QC201 : Advanced Math for Quantum Computing

We examine how quantum state vector S and measurement vector A yield the probability of 'aligned' outcomes via the squared magnitude of A†S, with examples.

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.

Understand orthonormality in bracket notation by analyzing unit vectors, orthogonality, and inner products. See how ket 0 and ket 1 form a complete orthonormal set with 1s and 0s.

Express any point as a linear combination of orthogonal unit vectors, rotate the axes, and rewrite coordinates with a new basis, illustrating basis vectors and coordinate conversion.

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.

Explore degrees of freedom as real numbers describing a system’s state, from temperature to coordinates in 1D, 2D, and 3D spaces, and see how constraints reduce them toward qubit entanglement.

Identify two degrees of freedom for a single qubit by using a state vector with four real numbers (two complex numbers) subject to two constraints: unit vector and global phase.

Distinguish global vs relative phase: state vectors differing only by global phase can be physically identical, but relative phase matters in superposition with ket 1, affecting normalization and equivalence.

Map qubit state vectors to the Bloch sphere using theta and phi, via the Hadamard basis with ket-plus, ket-minus, and ket i; the sphere visualizes states, not physical space.

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 of matrices by copying the second matrix for each element of the first and forming a product with that element, with hands-on exercises.

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 tensor products with a 4-qubit system, forming subsystems on qubits 1&3 and 2&4. Follow fixed qubit ordering and bracket notation to write the full state as a tensor product.

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

Examine how entanglement depends on how we decompose a multi-qubit system into subsystems and how tensor products indicate unentangled states, illustrated by a four-qubit example in bracket notation.

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.

Explores cryptography with entangled qubits, using Bell states to generate shared secrets between Alice and Bob via measurements in standard and Hadamard bases, and discusses eavesdropping by Eve.

Learn to compute the expected value of a measurable property by mapping apparatus orientations to real numbers, using orthonormal measurement bases, and applying probability-weighted sums.

Explore how the expected value is computed via a measurable matrix, derive bra-ket conjugates, and recognize the hermitian M with eigenvectors as apparatus orientations and eigenvalues as outcomes.

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

Deconstruct unitary matrices from an orthonormal vector set to reveal eigenvectors and eigenvalues, and prove M times adjoint equals I with eigenvalues of magnitude one, as e power i theta.

Prove the no-cloning theorem by assuming a universal cloning unitary and deriving a contradiction. Demonstrate that a reversible operation cannot clone an arbitrary quantum state.

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.

Explore Bell's theorem and entanglement using the Bell state, showing how two qubits share a state across distance and EPR pairs challenge local hidden states.

Derive the probability that two qubits in a Bell (EPR) pair yield aligned measurements given apparatus angles theta1 and theta2, showing it equals cos^2(theta1 − theta2) and supporting Bell's theorem.

Derive Bell's theorem by comparing measurement agreement probabilities for entangled qubits under shared state versus local state assumptions, showing local states cannot match experimental results.

Build the math foundation to understand how quantum computing works. Analyze basis vectors, degrees of freedom, and bracket notation for multi-qubit systems; explore dense coding and teleportation with math.