🚧 Draft — not final. This page was auto-generated as day-one scaffolding for the project and still needs a human pass before it reads as finished. The blanks and placeholders below are intentional — someone who runs this project should fill them in and delete this notice.

Notes — linear algebra warm-up

Quantum computing is linear algebra with a physics accent — so this is worked, not skimmed. Each idea gets a section in my own words plus one tiny example done by hand. No qubits until this page is solid; skipping it is why most quantum-computing attempts stall a chapter later.

Complex vectors & the qubit-to-be

A qubit's state is a length-1 vector of two complex numbers: α|0⟩ + β|1⟩, with |α|² + |β|² = 1. The squared magnitudes are probabilities — that's why the length is always 1.

• By hand: check that (1/√2, 1/√2) and (1/√2, −1/√2) are both valid states, and orthogonal.

Inner products & orthogonality

⟨a|b⟩ measures overlap; 0 means "perfectly distinguishable." Measurement bases are orthonormal sets.

• By hand: ⟨0|1⟩ = 0, ⟨0|0⟩ = 1.

Unitaries (every gate is one)

A quantum gate is a unitary matrix U — it preserves length, so probabilities still sum to 1. Reversible by definition (U†U = I): a fact with big consequences.

• By hand: verify the Hadamard H is unitary, and that H·H = I (it's its own inverse).

Tensor products (how one qubit becomes many)

Two qubits live in a 4-dimensional space, built by ⊗. This is where entanglement will hide.

• By hand: |0⟩⊗|0⟩ = (1,0,0,0)ᵀ; write out |0⟩⊗|1⟩ and |1⟩⊗|1⟩.

Explain-back test

Say why a quantum gate must be reversible, using only the word "unitary" and the phrase "length 1."