How to perform measurement of a qubit?

Question

I am trying to implement the Deutsch algorithm. My steps were:

1. Write down $|01\rangle$ in a matrix form $A$;
2. Apply $H^{\oplus2}$ gate to $A$ matrix;
3. Multiply it with $U_f$ matrix;
4. Apply $H$ to the first qubit;

After the above steps, I got a matrix:

array([[ 0.        ],
       [ 0.        ],
       [-0.70710678],
       [ 0.70710678]])

How should I perform a measurent step on the first qubit to get the answer?

Answer 1

You haven’t specified the order of entries in your vector. In general, the $i$’th entry is obtained with probability $|x_i|^2$.

Let’s consider two examples. In the first, the order of entries is $|00\rangle, |01\rangle, |10\rangle, |11\rangle$. The two final entries will be obtained with probability $1/2$ each, and so the first qubit will be measured as $|1\rangle$. If you measure only the first qubit, then you will still always get $|1\rangle$, and the other qubit will remain in the superposition $\frac{1}{\sqrt{2}}(|1\rangle-|0\rangle)$.

If the order of entries is instead $|00\rangle, |10\rangle, |01\rangle, |11\rangle$, then once again the two final entries will be obtained with probability $1/2$ each. This means that the first qubit will be measured as $|0\rangle$ or $|1\rangle$ with equal probability.