and quantum state evolution. The report integrates mathematical formalism, circuit-level realization, and physical interpretation, providing a coherent understanding of the role of Pauli operators in quantum information processing.
Description This laboratory work presents a practical and theoretical study of Pauli matrices (X, Y, Z) and their fundamental role in quantum teleportation implemented on IBM Quantum hardware. The work explores the mathematical structure of single-qubit operations in a two-dimensional Hilbert space and their geometric interpretation as rotations on the Bloch sphere. A Bell state is prepared using Hadamard and CNOT gates, forming the entangled resource required for teleportation. The protocol is analyzed in matrix form, demonstrating how measurement outcomes correspond to Pauli correction operators (I, X, Z, XZ). The study emphasizes that quantum teleportation represents controlled amplitude and phase correction based on classical measurement results rather than physical transfer of matter. Special attention is given to the applied implementation on IBM Quantum, where Pauli operators are realized through physical microwave pulses corresponding to exponentials of SU(2) generators. While the work operates at the level of quantum circuits, it highlights the underlying algebraic structure of the Lie algebra su(2), which governs qubit rotations and quantum state evolution. The report integrates mathematical formalism, circuit-level realization, and physical interpretation, providing a coherent understanding of the role of Pauli operators in quantum information processing.
We formulate the standard quantum teleportation protocol within the framework of deformation quantization, employing the Stratonovich--Weyl (SW) correspondence and the associated star product on the coadjoint orbits of SU(2). Each qubit is represented by a spherical phase space S2 endowed with the Kirillov--Kostant--Souriau (KKS) symplectic structure. Within this setting, we explicitly construct the phase space representations of the entangled Bell resource, the controlled-NOT and Hadamard operators, the projective measurements, and the conditional Pauli corrections. We demonstrate how the quantum information initially encoded in the Wigner function for the input qubit is redistributed through bipartite and tripartite phase space correlations. Upon consideration of a conditional unitary transformation, the measurement outcome produces a Wigner function on the receiver's coadjoint orbit that is related to the input by an outcome-dependent rotation, which can subsequently be inverted through the implementation of an appropriate Pauli correction. The covariance of the SW correspondence provides a geometric interpretation of this process, in which quantum information is redistributed and reconstructed without transferring the underlying symplectic geometry. Our formulation can also be extended to study quantum teleportation in more general phase space geometries equipped with suitable star products.
Abstract In the field of quantum computing, the study of single-qubit quantum gates is fundamental to constructing quantum circuits and algorithms. Single-qubit gates, such as the X, Y, and Z gates, enable precise manipulation of qubit states, which can be visualized on the Bloch sphere. The use of matrix representations, including Pauli matrices, provides a rigorous mathematical framework for analyzing gate operations and their effects within quantum circuits. This work also explores the application of the Quantum Fourier Transform (QFT) and introduces the Hadamard test subroutine as a practical method for evaluating quantum states. The presented algorithms were successfully implemented on a 5-qubit IBM quantum computer, demonstrating both the theoretical foundations and practical significance of quantum gates and Pauli matrices in modern quantum computation and their potential role in secure information processing.
Abstract This laboratory work presents a practical implementation of quantum teleportation and superdense coding 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 protocols and hybrid quantum-classical workflows. In Laboratory Work No. 5, participants implement the quantum teleportation protocol step by step. The procedure begins with the preparation of a randomly generated qubit by directly assigning complex probability amplitudes. The state vector is normalized to ensure physical validity, defining the amplitudes corresponding to the computational basis states ∣0⟩|0\rangle∣0⟩ and ∣1⟩|1\rangle∣1⟩. Once the qubit is prepared for teleportation, an entangled Bell pair is generated to establish a shared quantum channel between two parties, conventionally referred to as Alice and Bob. The protocol proceeds with joint operations and measurement of two classical bits on Alice’s side. The measurement outcomes determine conditional unitary corrections applied to Bob’s qubit, enabling the reconstruction of the original quantum state without physically transmitting the qubit itself. The laboratory exercise includes the construction of a circuit with separate quantum registers and classical bits, explicit amplitude assignment for Alice’s qubit, and verification of successful state transfer through measurement statistics. In addition to teleportation, the conceptual framework of superdense coding is discussed as a complementary protocol demonstrating how entanglement enhances classical information capacity. Together, these experiments illustrate the operational role of entanglement as a quantum communication resource. The guide emphasizes practical circuit design, normalization of quantum states, entanglement generation, conditional gate operations, and execution on both simulators and real quantum hardware. Through this laboratory work, participants gain experimental insight into two cornerstone protocols of quantum information science.
FINDING: Quantum teleportation is a protocol for transferring an unknown quantum state via shared entanglement and classical communication — not a transport of matter/energy. | MATH: Bell basis: {|Φ⁺⟩, |Φ⁻⟩, |Ψ⁺⟩, |Ψ⁻⟩} = (1/√2)(|00⟩ ± |11⟩), (1/√2)(|01⟩ ± |10⟩). Teleportation circuit: CNOT + Hadamard + 2 classical bits → Pauli correction {I, X, Z, XZ}. State transfer fidelity = 1 (ideal), requires 2 ebits + 2 cbits per qubit. | CONNECTION: Bell states form an orthonormal basis in ℂ²⊗ℂ² — a 4D root system isomorphic to D₂ (dihedral symmetry, order 8). The CNOT+Hadamard sequence corresponds to a Clifford group element, whose action on the Pauli group preserves the symplectic structure over GF(2)² — a lattice symmetry (square lattice, Z²). The 2-bit classical correction maps to the four Pauli operators, which are the vertices of a tetrahedron in the Bloch sphere — a crystallographic point group (T_d, order 24). The ratio 1/√2 appears in all Bell states — this is the cosine of 45°, linkin Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: Quantum teleportation relies on shared entanglement and classical communication, not instantaneous matter transfer; mathematical core is Bell state measurement and unitary transformation. | MATH: Bell state basis: ∣Φ±⟩=21(∣00⟩±∣11⟩), ∣Ψ±⟩=21(∣01⟩±∣10⟩). Teleportation protocol: Alice performs Bell measurement on her qubit and entangled pair, sends 2 classical bits to Bob, who applies Pauli correction I,X,Z,XZ based on outcome. Fidelity = ∣⟨ψ∣ρout∣ψ⟩∣2. No faster-than-light signaling: classical bits limit speed to ≤ c. | CONNECTION: No direct geometric ratios (0.382, 0.618, 0.786, 1.618, 2.618) appear. However, Bell states are maximally entangled, corresponding to the four vertices of a regular tetrahedron in the Bloch sphere representation (SO(3) symmetry). The Pauli matrices X,Y,Z generate the Clifford group, which is the sy Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Abstract This practical guide provides a detailed introduction to quantum logic gates and unitary operators as implemented on the IBM Q platform. It focuses on the fundamental gate structures used in quantum programming, particularly within the Composer interface, and provides screenshots and detailed descriptions of each operator’s functionality. The guide emphasizes the operational role of gates in controlling qubit states and constructing quantum circuits. To enhance understanding, the behavior of gates is visualized on the Bloch sphere, demonstrating how single-qubit operations rotate qubit states around the XXX, YYY, and ZZZ axes, as well as how phase shifts and amplitude adjustments modify the qubit. These operations are essential for measurement, state preparation, and precise manipulation of quantum systems. The relevance of this study is underscored by the rapid development of quantum technologies and the increasing accessibility of quantum computing platforms like IBM Q. A thorough understanding of quantum gates and their implementations is critical for advancing quantum algorithm design and for training specialists capable of deploying quantum computational solutions in fields ranging from cryptography to complex physical system simulations. This guide provides both theoretical insights and practical exercises, bridging the gap between abstract quantum mechanics and applied quantum programming, thereby equipping readers with foundational skills for effective circuit construction and quantum computation.