Roland does a really great job of putting into a couple of pictures the move from random to quantum walks when working in the Tunny-era “sign” (bit!) basis (of –1 and 1)
Not only that, he captures in a theoretical manner that we have seen exhibited in more elementary form as 1950s rotor-based cryptosystems. Whereas the random walk concerns the distance from the “measure” of the constant functions (i.e. the world of the stationary distribution) to the first E-v, the quantum walk takes those same ideas and treats them “circularly”.
That is, the measure is now a refined-measure (capital delta) – the length of the arc (actually fourier transformed) rather than the distance left over after one takes away the 1d shadow cast in the amplitude world). What’s more, in that quantum world where one is moving – via convolution – in phase spaces along arcs in the spectral basis, a “unit” motion may induce an anti-symmetry swap – in which a coin flip associated with the motion impulse in some direction along an arc THEN causes that intermediate phase to flip to its complex conjugate or NOT – depending on the value of the coin flip.
Of course, we saw that in rotor writing theory, too!
where the author postulates an Alfred Small-style rotor looping (i.e. 2 walk steps, where any “step” involves all n rotors)
In the first column, you see 2 groups of 13 characters to which 7 or 8 feedback lines are added – in each group.You also see a second column, also of two groups, where you should imagine that it’s a copy of the first rotor, rotated by 180 so that each half is opposite its anti-symmetric peer on the other wheel (i.e. upper group is opposite lower group , etc). The coin flipper (or plug setting!) , for each output path in the first rotor then decides which of two wires is used – the one to the upper or lower group.
But more than just the controlled swap feature, of the quantum walk, on display. There is the phase conjugacy too. For each group in the column has 10 switching points, which you should think of as 20 input connections. Huh? taking the lowest group of the first column and the lowest square (switching point) of the alphabet A-M (interspersed with 7 additional input lines from the small feedback) the switch is acting on letter at distance (A-M)/2 (somewhere around F). The next switch point up is acting either on F-1 (E) or F+1 (G). The next one further up is F-2 (D) and F+2 (H).If you trace the linear wiring and then realize each connects to a pin/pad on the wheel, one sees the phase space geometry in the input wiring plan, and the output switch reflect the swap operation (as the arc-length motion from 1 to etheta1 may swap over to e-theta1
So, there! we were able to talk all about eigenvalues without requiring you to do lots of boring matrix fiddling. Furthermore one sees how in the quantum world the notion of the limiting distribution is distinct – and is a summation of the contribution of *all* the eigenvalues in the eigenspectrum (reflecting the k-long “spectral impulse” associated with each k-long path in what is now a “spectral basis”). Since in quantum spaces one has entanglement states (which is “non-linear”), SEE HOW one captures how the terms of each generated wave functions for each step on the evolution can be co-dependent in that unique and quantum-mechanical –only world. This is quantum mixing (and NSA/GCHQ/IBM quantum searching, moreover).