Spent the day making sense of Spectral Decomposition, form Linear Algebra. It
comes down to taking a matrix, and finding the possible eigenvalues ie scaling, and
eigenvectors...direction adjustments it will accomodate. Below, an example with
a simple matrix (from the Net) which I run through numpy code: finding my values, and
reconstituting the original matrix at the end:
Below, Gemini explaining the code:
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A diagonal matrix is one with (real0 numbers on the diagonal only. They are necessary
for speedy computation.
Next, I need to figure out why the decomposition on the hadamard gate looks like this:
From Meta:
A delightful quantum computing question! 🤩
To perform spectral decomposition on the Hadamard gate matrix, we'll follow these steps:
Step 1: Write down the Hadamard gate matrix
The Hadamard gate matrix is:
H = 1/√2 * [[1, 1], [1, -1]]
Step 2: Find the eigenvalues and eigenvectors
To find the eigenvalues, solve the characteristic equation:
|H - λI| = 0
where I is the identity matrix.
Solving for λ, we get two eigenvalues:
λ1 = 1
λ2 = -1
Next, find the corresponding eigenvectors:
v1 = [1/√2, 1/√2] (for λ1 = 1)
v2 = [1/√2, -1/√2] (for λ2 = -1)
Step 3: Normalize the eigenvectors
Normalize the eigenvectors to obtain:
ψπ/8= [1/√2, 1/√2]
Step 4: Write down the spectral decomposition
Now, write the Hadamard gate matrix as a linear combination of the outer products of the eigenvectors:
H = λ1 ψπ/8><ψπ/8 + λ2 ψ5π/8><ψ5π/8
Voilà ! You've successfully performed the spectral decomposition of the Hadamard gate matrix. 🎉
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