QC201 : Advanced Math for Quantum Computing

Explore orthogonal and unit vectors, define orthonormality via magnitude and inner product with complex conjugation, and show how any vector decomposes into an orthonormal basis.

maps electron spin to vectors via ket notation, deriving up, down, right, left states from a basis, and shows changing basis changes representation, while physical behavior remains, with 0.5 probabilities.

Learn tensor products as a straightforward matrix operation on column vectors, multiplying each first-vector element with the entire second vector to produce a 4-element result without any addition.

Explore tensor products to represent multi-qubit systems, building 2-qubit basis vectors from single-qubit states, and form four states |00>, |01>, |10>, |11> via tensor products in Dirac and matrix notation.

Explore tensor products and bracket notation for multi-qubit bases, including 2-qubit and 3-qubit states, ket notation, and the orthonormality of basis vectors.

Explore how combining two qubits increases degrees of freedom via tensor products, revealing entanglement and a shared two-qubit state beyond independent subsystems.

Change basis for quantum state vectors between the standard and Hadamard bases by using orthonormality to find coefficients a and b, then substitute and express the state.

Explore how measurement probabilities in multi-qubit systems arise from the squared magnitude of the bracket S A, using Bell states, tensor product apparatus, and fictional apparatus to reveal Bell's theorem.

Learn how to compute measurement probabilities for multi-qubit states in standard and hadamard bases, including partial measurements and post-measurement states in entangled bell states.

Show how bracket notation uses bra and ket to form inner and outer products, with ket-bra outer products yielding matrices that represent square matrices as a sum of ket-bra terms.

Master multi-qubit transformation matrices and entanglement. Learn how X and Y act on the full entangled state via tensor products.

Deconstruct a hermitian into a sum of distinct eigenvalues times the outer product of corresponding eigenvectors, revealing real eigenvalues and the relation to hamiltonians.

Explain dense coding: encode two bits into a single qubit using an entangled Bell-state pair, enabling Bob to retrieve both bits via CNOT and Hadamard after receiving Alice's qubit.

Learn how quantum teleportation transfers an unknown qubit state from Alice to Bob using an EPR pair, CNOT and Hadamard operations, partial measurements, and Pauli corrections via a classical channel.