Linear Algebra for Quantum Computing
Linear algebra guide for quantum computing, based on supplementary material of HKU COMP3366.
Table of Contents
Cauchy-Schwarz Inequality
For any vectors \(\mathbf{v}, \mathbf{u} \in \mathbb{C}^{n}\), the following inequality holds:
\[ \|\mathbf{v}\| \|\mathbf{u}\| \ge |\lang{\mathbf{v}, \mathbf{u}}\rang | \]
1. Matrix
Monotonicity of Rank
For arbitrary matrices \(A,B\) such that \(AB\) is a legitimate matrix, the following holds
\[ \mathrm{rank}(AB) \le \min(\mathrm{rank}(A), \mathrm{rank}(B)) \]
1.1. Hermitian Matrices
A matrix \(H\) acting on \(\mathbb{C}^{d}\) is Hermitian if and only if \(H^{\dagger} = H\)
1.2. Unitary Matrices
Unitary as a Change of Basis
For any arbitrary unitary matrix \(U\) on \(\mathbb{C}^{d}\), there exists 2 orthonormal bases \(\{\ket{\phi_{i}}\}_{d}\) and \(\{\ket{\psi_{i}}\}_{d}\) such that
\[ U = \sum_{i=1}^{d} \ket{\psi_{i}}\bra{\phi_{i}} \]
1.3. Diagonalization and Eigensystems
A core feature of Hermitian matrices is that they can be diagonalized.