Linear algebra refresher

Quantum states are vectors; evolution before measurement is described by unitary matrices (gates).

Vectors and inner products

For column vectors $u, v \in \mathbb{C}^d$, the inner product is

where $u^\dagger$ is the conjugate transpose. The Euclidean norm is $\lVert v \rVert = \sqrt{\langle v \vert v \rangle}$. Valid pure quantum states satisfy $\lVert \psi \rVert = 1$.

Linear operators

A gate on $n$ qubits is a unitary $U \in \mathbb{C}^{2^n \times 2^n}$ with $U^\dagger U = I$. Composition of gates corresponds to matrix multiplication applied in the correct order for the circuit diagram.

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Dirac notation