Bloch sphere

Until now, we have seen qubits only in the braket notation. To get a more visual and geometrical understanding of a single qubit, Arjen Vaartjes (QuTech) will introduce the Bloch Sphere. This three dimensional sphere of radius 1, can also help us understand the action of unitary gates and measurement on a single qubit.

Prerequisite knowledge

Bra-ket notation
Basic linear algebra

Further thinking

Knowing that the bit flip gate X takes |0> to |1>, which of the below could be a valid matrix representation for the bit flip gate?

$A) \begin{pmatrix} 0 & 0\\ 0 & 1 \end{pmatrix},\quad B) \begin{pmatrix} 0 & 0\\ 0 & 1 \end{pmatrix},\quad C) \begin{pmatrix} 1 & 0\\ 0 & -1 \end{pmatrix},\quad D) \begin{pmatrix} 0 & 1\\ 1 & 0 \end{pmatrix}$

Given that H (the Hadamard gate) acting on |0> produces |+>, i.e. H|0>=|+>, and that H|1> = |->, which of the following could be a valid matrix representation for the Hadamard gate?

$A)\; \frac{1}{\sqrt{2}}\begin{pmatrix} 0 & 1\\ 1 & 0 \end{pmatrix},\quad B)\; \frac{1}{\sqrt{2}}\begin{pmatrix} 1 & -1\\ -1 & 1 \end{pmatrix},\quad C)\; \frac{1}{\sqrt{2}}\begin{pmatrix} 1 & 1\\ 1 & -1 \end{pmatrix},\quad D)\; \frac{1}{\sqrt{2}}\begin{pmatrix} 1 & 1\\ -1 & -1 \end{pmatrix}$

Solutions

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Fundamentals

Bloch sphere

Prerequisite knowledge

Further thinking

Further reading