The projection of the state vector onto one of the axes of its Hilbert
space shows the contribution of that axis's eigenstate to the whole
state. A classical bit's state vector can only lie along one of the
two axes. The state of a qubit can be any vector | *X* in the
Hilbert space with
*X*| *X* = 1; such a state vector is
called *normalized*. The inner product of a vector
| *X* = (*x*_{0}, *x*_{1},..., *x*_{N-1})^{T} with itself is
| *x*_{0}|^{2} + | *x*_{1}|^{2} +...+ | *x*_{N-1}|^{2}, where
| *a* + *ib*|^{2} = *a*^{2} + *b*^{2}. More generally
*X*| *Y* = *x*_{i}^{*}*y*_{i}, where
(*a* + *ib*)^{*} is *a* - *ib*.

Let *x*_{1} be the eigenstate corresponding to the 1 state, and let
*x*_{0} be the eigenstate corresponding to the 0 state. We can write
any state | *X* as
*w*_{0}| *x*_{0} + *w*_{1}| *x*_{1},
where
*w*_{0}, *w*_{1} are the complex projections of | *X* onto
the eigenstates such that
| *w*_{0}|^{2} + | *w*_{1}|^{2} = 1. When the
qubit with state vector *X* is measured, we are guaranteed to find it
to be in either the state
1| *x*_{0} +0| *x*_{1} = | *x*_{0} or the state
0| *x*_{0} +1| *x*_{1} = | *x*_{1}.

More generally, the Hilbert space of an *N*-state quantum system is
. As with the two state system, when we measure our
*N*-state quantum system we will always find it to be in exactly one
of the eigenstates. The system is allowed to exist in any complex
linear superposition of the *N* states between measurements. An
*N*-state quantum system with eigenstates
*x*_{0}, *x*_{1},..., *x*_{N-1} can be fully described by the vector

| *X* = *w*_{k}| *x*_{k}, where| *w*_{k}|^{2} = 1.

Our state vector can exist in a linear superposition of eigenstates,
but we can only measure the state vector to be in one of the
eigenstates. When the state vector is observed, it makes a sudden
discontinuous jump to one of the eigenstates. When measurement is
performed the state vector is said to collapse
[18]. For an *N*-state quantum system with a
normalized state vector, the probability that the state vector will
collapse into the *j*th eigenstate is simply
| *w*_{j}|^{2}. The
coefficient *w*_{j} is called the *amplitude* of eigenstate
| *x*_{j}.

We can construct a quantum memory register out of the qubits described
in the previous section. Just as in a classical computer, a quantum
computer will perform calculations by manipulating its memory register
from some start state to some final state. Note that a quantum
register composed of *N* qubits requires 2^{N} complex numbers to
completely describe its state vector, as an *N*-qubit register has
2^{N} basis states.