# Solving Quantum Computation: Diagonalizing Y & Z Operators

• James Jackson
In summary, the conversation discusses the definition and representation of operators Y and Z on a quantum computation system. The question at hand is how to write Z in diagonal form and Y in Dirac form. The conversation concludes that the Dirac form of an operator is given by a sum of projection operators, while the diagonal form is a sum of the operator's eigenvalues and eigenvectors. It is noted that the Dirac form is the same as the diagonal form for a diagonal matrix.
James Jackson
I'm just looking at another quantum computation question. It is stated like so:

The operators Y and Z on $$C^2$$ are defined by:

$$Y|0\rangle =i|1\rangle ; Y|1\rangle = -i|0\rangle$$
$$Z|0\rangle = |0\rangle ; Z|1\rangle = -|1\rangle$$

Write Z in diagonal form
Write Y in Dirac form with respect to the basis $$\{ 0\rangle , |1\rangle\}$$

Now, I'm confusing myself something silly. I'm under the impression that the diagonal form of an operator is given by:

$$A=\sum \lambda_{n}|n\rangle\langle n|$$

where $$|n\rangle$$ are the eigenvectors and $$\lambda_n$$ are the eigenvalues of A.

But I would also take this to be the Dirac form, so I'm clearly missing something.

The eigenvalues of Z are clearly $$\{1,-1\}$$ with eigenvectors $$\{ |0\rangle ,|1\rangle\}$$, so the diagonal form is $$Z=|0\rangle\langle 0|-|1\rangle\langle 1|$$.

I suppose my question breaks down to 'What is meant by the Dirac form of an operator?'

Any hints?

Edited to remove me being stupid and working out eigenvectors incorrectly.

Edit: Or, by Dirac form of an operator, do they mean the matrix representation which, for Y, is given by:

$$Y=\left(\begin{array}{cc}0&-i\\i&0\end{array}\right)$$

Last edited:
My guess is that by "diagonal form" they mean write Z as a diagonal matrix. And by "Dirac form" they mean write Y as a sum of projection operators as you did for Z. But I agree with you, in that I would say the "Dirac Form" version of Z written in terms of eigenvectors is also in "diagonal form".

To anyone who's remotely interested, it transpires that given an operator, A, in matrix respresentation, then the Dirac form of the operator is:

$$\hat A=\sum_{ij} A_{ij} |i\rangle\langle j|$$

and given the eigenvalues $\lambda_i$ and eigenvectors $|\lambda_i\rangle$ of an operator, the diagonal representation is (of course):

$$\hat A = \sum_i \lambda_i |\lambda_i\rangle\langle\lambda_i |$$

As a 'learning point', note that the Dirac form of an operator is identical to the diagonal form for a diagonal matrix. An obvious result, but perhaps it will be useful for others.

## 1. What is quantum computation and why is it important?

Quantum computation is a type of computing that uses the principles of quantum mechanics to perform operations on data. It is important because it has the potential to solve complex problems that are currently infeasible for classical computers to solve efficiently.

## 2. What are Y and Z operators in quantum computation?

Y and Z operators are two of the three fundamental Pauli operators used in quantum computation. Y operator represents a rotation around the y-axis by 180 degrees, while Z operator represents a rotation around the z-axis by 180 degrees.

## 3. Why is diagonalizing Y and Z operators important in quantum computation?

Diagonalizing Y and Z operators is important because it allows us to simplify complex quantum computations by breaking them down into simpler operations. It also helps us understand the behavior of quantum systems and design more efficient quantum algorithms.

## 4. How do you diagonalize Y and Z operators in quantum computation?

To diagonalize Y and Z operators, we use a process called quantum gate decomposition, which breaks down the operators into smaller operations that are easier to manipulate. This involves using a combination of quantum gates, such as Hadamard, CNOT, and Phase gates.

## 5. What are the applications of diagonalizing Y and Z operators in quantum computation?

Diagonalizing Y and Z operators has various applications in quantum computation, such as in designing quantum error correction codes, implementing quantum algorithms, and studying quantum entanglement. It also plays a crucial role in quantum computing hardware development and quantum simulation.

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