Joseph Van Name comments on Joseph Van Name’s Shortform

[Joseph Van Name]

I have originally developed a machine learning notion which I call an LSRDR ($L_{2,d}$

-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 $A_1,\dots ,A_r,B_1,\dots ,B_r$ are complex matrices, then define superoperators $\Phi (A_1,\dots ,A_r)$ and $\Gamma (A_1,\dots ,A_r;B_1,\dots ,B_r)$ by setting

$\Phi (A_1,\dots ,A_r)\left(X\right)={\sum }_{k=1}^r{A}_{k}X{A}_k^{\ast }$ and $\Gamma (A_1,\dots ,A_r;B_1,\dots ,B_r)={\sum }_{k=1}^r{A}_{k}X{B}_k^{\ast }$ for all $X$.

Given tuples of matrices $(A_1,\dots ,A_r),(B_1,\dots ,B_r)$, define the L_2-spectral radius similarity between these tuples of matrices by setting

$\Vert (A_1,\dots ,A_r)\simeq (B_1,\dots ,B_r){\parallel }_2$

$=\frac{\rho \left(\Gamma \right(A_1,\dots ,A_r;B_1,\dots ,B_r\left)\right)}{\rho \left(\Phi \right(A_1,\dots ,A_r\left){)}^{1/2}\rho \right(\Phi (B_1,\dots ,B_r){)}^{1/2}}$.

Suppose now that $A_1,\dots ,A_{4n}$ are matrices where $N=\Phi (A_1,\dots ,A_{4n})$. Let $1\le d\le n$. We say that a tuple of complex $d$ by $d$ matrices $(X_1,\dots ,X_{4n})$ is an LSRDR of $A_1,\dots ,A_{4n}$ if the quantity $\parallel (A_1,\dots ,A_{4n})\simeq (X_1,\dots ,X_{4n}){\parallel }_2$ is locally maximized.

Suppose now that $X_1,\dots ,X_{4n}$ are complex $2\times 2$-matrices and $(X_1,\dots ,X_{4n})$ is an LSRDR of $A_1,\dots ,A_{4n}$. Then my computer experiments indicate that there will be some constant $\lambda$ where $\lambda \Gamma (A_1,\dots ,A_{4n};X_1,\dots ,X_{4n})$ is similar to a positive semidefinite operator with eigenvalues $\{0,\dots ,n+1\}$ and where the eigenvalue $j$ has multiplicity $3\cdot C(n-1,k)+C(n-1,k-2)$ where $C(\cdot ,\cdot )$ denotes the binomial coefficient. I have not had a chance to try to mathematically prove this. Hooray. We have interpreted the LSRDR $(X_1,\dots ,X_{4n})$ of $(A_1,\dots ,A_{4n})$, and I have plenty of other examples of interpreted LSRDRs.

We also have a similar pattern for the spectrum of $\Phi (A_1,\dots ,A_{4n})$. My computer experiments indicate that there is some constant $\lambda$ where $\lambda \cdot \Phi (A_1,\dots ,A_{4n})$ has spectrum $\{0,\dots ,n\}$ where the eigenvalue $j$ has multiplicity $3^{n-j}\cdot C(n,j)$.