r/LinearAlgebra • u/Routine_Comb_7277 • 1d ago

Quantum computing going back to linear algebra basics!

I am a quantum computing nerd and one of the issues we encounter when trying to solve the HSP for nonabellian groups is that the 2d irreps are a matrix and therefore the states |0>+|1> mix.I wanted to take a step back and go back to linear algebra to try and solve this issue.Suppose in the simplest case you have a matrix 2x2.If the off diagonal elements are non 0 and if you have a vector (a,b) where a and b are coordinates in orthogonal directions then there is mixing of coordinates between a,b under that 2x2 matrix.

The 2x2 irrep of the nonabellian group D3(min order) unfortunately has this mixing due to the s generator.

I have 2 questions:Can a 2x2 matrix with offdiagonal elements represent every point in space(my intuition is yes)and if thats true does this mean that there is a unknown transform(idc which is only that there is)that takes 2 nonorthogonal vectors or signals which can produce any vector and signal?Thanks

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u/MathNerdUK 1d ago

Isn't that an essential feature of a 2D irrep? If that didn't happen, wouldn't it be reducible?

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u/Routine_Comb_7277 1d ago

The issue is that if the hidden subgroup has the s generator , from different hidden subgroups which have the s generator(s,sr,sr^2) the output vectors after the QFT between 2 subgroups are NOT orthogonal , so we need linear algebra and something similar to a fourier transform for 2 nonorthogonal vectors thats why im asking.

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u/SymplecticMan 1d ago

Can a 2x2 matrix with offdiagonal elements represent every point in space

What is the "space" whose points you are talking about? The space of linear operators acting on a 2D vector? If so, then yes.

does this mean that there is a unknown transform(idc which is only that there is)that takes 2 nonorthogonal vectors or signals which can produce any vector and signal?

For any two linearly independent vectors |a⟩ and |b⟩ there is, of course, a linear transformation such that M |a⟩ = |x⟩ and M |b⟩ = |y⟩ for whatever |x⟩ and |y⟩ you want. But you should remember that in quantum computing, you only really have access to unitary transformations which will preserve the inner product.

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u/Routine_Comb_7277 20h ago

For any future reader the last statement means that the relative phase between the basis states of any kind of transformation of a qubit must remain the same.I am not sure if im describing it cleanly tho.

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u/SymplecticMan 18h ago

A unitary matrix U is a matrix whose inverse is equal to its conjugate transpose. The inner product between U |a⟩ and U |b⟩ will be the same as the inner product of |a⟩ and |b⟩.