In this note, I will continue to demonstrate not only the ways in which LSRDRs (-spectral radius dimensionality reduction) are mathematical but also how one can get the most out of LSRDRs. LSRDRs are one of the types of machine learning that I have been working on, and LSRDRs have characteristics that tell us that LSRDRs are often inherently interpretable which should be good for AI safety.

Suppose that is the quantum channel that maps a qubit state to a qubit state where we select one of the 6 qubits at random and send it through the completely depolarizing channel (the completely depolarizing channel takes a state as an input and returns the completely mixed state as an output). Suppose that are by matrices where has the Kraus representation .

The objective is to locally maximize the fitness level where the norm in question is the Euclidean norm and where denotes the spectral radius. This is a 1 dimensional case of an LSRDR of the channel .

Let when is selected to locally maximize the fitness level. Then my empirical calculations show that there is some where is positive semidefinite with eigenvalues and where the eigenvalue has multiplicity which is the binomial coefficient. But these are empirical calculations for select values ; I have not been able to mathematically prove that this is always the case for all local maxima for the fitness level (I have not tried to come up with a proof).

Here, we have obtained a complete characterization of up-to-unitary equivalence due to the spectral theorem, so we are quite close to completely interpreting the local maximum for our fitness function.

I made a few YouTube videos showcasing the process of maximizing the fitness level here.

Spectra of 1 dimensional LSRDRs of 6 qubit noise channel during training

Spectra of 1 dimensional LSRDRs of 7 qubit noise channel during training

Spectra of 1 dimensional LSRDRs of 8 qubit noise channel during training

I will make another post soon about more LSRDRs of a higher dimension of the same channel .

I have originally developed a machine learning notion which I call an LSRDR (L2,d

-spectral radius dimensionality reduction), and LSRDRs (and similar machine learning models) behave mathematically and they have a high level of interpretability which should be good for AI safety. Here, I am giving an example of how LSRDRs behave mathematically and how one can get the most out of interpreting an LSRDR.

Suppose that n is a natural number. Let N denote the quantum channel that takes an n qubit quantum state and selects one of those qubits at random and send that qubit through the completely depolarizing channel (the completely depolarizing channel takes a state as input and returns the completely mixed state as an output).

If A1,…,Ar,B1,…,Br are complex matrices, then define superoperators Φ(A1,…,Ar) and Γ(A1,…,Ar;B1,…,Br) by setting

Φ(A1,…,Ar)(X)=∑rk=1AkXA∗k and Γ(A1,…,Ar;B1,…,Br)=∑rk=1AkXB∗k for all X.

Given tuples of matrices (A1,…,Ar),(B1,…,Br), define the L_2-spectral radius similarity between these tuples of matrices by setting

∥∥(A1,…,Ar)≃(B1,…,Br)∥2

=ρ(Γ(A1,…,Ar;B1,…,Br))ρ(Φ(A1,…,Ar))1/2ρ(Φ(B1,…,Br))1/2.

Suppose now that A1,…,A4n are matrices where N=Φ(A1,…,A4n). Let 1≤d≤n. We say that a tuple of complex d by d matrices (X1,…,X4n) is an LSRDR of A1,…,A4n if the quantity ∥(A1,…,A4n)≃(X1,…,X4n)∥2 is locally maximized.

Suppose now that X1,…,X4n are complex 2×2-matrices and (X1,…,X4n) is an LSRDR of A1,…,A4n. Then my computer experiments indicate that there will be some constant λ where λΓ(A1,…,A4n;X1,…,X4n) is similar to a positive semidefinite operator with eigenvalues {0,…,n+1} and where the eigenvalue j has multiplicity 3⋅C(n−1,k)+C(n−1,k−2) where C(⋅,⋅) denotes the binomial coefficient. I have not had a chance to try to mathematically prove this. Hooray. We have interpreted the LSRDR (X1,…,X4n) of (A1,…,A4n), and I have plenty of other examples of interpreted LSRDRs.

We also have a similar pattern for the spectrum of Φ(A1,…,A4n). My computer experiments indicate that there is some constant λ where λ⋅Φ(A1,…,A4n) has spectrum {0,…,n} where the eigenvalue j has multiplicity 3n−j⋅C(n,j).