Quantum mechanics uses two special kinds of grid, and mixing them up is the most common early stumble. They have completely different jobs.
Unitary grids are the verbs. They do things.
Hermitian grids are the questions. They are the things you can ask a qubit.
Unitary: the verbs
Every gate is one. The definition buys you the two properties from the gates lesson:
- You can undo it. Nothing is ever lost until you look.
- The total stays 1. A valid qubit stays valid.
Think of them as turns of the globe. They rotate; they never stretch or squash.
Hermitian: the questions
Anything you can measure is one of these, and there is a neat reason why.
Hermitian grids always have ordinary, real answers — no imaginary bits. Since a measurement gives you a number you read off a screen, and screens do not display imaginary numbers, anything measurable has to be this kind.
There is a bonus, too. The possible answers to a Hermitian question never overlap, which is exactly the condition from the last lesson for being able to tell them apart properly.
How the two are related
Any verb can be written as U = e^(−iHt), where H is one of the question-grids.
That is not just bookkeeping. In physics H is the energy of the system, and the equation says: let a system sit there for time t, and it performs a gate.
Which is literally how real hardware works. There is no "apply H gate" button. There is a precisely timed pulse of microwaves, and the timing is t.
Worth remembering
- Unitary grids are the verbs: undoable, and they keep the total at 1.
- Hermitian grids are the questions: their answers are always ordinary real numbers.
- X, Y and Z are both, which is why they appear everywhere.
- U = e^(−iHt) links them, and it is why gates are really just carefully timed pulses.