Superposition

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 $\vert X\rangle$ in the Hilbert space with $\langle X\vert X\rangle = 1$; such a state vector is called normalized. The inner product of a vector $\vert X\rangle = (x_{0},x_{1},\ldots, x_{N-1})^{T}$ with itself is $\vert x_{0}\vert^{2} + \vert x_{1}\vert^{2} + \ldots + \vert x_{N-1}\vert^{2}$, where $\vert a+ib\vert^{2} = a^{2} + b^{2}$. More generally $\left\langle X\vert Y\right\rangle = \sum_{i = 0}^{N-1} x_{i}^{*}y_{i}$, 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 $\vert X\rangle$ as $w_{0}\vert x_{0}\rangle + w_{1}\vert x_{1}\rangle$, where $w_{0}, w_{1}$ are the complex projections of $\vert X\rangle$ onto the eigenstates such that $\vert w_{0}\vert^{2} + \vert w_{1}\vert^{2} = 1$. When the qubit with state vector $X$ is measured, we are guaranteed to find it to be in either the state $1 \vert x_{0} \rangle + 0 \vert x_{1} \rangle = \vert x_{0}\rangle$ or the state $0 \vert x_{0} \rangle + 1 \vert x_{1} \rangle = \vert x_{1}\rangle$.

More generally, the Hilbert space of an $N$-state quantum system is $\Complex^{N}$. 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 $x_{0}, x_{1},\ldots, x_{N-1}$ can be fully described by the vector

$$\vert X \rangle = \sum_{k = 0}^{N-1} w_{k} \vert x_{k} \rangle, \textrm{ where } \sum_{k = 0}^{N-1} \vert w_{k}\vert^{2} = 1.$$

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 $\vert w_{j}\vert^{2}$. The coefficient $w_{j}$ is called the amplitude of eigenstate $\vert x_{j}\rangle$.

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.