L15 Entanglement IBM Quantum Circuit, Entanglement Swapping, Midterm Review

Overview

Hiu-Yung Wong completes quantum teleportation by explaining IBM Quantum/Qiskit execution, Bell-basis measurements implemented with Hadamard gates, and classical feed-forward corrections, then shows how teleporting one member of an entangled pair swaps entanglement to a distant qubit. The midterm review revisits state vectors, inner products, normalization, Pauli operators, adjoints, eigenvalue diagonalization, basis changes, tensor products, quantum gates, and trace, ending with a two-qubit circuit worked in both ket and matrix notation.

Key takeaways

Quantum teleportation combines shared entanglement, two classical measurement results, and conditional X and Z corrections to transfer a qubit state; it does not copy the original state.
Applying a Hadamard before a computational-basis measurement is equivalent to measuring in the |+⟩, |−⟩ basis, allowing standard hardware measurements to implement a different measurement basis.
Entanglement swapping uses teleportation to make Bob and Carlos entangled even though Alice and Carlos held the original entangled pair.
For an ideal teleportation circuit with input |0⟩, Bob’s output qubit must be |0⟩; deviations on IBM Quantum hardware can reflect noise and execution conditions.
Midterm calculations rely on consistent complex conjugation, normalization, basis ordering, and left-to-right circuit sequencing; the review’s worked example applies these in both ket and matrix form.

Chapters

0:00 IBM Quantum Assignment: Run the Teleportation Circuit on Real Hardware

Hiu-Yung Wong asks students to submit the circuit, measurement histogram, and any required calculation from the teleportation assignment.
Running on a real IBM Quantum device exposes hardware noise, queue times, circuit layouts, and transpilation into native gates.
A simulator may produce cleaner results, but the assignment is intended to give students experience with real-device execution.

2:21 Preparing the Teleportation Resource and Conditional X Correction

The circuit initializes its qubits to |0⟩; the unknown input state is modeled as |0⟩ for this example.
Bob’s qubit pair is entangled into a Bell state, with one qubit sent to Alice so they share an entangled resource.
After Alice measures in the computational basis, classical feed-forward conditionally applies an X gate to Bob’s qubit.
Because the measurement is stored in one bit of a three-bit classical register, the circuit checks the corresponding register values rather than an isolated bit.

11:15 Hadamard Gates Turn Computational Measurements into Bell-Basis Measurements

Applying H before a computational-basis measurement implements a measurement in the |+⟩, |−⟩ basis.
Wong expands a general two-qubit state in both bases to show how measurement outcomes correspond between them.
The outcome mapping is |+⟩ ↔ 0 and |−⟩ ↔ 1 when the Hadamard-plus-computational-measurement circuit is used.
Expanding and regrouping amplitudes provides a direct check of how the measured outcome changes the remaining qubit’s state.

23:05 Classical Feed-Forward Applies the Teleportation Z Correction

A measurement outcome of 1 after the Hadamard requires a Z correction on Bob’s qubit; the other measurement result requires no Z gate.
The circuit checks classical-register values 4 and 6 because either can encode the relevant measured bit as 1, while another measured bit varies.
For the example input |0⟩, Bob should recover |0⟩ in an ideal noiseless run, even though intermediate measurement bits vary.
Teleporting a known basis state does not violate no-cloning: the no-cloning theorem forbids copying an arbitrary unknown state while retaining the original.

31:28 Entanglement Swapping Transfers Correlations from Alice–Carlos to Bob–Carlos

The setup begins with Carlos entangled with Alice while Alice also shares a teleportation resource with Bob.
Teleporting Alice’s qubit state to Bob leaves Bob and Carlos entangled, enabling entanglement between distant parties.
In the illustrated example, measuring Carlos and Bob yields correlated outcomes such as 00 or 11; other entangled states can instead produce anticorrelation.
The shared outcomes are correlations, not copies of an arbitrary unknown state, so the process does not violate no-cloning.

38:03 Midterm Review: State Vectors, Superposition, and Measurement Probabilities

A quantum state is represented by a vector whose coefficients multiply basis states; indexing from 0 through n gives n + 1 entries.
Measurement probabilities are the squared magnitudes of amplitudes, including complex amplitudes such as 3 + 2i.
For bra-ket inner products, taking a ket to a bra requires complex conjugation of its coefficients.
An orthonormal basis has unit self-inner-products and zero inner-products between distinct basis vectors.

41:33 Normalization, Basis Changes, and Complex Inner-Product Rules

Normalize a state by dividing its vector by its length, just as a three-dimensional vector is divided by √(a² + b² + c²).
Changing basis is performed by substituting one basis representation for another and tracking the resulting coefficients.
The inner-product identity ⟨A|B⟩ = ⟨B|A⟩* requires complex conjugation; the values are equal without it when the relevant inner products are real.
Wong recommends checking normalization and conjugation explicitly to avoid common calculation errors under exam time pressure.

43:33 Pauli Matrices, Anticommutation, and the Adjoint Operation

The Pauli matrices σx, σy, and σz are key qubit operators, and their anticommutator obeys {σi, σj} = 2δijI.
The Kronecker delta δij equals 1 when i = j and 0 otherwise, compactly expressing the Pauli anticommutation rule.
A matrix adjoint takes the complex conjugate and transpose, matching the conjugation-and-transposition rules for bras and kets.
A Hermitian matrix satisfies A† = A and has real eigenvalues; a unitary matrix preserves inner products.

49:18 Eigenvalues, Diagonalization, and Coordinate Transformations

An eigenvector retains its direction under a matrix operation, while its eigenvalue gives the multiplying scalar; eigenvalues can be found from det(A − λI) = 0.
The review expects eigenvalue or diagonalization problems no larger than 2 × 2.
For a basis-change matrix U, vector coordinates transform as v′ = Uv, while operator matrices transform by a basis-change sandwich using U and U†.
Representing a matrix in its eigenbasis diagonalizes it because the eigenvectors become the coordinate basis.

53:42 Projection Operators and Tensor Products for Multiple Qubits

An outer product |v⟩⟨v| combines a column ket and row bra to form an operator that projects onto a state.
Projection operators connect state amplitudes to the probabilities of obtaining measurement outcomes.
For multiple qubits, tensor products build larger state spaces and operators by replacing each matrix entry with a scaled copy of the other matrix.
Careful basis ordering is essential when arranging tensor-product results for a multi-qubit system.

55:52 Quantum Gates: X, CNOT, SWAP, and XOR as Modulo-Two Addition

The X gate exchanges computational basis states: |0⟩ becomes |1⟩ and |1⟩ becomes |0⟩.
A CNOT applies X to its target only when its control qubit is 1; its classical action is target ⊕ control.
XOR can be understood as addition modulo 2, or as testing whether the number of 1s is odd—an interpretation useful for algorithms.
Wong also reviews the SWAP gate and emphasizes identifying the control and target wires before applying a gate.

59:10 Trace Cyclicity Explains Why Trace Equals the Sum of Eigenvalues

The trace of a matrix is the sum of its diagonal entries, and trace is unchanged under cyclic rearrangement: tr(AB) = tr(BA).
Ordinary matrices generally do not commute; cyclic rearrangement is valid here because the expression is inside a trace.
A matrix transformed into its diagonal eigenbasis has eigenvalues on its diagonal, so its trace equals their sum.
The derivation uses a unitary basis transformation and the identity U†U = I.

1:03:30 Phase Gates, Toffoli Controls, and Hadamard Transformations

The one-qubit phase-shift gate becomes the Z gate when its phase angle is π.
A Toffoli gate flips its target only when both control bits are 1, matching the Boolean condition a · b = 1.
A control marked with an open circle activates when that control is 0, rather than 1.
The Hadamard gate is its own inverse, creates superpositions, and connects binary-vector inner products to parity through modulo-two arithmetic.

1:07:10 ChatGPT Practice: Evaluate a Two-Qubit Circuit Step by Step

Wong prompts ChatGPT to generate an introductory two-qubit circuit for students to evaluate, then stresses solving it independently rather than relying on a supplied answer.
The example starts from |00⟩, applies H to the most-significant qubit, then applies CNOT and a final X operation.
Applying H and CNOT produces an entangled superposition; applying X to the appropriate qubit changes it to a different entangled state.
The exercise reinforces reading circuit order and applying each gate to the correct qubit.

1:12:13 Matrix-Form Circuit Calculation and Midterm Preparation

The input |00⟩ is represented by the column vector (1, 0, 0, 0)ᵀ, with gates multiplying the state vector from the left.
The first operation is H ⊗ I; the CNOT matrix is then applied using the course’s qubit-ordering convention.
Students should verify that the final state is normalized and that its nonzero amplitudes match the ket-notation calculation.
Wong expects a deeper circuit question on the midterm, but says it will not require an impractical three- or four-qubit calculation.

Summary, takeaways, and chapters were generated by AI from the video's transcript and may contain errors. The video belongs to its creator, Quantum Computing, TCAD, Semicond by Hiu-Yung Wong.