g(...) = g(
    T[t-1], p[t-1], g(
    T[t-2], p[t-2], g(
    T[t-3], p[t-2], g(
    ...
    T[t-n], p[t-n], g(0)
)))

Right, I think I understood it now. Just to be clear, this is your setup:

1. We insert white ball into a machine
2. Machine spits out either a blue ball or a red ball
3. It’s not possible to whether a blue ball or a red ball will output from the machine given that we only study the white ball, since all white balls are 100% identical
4. We then study the past of the white ball, and we discover that all white balls are dispensed from either a dispenser A and a dispenser B
5. Despite the nonexistence of any transmission of information between a dispenser and the machine, there is a 100% correlation between the dispenser of a white ball and the color of the ball of the output (for example, the machine will always output a blue ball when a white ball comes from dispenser A, despite this white ball being identical to one which came from dispenser B)

The ball itself doesn’t matter, its past does. Or in the case of qubits, its current value and its context.

I don’t know if this came up when you described the non-Markovian example, but this reminded me of a Turing machine. Like, a Turing machine T can be described by f:Q_i → Q_j, where Q_i and Q_j are possible states of T. A regular Turing machine acting through time could be described by f:Q[t] → Q[t+1] and with some boundary condition Q[0]. However, in your example, this hypothetical machine X can also “read” its own past. So, X would be described as g:{Q[t], {Q[t-1], Q[t-2], Q[t-3] … Q[0]} → Q[t+1].

I think I would understand the math of your book, but not a lot about the concepts where it’s applied. I hit a limit on my knowledge when you mentioned quantum gates and matrices on your comment. I don’t know anything about those. I’m going to read more on that both because of curiosity and because I’m reallyyyyy rusty with my math skills.

Thanks for the write-up. I have a lot of review and studying to do. If you don’t mind me asking, do you work mostly on the theory or do you also deal with the practical reality of quantum computing?

Also

some GNU Octave code at the end of each chapter just to give you some hands-on with the math

Based