In general, a Hilbert space of wave functions.
For quantum-C, in particular,
finite dimensional complex vector space with an inner product
that are spanned by abstract wave function such as $|\to \rangle$
will be sufficent

what is spanned?

Quantum state spaces and transformations acting on them can be descriged 
in terms of vectors and matices
or bra/ket notation


Kets like $|x \rangle$ denote column vectors and typically quantum states

qubit 

qubit is a nuit vector in a two dimensional complex vector space 
where a paticular basis, e.g., $|0 \rangle$  and $| 1 \rangle$  

$|0 \rangle$  and $| 1 \rangle$  correspond to 0 and 1 in classical comuter

However qubits is a superposition of 
$|0 \rangle$  and $| 1 \rangle$  that is, 
$a |0 \rangle + b | 1 \rangle$  that is, 
the  prob that measurment is $|0 \rangle$ is $|a|^2$ 
and $| 1 \rangle$ is $|a|^2$ 


example of state
the position on the line $| x \rangle$, 
and joint state of a particle at posiion x with a con in state
$| L \rangle$, 

$| L, x \rangle$, 


 J. Kempe (2003) 

"Imagine a particle on a line whose position is described
by a wave-packet 
$| \psi_{x_0} \rangle$ 
localized around a position $x_0$,
i.e. the function 
$\langle x | \psi_{x_0} \rangle$ 
corresponds to a wave-packet
centered around $x_0$.

Now let $P$ be the momentum operator.
The translation of the particle, corresponding to a step
of length $l$ can be represented by the unitary operator
$U_l = \exp (-iPl/\hbar)$ so that 
$U_l$ | \psi_{x_0} \rangle U_l$ | \psi_{x_0-l} \rangle$.


[C7] A. Ambainis [c7]
We consider the state space consisting of states
$| n,0 \ranle$ and $| n,1\ranle$