Quantum Computing

Complex Vector Spaces

Complex Vector Spaces

Complex matrices

Matrix Multiplication

\[* : \mathbb{C}^{m \times n} \times \mathbb{C}^{n \times p} \rightarrow \mathbb{C}^{m \times p}\]

Linear maps

Isomorphism

Isomorphism Example

\[\begin{bmatrix} x & y \\ -y & x \end{bmatrix}\]

Basis and Dimension

implies $c_0 = … = c_{n-1} = 0$.

Change of basis

Hadamard Matrix

\[\left\{ \begin{bmatrix} \frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}}\\ \end{bmatrix}, \begin{bmatrix} \frac{1}{\sqrt{2}}\\ -\frac{1}{\sqrt{2}}\\ \end{bmatrix} \right\}\]

is the Hadamard matrix

\[H = \frac{1}{\sqrt{2}} \begin{bmatrix} 1 & 1 \\ 1 & -1 \\ \end{bmatrix}\]

Inner Products and Hilbert Spaces

\[\langle -,- \rangle = \mathbb{V} \times \mathbb{V} \rightarrow \mathbb{C}\]

Norm and Distance

\[| | : \mathbb{V} \rightarrow \mathbb{R}\]
defined as $ V = \sqrt{\langle V,V \rangle}$
\[d( , ): \mathbb{V} \rightarrow \mathbb{R}\]

where

\[d(V_1, V_2) = |V_1 - V_2| = \sqrt{\langle V_1-V_2 \rangle, \langle V_1-V_2 \rangle}\]

Orthogonal Basis

Eigenvalues and Eigenvectors

\[AV = c\cdot V\]

Hermitian Matrices

\[\langle AV, V' \rangle = \langle V, AV' \rangle\]

Unitary Matrices

\[A * A^{-1} = A^{-1}*A = I_n\] \[U * U^\dagger = U^\dagger * U = I_n\]

Tensor Products


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