Quantum Matrix Calculation

In summary, the given conversation discusses the calculation of an expression for eta from a given psi, resulting in a 4x4 matrix. However, finding eta squared while maintaining a 4x4 matrix proves to be challenging. Various attempts are made, but it is eventually suggested to square the matrix formed by eta, which will result in the scalar product between the basis vectors.
  • #1
quixi
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0

Homework Statement



From a given [itex] \left| \psi \right> [/itex] I have calculated an expression for [itex]\eta[/itex] which results in a 4x4 matrix (as required) however I now need to find [itex]\eta^{2}[/itex] but don't know how to do this with still keeping a resulting 4x4 matrix.

Homework Equations



[tex]\eta = \alpha \left| 0000 \right> + \beta \left| 0001 \right> + ... + \theta \left| 1111 \right> [/tex] etc.

The Attempt at a Solution



I'm pretty sure I can't just do:

[tex]\eta^{2} = \alpha^{2} \left| 0000 \right> + \beta^{2} \left| 0001 \right> + ... + \theta^{2} \left| 1111 \right> [/tex] etc.

But if I do [itex]\eta^{2}[/itex] I'll end up with:

[tex]\eta^{2} = \alpha^{2} \left| 00000000 \right> + \beta^{2} \left| 00010001 \right> + ... + \theta^{2} \left| 11111111 \right> [/tex] etc.

which isn't any good, I need a 4x4 matrix not a 8x8 matrix.

Perhaps:

[tex]\eta^{2} = \eta \eta*[/tex]

but I still don't think that would do.

Hmm. :frown:
 
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  • #2
The eta you gave is no 4x4 matrix. It is vector in a 2^4 dimensional space.
If you are give [tex]\psi[/tex] then [tex]\eta[/tex] is defined as [tex]\eta = |\psi><\psi|[/tex]. Then you can square the matrix which will result in having the scalar product between your basis vectors.
 

What is quantum matrix calculation?

Quantum matrix calculation is a computational method that uses principles from quantum mechanics to solve complex problems. It involves representing data as quantum states and manipulating them through quantum operations to obtain a solution.

How does quantum matrix calculation differ from classical matrix calculation?

Quantum matrix calculation differs from classical matrix calculation in several ways. Firstly, classical matrix calculation operates on classical bits, while quantum matrix calculation operates on quantum bits. Additionally, quantum matrix calculation utilizes quantum gates, which can perform multiple operations simultaneously, while classical matrix calculation uses classical gates that can only perform one operation at a time.

What are the potential applications of quantum matrix calculation?

Quantum matrix calculation has a wide range of potential applications, including in quantum chemistry, cryptography, optimization problems, and machine learning. It can also be used to improve the efficiency of classical algorithms and simulations, making it a valuable tool in various scientific fields.

What are the limitations of quantum matrix calculation?

One of the main limitations of quantum matrix calculation is the high level of complexity and resource requirements for implementing quantum algorithms. Additionally, the technology is still in its early stages, and there are challenges in controlling and isolating quantum systems, which can lead to errors in calculations.

How is quantum matrix calculation advancing scientific research?

Quantum matrix calculation has the potential to revolutionize scientific research by providing faster and more efficient solutions to complex problems. It can also help scientists gain a better understanding of quantum systems and phenomena, leading to breakthroughs in various fields of study.

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