# Why adding H-gate is referred as changing the basis of measurement?

I study Qiskit. When I measure a qubit by including qc.measure(qb,cb) instruction, I measure it in the computational basis. But if I apply the Hadamard gate adding qc.h(qb) before the measurement qc.measure(qb,cb), it is often referred to as measuring in X-basis. In my mind measuring in a different basis should involve some changes in the measurement apparatus, not in the state of the system being measured (as happens in this case). I understand that computationally, applying the Hadamard gate produces the same outcome as the actual measurement in a different basis, but I'm just curious why it is call the 'change of basis'. Is it just some informal naming convention or does it have some hidden meaning?

## 2 Answers

There are many ways to see that (one can invoke group theory, for example), but the way that I find easier is the following.

Let q denote a system of a single qubit. Consider then two situations:

1. q is in the state $$| 0 \rangle$$ or $$| 1 \rangle$$ and we measure q in the x-basis, $$\{|+\rangle, |-\rangle\}$$. Because $$| 0 \rangle = \frac{1}{\sqrt{2}} (| + \rangle + | - \rangle)$$ and $$| 1 \rangle = \frac{1}{\sqrt{2}} (| + \rangle - | - \rangle)$$, we get as outcome $$| + \rangle$$ with probability $$1/2$$ or $$| - \rangle$$ with the same probability.

2. q is in the state $$| + \rangle$$ or $$| - \rangle$$ and we measure in the standard basis, $$\{|0\rangle, |1\rangle\}$$. Because $$| + \rangle = \frac{1}{\sqrt{2}} (| 0 \rangle + | 1 \rangle)$$ and $$| - \rangle = \frac{1}{\sqrt{2}} (| 0 \rangle - | 1 \rangle)$$, we get as outcome $$| 0 \rangle$$ with probability $$1/2$$ or $$| 1 \rangle$$ with the same probability.

Now, note that if we define the map $$H: |0\rangle \to |+\rangle, |1\rangle \to |-\rangle$$, situation 1 turns exactly into situation 2! But note that the map $$H$$ is just the definition of the Hadamard gate. Therefore, the Hadamard gate makes the two situations equivalent.

Obs 1. Conversely, if the map was $$H: |+\rangle \to |0\rangle, |-\rangle \to |1\rangle$$ it would turn situation 2 exactly into situation 1. This can also be used as the definition of the Hadamard gate.

Any unitary operator can be understood as a change of basis in which we describe operators in quantum theory.

Specifically, the Hadamard gate is a unitary that sends the basis $$\{\vert 0\rangle , \vert 1\rangle\}$$ into the basis $$\{\vert + \rangle, \vert - \rangle\}$$. If you note, we changed the eigenbasis of Z into the eigenbasis of X.