GenZedong

The digital PDF is about 150 pages, so it's not incredibly long. (Page count may be different for physical version.) It is also written so that one can follow along even if they have no technical background, although it will be harder to do so. The mathematics used are explained, with some GNU Octave code at the end of each chapter just to give you some hands-on with the math. So if you want to learn it in the process of reading it, you can, but someone who already knows the basics of quantum computing could probably skim through the whole first third of the book, with only the latter 100 pages actually assuming you know the mathematics enough to be able to simulate a quantum circuit.

Also, as per your notes, " non-Markovian (ie: non-Stochastic)", non-Markovian does not mean non-stochastic.

Consider that you apply a logic gate to a bit. The outcome effect on the bit will depend upon two things: (1) the definition of the logic gate, given by a truth table, and (2) the present value of the bit. You could thus imagine that the present bit's value at time t is p[t], its future value at time t+1 is p[t+1], and the truth table is given by T[t] describing the interaction or logic gate at time t, and then you could define this transition in terms of:

p[t+1]=f(T[t],p[t])

i.e., some function that takes the truth table and the present state of the bit as an input, and outputs the future / altered state of the bit.

This is the structure of a Markovian transition law. It takes what is effectively a truth table describing the effect of an interaction (represented by a matrix) as an input, as well as the present state of the system, into a function, and the function outputs the altered state of the system.

A non-Markovian law takes a different form.

p[t+1]=f(T[t],p[t],g(...))

If you want to evolve p[t] to p[t+1], you additionally need a second function g(...), which is defined recursively into the past, meaning it expands out into...

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

Every layer you expand g(...) goes one time interval back into the past, until it reaches a base case where the expansion stops.

That is to say, if you want to compute how a logic gate stochastically perturbs a bit's value in a quantum circuit, you need a function which takes into account not just the current truth table for the current logic gate (the operator given by U) and the current statistical distribution of the bits (given by |ψ|²), but all previous ones as well, going back to the beginning of the circuit.

Let me give an analogy. Imagine you have a machine which you pass a white ball into and sometimes outputs a red ball or a green ball. You are not sure what determines whether or not it outputs a green or red, so you study the white ball very closely, but no matter how detailed your measurements are, you cannot find anything that distinguishes one white ball from another. They all seem truly identical.

But then, later, you discover that there seems to be a perfect correlation between whether or not the white ball comes from dispenser A or dispenser B as to whether or not the machine will transform it into a red ball or a green ball. The "cause" of the machine outputting a red or green ball seems to have no relevance to the actual details of the white ball itself, but rather, where the white ball came from in the past.

The non-Markovian stochastic laws in quantum theory behave in a similar fashion. The statistical laws don't just care about the current statistical state of the system, but also where it came from, meaning, they also take into account its historical evolution as well. This means the same logic gate can have different behavior if the gates preceding them in quantum circuit are different, or if the bits begin with different initial values.

The logic gates are described by unitary operators which are complex-valued, and so they don't directly tell you the stochastic perturbation each logic gate applies to the bits. Stochastic perturbations are described by a stochastic matrix which are real-valued, positive, and all columns must sum to 1. So, you have to somehow compute what the correct stochastic matrix is from the unitary operators.

It's proved in the academic literature many times over that it's impossible to assign a single real-valued stochastic matrix to each unitary matrix that remains always positive where columns always sum to 1. The only possible way around this is to allow the assignment of different stochastic matrices to the same logic gate under different conditions. That is to say, the behavior of the same logic gate can change if the surrounding context around the logic gate changes.

The simplest example of this is the Hadamard gate in a quantum circuit that starts in a degenerate distribution. If you apply the Hadamard gate once, it behaves like a fair coin flip. If you flip a coin twice, the first time, the outcome is a uniform distribution, and the second time, the outcome is also a uniform distribution. But for the Hadamard gate, it only gives you a uniform distribution if you apply it once. If you construct a quantum circuit consisting of two Hadamard gates, it gives you a degenerate distribution.

There is no stochastic matrix that reproduces this behavior. You have to assign two separate stochastic matrices to the two separate instances of the two Hadamard gates: the first one moves it to a uniform distribution, [0.5 0.5; 0.5 0.5], and the second moves it to a degenerate distribution, either [1 1; 0 0] or [0 0; 1 1] depending upon whether the bit was originally initialized to 0 or 1. The book gives a general formula for computing the correct stochastic matrix for any logic gate, but the formula requires you to take in as input all previous gates and statistical distributions of the bits in the circuit up to that point.