In Quantum Computing, a qubit is the basic unit of information, with two states: and
Formally the sate of a qubit is a linear combination:
\ket{\psi} = \alpha \ket{0} +\beta \ket{1}\quad \forall \alpha ,\beta \in \mathbb{C} $$ But we cannot observe this directly. When measured, we either get $\ket{0}$ or $\ket{1}$. Assume $|\alpha|^2+|\beta|^2=1$ unelss we are on a branch. Taking polar form\begin{aligned} \alpha =r_\alpha e^{\phi_\alpha } \ \beta =r_\beta e^{\phi_\beta } \ \end{aligned}
### Global phase unity(?) factor We can't change the probabilities, or the phase. the ony transformation we can do is rotate all phases. Let $\phi$ be the global phase, of alpha. -- this can be changed Let $\theta$ be the relative phase: $\theta = \phi_\beta - \phi_\alpha$ -- this cannot be changed\ket{\psi} = e^{i\phi} (r_\alpha \ket{0} + r_\beta e^{i \theta } \ket{1}\quad)
### Standard form\ket{\psi} = cos\left( \frac{\theta}{2} \right)\ket{0} + sin\left( \frac{\theta}{2} \right) e^{i \phi }\ket{1}\quad
These can be represented on a 2-sphere with the north pole being $\ket{ 0 }$ and south being $\ket{ 1 }$ On the equator, we call Hadamar basis $(\phi,\theta) = $ ### Hadamard basis $\ket{ + } = (\phi=0,\theta=\frac{\pi}{2})$ $\ket{ - } = (\phi=0,\theta=\pi)$ $\ket{ i }$ is a clockwise rotation $\ket{ -i }$ is an anticlockwise rotation. See https://en.wikipedia.org/wiki/Hadamard_transform ### Conjugate transpose Let $A\in \mathbb{C}^{m{\texttimes}n}$ Then the conjuagate transpose is the transpose composed with an element-wise [[Conjugate (Complex Number)|conjugation]]. Written $A^*$ ### [[Unitary Matrix]]