Generally, quantum computing only uses two states, often spin states. Each qubit is then a state of the form$$| \psi \rangle = \alpha |\downarrow\rangle + \beta | \uparrow\rangle $$
With the following constraint from the usual normalization condition$$|\alpha|^2 + |\beta|^2 = 1$$ To represent those states, we can simply represent states $|0\rangle$ and $|1\rangle$ as two different colors. In our case, let's say blue (#0000FF) and red (#FF0000). A general state will simply be a mixture of the two with proportions $|\alpha|^2$ and $|\beta|^2$. As this leaves out the phase of the system, let's represent them as two squares of the appropriate colors with the tilt of each representing their phase. Since squares are invariant under $90^\circ$ rotations, we'll also add a single dot on each squares to indicate their orientation.
To represent two entangled qubit, we'll just place the different states of each particles together and link them to represent the entanglement. For instance, for the state $(|0\rangle \otimes |1\rangle + |1\rangle \otimes |0\rangle)$
Since this is a quantum system with the Hilbert space $\mathbb C^2$, we can simply represent operators as $2\times2$ complex matrices.
Logical gates will be represented as $2N \times 2M$ matrices, taking in $N$ qubits and outputing $M$ qubits.
Last updated : 2017-09-20 13:52:13