Let N be the quantum channel on 6 qubits which selects one out of these 6 qubits and then sends that qubit through the completely depolarizing channel (recall that the completely depolarizing channel replaces a quantum state with the completely mixed state).
Suppose that (A_1,...,A_{24}) are the 64 by 64 matrices in a Kraus representation of the channel N. Then N(X)=\sum_{k=1}^{24} A_k X adjoint(A_k) for all 64 by 64 complex matrices X.
Then the goal is to find a complex vector (z_1,...,z_{24}) is 24 dimensional complex Euclidean space where the fitness level rho(z_1A_1+...+z_rA_r)/||(z_1,...,z_r)|| is locally maximized. Here, rho denotes the spectral radius. We maximize the fitness level 6 times using gradient ascent and in each of these instances, and we show the process of gradient ascent in the visualization where each of the 6 instances of gradient ascent is assigned its own color.
For the visualization, we show the spectra of the matrices z_1A_1+...+z_rA_r. After training, the matrices z_1A_1+...+z_rA_r will be Hermitian matrices up to a constant phase factor, so their eigenvalues will all be on the real number line. Therefore, to make the visualization more interesting, we zoom in on the imaginary axis so that we can see how the gradient ascent converges. We even train and zoom in towards the point where the effects of floating point errors (we are using 64 bit floats) begin to confuse the adaptive learning rate algorithm. As a result, when the floating point errors become too significant at the end of training, the learning rate increases to the point where the training becomes unstable.
The notion of an LSRDR is my own, but the notions related to quantum channels in general are not my own. There is a lot going on here. Not only are the matrices z_1A_1+...+z_rA_r Hermitian after training, but we also have a simple characterization of the spectrum of z_1A_1+...+z_rA_r. Here
z_1A_1+...+z_rA_r has eigenvalues {0,...,6} where the eigenvalue j has multiplicity C(6,j) where C(n,k) denotes the binomial coefficient. In particular, in each of the 6 instances of gradient ascent, we arrive at the same spectrum after training.
Since we arrive at the same spectrum after training and since the spectrum is completely interpretable, we can conclude that the notions related to this visualization are not just sound mathematical notions but these mathematical notions are also compatible with one another. These compatible notions include that of the LSRDRs, quantum channels, and the particular noise channel that we use.
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