Why do we need different basis vectors?

Question

There is a basis consisting of $|0\rangle$ and $|1\rangle$ states, i.e. z-basis. Why do we need different basis (in quantum computation), like polar basis and others ($|+\rangle$ and $|-\rangle$)?

Answer 1

It is not clear for me what need means here.

Transformation between standard $\{|0\rangle, |1\rangle \}$ basis and a different basis is unitary, so instead of measuring in an arbitrary basis we can apply a proper unitary gate and measure in the standard basis; we can do all measurements in the standard basis, this is enough for quantum computation. If our quantum computer supports measurements in $\{|+\rangle, |-\rangle \}$ basis, then we can use this option to optimize computation or ignore.

We don't really need different bases, we just can use them if our quantum hardware supports measurements in different bases.

Answer 2

I would add the different basis are necesary for quantum tomography. If you want to describe quantum state competely, you have measure it in different basis. For example measuring states $\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle)$ and $\frac{1}{\sqrt{2}}(|0\rangle+\mathrm{e}^{i\frac{\pi}{2}}|1\rangle)$ in standard basis returns $|0\rangle$ and $|1\rangle$ both with probability 50 % but the second state has relative phase $\frac{\pi}{2}$ about which you know nothing. But mesuring in other basis allows you to get information about the phase. In plain words, this means that you look at a quantum states from different angles of view which allows you to see "all its properties".

Moreover, sometimes, it is more convenient to switch from standard basis to other one because of simplier calculations.