Hadamard Test Proofread Notes

[DrDavvyDaDonut]

Everywhere

• Capitalize Dirac
• Eachother is not a word
  • Split to "each other"

Bloch Sphere

A common technique for visualizing superposition states is the bloch sphere.

Bloch should be capitalized.

Measurement

Explain what an inner product is.

Well, when a particle that is put into superposition is measured, it collapses in to one of the states it is in a combination of.

Change to "when a particle in superposition is measured".

The ending of the sentence feels clunky, so you could try

• defining collapse in a second sentence
• changing the ending to something like "it collapses into one of its states"

For example if we were to take the |+> state above and tried to measure it sometimes we would get 0 and other times we would get 1 with certain probabilities.

This needs a comma after "For example" and "measure it". The ending seems like it's trying to say that the probabilities for 0 and 1 are dependent on the state, but the example has a known state. So explain the state dependent probabilities separately or have an undefined state as your example.

We can do an inner product of the hermitian (also called dagger operation) ...

Capitalize "hermitian".

The Hermitian of a generic state can be written in dirac notation as <psi| where this "backwards" ket is pronounced "bra psi".

The "backwards" ket is pronounced as "psi", and the state is "bra psi".

For example, lets say I wanted ...

You used "for example" twice in quick succession. Try using other phrases, like "for instance".

It should be "let's", not "lets".

By performing the same calculation in dirac notation and alternatively with linear algebra, we can see how they are equivalent processes and we end up with the same result.

"By performing the same calculation in Dirac notation and linear algebra we can see how they are equivalent processes."

To this end we multiply the coefficient by its complex conjugate to find the true probability.

"To this end" is a phrase used to describe steps taken towards a goal. This sentence is about combatting a potential problem, so it doesn't make sense.
Change it to something along the lines of "To combat this, we multiply"

For any state |psi> the probability of getting |0> plus the probability of getting |1> must be equal to 1 , because putting a particle into superposition doesn't create or destroy parts of the particle.

Get rid of the space after 1 and before the comma.
The correlation between creating and destroying parts of a particle and the probability equaling 1 is not immediately obvious. Explain it or get rid of it.

In order to find a probability of a particle being in a state, this always requires we multiply it by its complex conjugate.

What "this", "it", and "its" is referring to is unclear. Something more like "we must multiply the inner product by its complex conjugate" will be better.

Therefore, there can be states that are equivalent to eachother even though they have different constants in front of them when measured in a certain basis.

If the constants depend on the basis, then make that clear. As the sentence is, it makes it seem like there are only some bases that have constants that don't affect the measurement.

H-Gate

The Hadamard Gate ...

Gate should not be capitalized.

Tensor Product

Center figure 4.

CNOT01-Gate

CNOT01|01> result vector is wrong.

Eigenvalues and Eigenvectors

In the first paragraph, explicitly say that U is an arbitrary gate matrix.

This equation is effectively saying that a gate operating ...

Change to "This equation is saying that a gate operating ..."

... eigenvectors as columns arranged in a similar ordering to the Lambda matrix.

The same ordering. It can't just be similar, it has to be the same.

... eigenvector is normalized meaning <psi|psi> = 1.

Change to "... eigenvector is normalized (meaning <psi|psi> = 1."

Quantum Phase Estimation (QPE): The Problem Setup

Explicitly define Psi.

In the latex, Psi dagger looks like a 1 by n*n matrix. Make it a n by 1 matrix instead.

... where 0 <= psi k <= 1

Should not be <= 1.

Proof

The proof is imbedded in the page. I think it shouldn't be.

Looking at the real Hadamard test in Fig. 3 ...

Should be Fig. 5