Part 2) a little bit of those details

Now a little bit of math and those details =) "But don't be afraid, it's not much, it won't hurt."
Just the basics, what about the effective upcoming of classical quantum systems .
Let me show you an example.
We approximate an n-kbit quantum state p by performing random one-kbit Pauli measurements on T copies of p. That is, we measure each cubit of a quantum state in a random Pauli basis- X, Y , Z-to get a measurement result of +- 1.
The collapse of the wave function implies that this measurement procedure turns p into a random pure thing state . Each element represents a highly structured pure state with one qubit, and a total of nT of them. So, 3nT bits are enough to store the entire collection in classical memory.
Randomized measurements can be performed under real-world conditions. Physical experiments or with classical simulations. The data obtained can then be used to approximate the base state of the n-cubit p:
(...is a sequence of numbers) p ≈ 𝜎T (p) = 1/T ∑ from T to t=1 𝜎..1^(t) ⊗ - - - ⊗ 𝜎..n^(t) where 𝜎..i^(t) = 3|S..i^(t) ⟩⟨S..i^(T) | - I

And I denotes a 2 × 2 matrix . This classical shadow representation , exactly reproduces the density matrix in the limit : 𝑇 tends to infinity , but T = O(const^r log(n)/ 𝜖 2) provides a 𝜖-accurate
approximation of all reduced r-body density matrices (in the trace distance). This, in turn, means that we can use 𝜎T(p) to predict any function that depends only on the reduced density matrices, such as the expected values of the sum of local observables and the sum of entanglement entropies of small subsystems. Classical storage and postprocessing also remain available.
To summarize, the classical shadow formalism is equipped with an effective quantum-classical transducer, which allows classical machines to efficiently and reliably estimate properties of subsystems of any quantum state p.

In general after this I could write a model of construction and training of ML, present information about classical shadows, more detailed experiments, show the code of ML and so on. But that would be long, and it would be better to see it in the article, and now in conclusion I want to tell about training neural networks with classic shadows for quantum problems.

Generally there are several ways of combining classical shadows and neural networks.
Here I will show only one of such approaches: using the theory of neural tangent kernels . It is noteworthy that this theory allows us
effectively train different types of neural networks (superprecision/graphic/etc.) with an infinite number of neurons in each hidden layer (infinite width). Thus, let us take the data set . In the infinite width limit, we can analytically solve the neural network after training on the data set {x^ℓ, y^ℓ} where x^ℓ and y^ℓ are vectors of some size. Next I could give you an example of neural network training, but I won't because the mathematical part doesn't look good on Telegram, see for yourself on page 17.
Hopefully this makes neuronics help in quantum problems a little bit clearer.

But if you want more details, look at the article.
And that's it, thanks for reading ❤️.

And remember, if you look at the white rabbit, it will run the other way 🐰
#quantum_mechanics #neural_networks