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 | X$\rangle$ in the Hilbert space with $\langle$X| X$\rangle$ = 1; such a state vector is called normalized. The inner product of a vector | X$\rangle$ = (x₀, x₁,..., x_N-1)^T with itself is | x₀|² + | x₁|² +...+ | x_N-1|², where | a + ib|² = a² + b². More generally $\left\langle\vphantom{ X\vert Y}\right.$X| Y$\left.\vphantom{ X\vert Y}\right\rangle$ = $\sum_{{i = 0}}^{{N-1}}$x_i^*y_i, where (a + ib)^* is a - ib.

Let x₁ be the eigenstate corresponding to the 1 state, and let x₀ be the eigenstate corresponding to the 0 state. We can write any state | X$\rangle$ as w₀| x₀$\rangle$ + w₁| x₁$\rangle$, where w₀, w₁ are the complex projections of | X$\rangle$ onto the eigenstates such that | w₀|² + | w₁|² = 1. When the qubit with state vector X is measured, we are guaranteed to find it to be in either the state 1| x₀$\rangle$ +0| x₁$\rangle$ = | x₀$\rangle$ or the state 0| x₀$\rangle$ +1| x₁$\rangle$ = | x₁$\rangle$.

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 x₀, x₁,..., x_N-1 can be fully described by the vector

| X$$\rangle$$ = $$\sum_{{k = 0}}^{{N-1}}$$w_k| x_k$$\rangle$$, where$$\sum_{{k = 0}}^{{N-1}}$$| 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 jth eigenstate is simply | w_j|². The coefficient w_j is called the amplitude of eigenstate | 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.