Quantum Logic and Quantum Logic Gates: The beauty behind the mind and thinking process of a Quantum Computer, Part 2.

Can quantum logic actually be considered as a logic given our contemporary (Boolean) logical sense of reasoning? If so, then how so? Or is it simply an ad hoc derivative from experimental observations from the principles and processes of quantum mechanics? Is quantum logic valid or useful? What are the fundamental characteristics of logic? Is logic empirical? These and many more of such questions and debates has led to the development of countless proposition about quantum logic to the point where, truthfully speaking, it seems nobody knows exactly what quantum logic really is. As a matter of fact, the word “know” in this context is another deep, dark and tortuous philosophical tunnel that leads to more confusion. The point of asking these questions and taking the answers we come up with seriously is because logic forms the foundations to any scientific/technological structure we built thus far: For this reason, we cannot afford to be sloppy or hasty in assigning formalism and standardization to quantum logic and gates without proper experimental results, and a rigorous theoretical/mathematical framework to back it.

I think I can safely say that nobody understands quantum mechanics - Richard Feynman

If you missed the first part of this newsletter, please read for continuity: https://www.dhirubhai.net/pulse/quantum-logic-gates-beauty-behind-mind-thinking-process-felix-wejeyan-micpc

It is imperative at this point to understand that contemporary quantum programs are sequences of instructions that are interpreted by classical computers to perform a particular task. This task is usually in the form of a low-power signal provided by a Digital Signal Processor (DSP) which tries to control an isolated quantum system to perform computation(s). After which, the quantum system (computer) returns a very low-power signal (which usually requires several stages of amplification) back to the DSP, before the signal is sent back to a classical computer for interpretation of result. So, understand that contemporary quantum logic and gates are designed directly for classical computers to process, and then afterwards, the classical computer interprets the output signal from the quantum computer using same logic and gates.

What the quantum computer does with these signals is highly probabilistic, and how it does it is none of our business. We try to send in the right signals, and hope to get interpretable and understandable signals from the black box (the quantum computer). The input signals are designed by classical programs based on some combination of quantum logic, operations, and gates in the form of a quantum algorithm/program: So, understand that with contemporary quantum logic and gate, you are not communicating directly with the quantum computer, but providing it with appropriate signal(s) for required computation: and it is therefore expedient that we understand how to send and process these signals. ???

The fundamental unit of quantum logic is the qubit. A qubit is a coherent superposition of the basis state “ket-0” (|0>) and “ket-1” (|1> ): this is usually regarded as a two-state quantum-mechanical system. One might ask, why two-state? Isn’t this a corollary of the two digits of classical logic 0s and 1s? ?This is a tricky question, whose answer can neither be “yes” nor “no”. The sum of the probability amplitude of a qubit’s basis states is always normalized (that is, equals to 1). Note that theoretically, there can be an infinite number of basis states that forms a qubit, and the sum of their probability amplitude would always equal to 1. There are some observable quantum-mechanical systems that possess a two-level/two-state characteristics, for example the spin of an electron (spin up and spin down) or the polarization of a single photon (left-handed and right-handed circular polarization).

“I wouldn't be that stuck with binary logic. We still don't know if it is underlying the fundamental physics” – Michael Grochol

In choosing the number of level-state with which to describe a qubit, we are faced with two options: we can either chose infinities (which, like complex numbers once was, is another scary and mysterious mathematical concept which the mathematicians, thanks to Dedekind’s cut, are still trying to understand and manipulate), or chose a two-state level (which is very similar to classical logic, and embodies some fundamental quantum properties such as spin and polarization). At this point, the choice of a two-level system for the qubit should be quite obvious. It is believed that with superdense coding, the idea of quantum entanglement can be used to fully encode up to two bits in one qubit. In other words, classical logic can be a subset of quantum logic.

The problem in nullifying "classical" logic is that you draw the fundamentals from underneath mathematics, and hence you can't even formulate anything anymore. You need classical logic to even be able to say what is a Hilbert space. And once you've accepted classical logic, you're "stuck" with it. You can build mathematical theories on top of that, but you cannot pull the floor from under your construction or everything you have been building falls down too… You cannot build quantum logic upon quantum logic because you can't even construct natural numbers. You don't have axioms, you don't have theorems, you don't have demonstrations, and you don't have results – Patrick Van Esch.

Quantum logic does not provide much solid grounds to build mathematical theories on, rather we are to base our foundations on probabilities. Although we know that professional gamblers can pull this off as a way of living, but as Einstein said, “God does not play dice with the universe”. So how do we not play dice while playing a dice game with quantum computation? We represent the classical bit 0 by a vector |0>?and 1 by |1> . A qubit, which is a superposition of both basis states, can either remain in such state or collapse to either of the two basis states. A collection of qubits is called a register, but note that a qubit cannot be concatenated.

The qubit, as a basic unit of quantum logic, can undergo a limited number of operations. It can be sent through a quantum channel, initialized or reinitialized, measured by collapsing the coherent state into its basis state, or operated on by quantum logic gates which forms the building blocks of the quantum circuit of a quantum computer (this is essentially the brain of the quantum computer).

An experimenter is an 'explorer' who uses maps to go to the edge of what is reliably known. The explorer's problem is that at some point, he might push beyond the map, usually resulting in more questions than answers. Meanwhile, the mathematician tries to extrapolate based on the existing map. Back onshore we may be able to determine how reliable an effect is (with math/logic), however out on the boat, there may be no common reference point (causality) available to make sense of anything. So going 'quantum' (if you're out there in the boat) means the map is near-useless. So, you let the boat do its own thing and note what the boat is doing. You can indeed map what the boat does, but you can't really recreate it as a logic ladder, because you never know whether the Cat is dead or alive. So, you design using a crazy amount of fault-tolerance (probabilities) to accomplish your end-goal either way regardless of the particular condition of the Cat :) – Jeffery D Krause

For an observer at the Boolean Island (onshore), the mathematicians are the map designer, using experimental and logical extrapolations to map out the entire topology of the Island. The physicist (and computer scientist in this case) are the voyager trying to navigate the sea of the quantum world, extrapolating the Boolean map into new logical domains and trying to tie this logic to that which is already known at shore – Boolean Island. And the Imperialist/Capitalist are the pirates of the Quantum Caribbean Sea, trying to take advantage of whatever treasures the physicist finds, and due to their desperation, are therefore ready to believe anything, even if it is complete hokum. Don’t be mistaken: the physicist and computer scientist/engineers are doing a great job in trying to understand the concept of quantum logic, gates, computations and its applications, given the contemporary knowledge, resources and technology available; it is the capitalistic and marketing forces that over-hype the present infant nature of quantum computers and programs.

Now that we have translated quantum logic into a mathematical formation that we can manipulate using classical computers, our next task is how do we use this language to speak directly to the quantum computer? What quantum mechanical processes, can we take advantage of in order to effectively communicate with a quantum computer and manipulate its internal states to make computations whose results can be relayed back to us? Luckily enough, there seem to be a few that stands out, and these few seems to be Boolean friendly as well – serendipity, eh? ?Examples are the Pauli’s X, Y and Z, Hadamard, Identity, Phase, SWAP, CNOT, Toffoli and many more gates. These gates act as unitary operators on Hilbert space that preserves their inner product. An oversimplification would be that the quantum gates use Boolean logical structures to map probabilistic areas of Hibert space of an isolated quantum system.

Remember that contemporary quantum algorithms do not communicate directly with the quantum computer; so just as the classical logic gate is used to directly control a computer circuit, a quantum gate doesn’t. Rather, it is a farfetched language that we hope the quantum computer would understand because we have experienced some unusual quantum-mechanical processes that are similar to the logic and gates we are applying. This is what we have at the moment; a hybrid collaboration between the classical computer and the quantum computer.

Although, it is believed that our understanding of quantum computation is in its infancy, but is this the best approach we can come up with? Is there a way we can communicate directly with the quantum computer without using corollary of classical logic? What language would we need to develop to do so? What are the modalities we need to change or readjust to achieve this? If achieved, how useful would this be? And for what purpose? What kind of machines and Artificial Intelligence can we build from this technology? Given that quantum logic has refused to play by the deterministic rules of propositional logic (classical computation), can a quantum neural network (QNN) be able to decide for itself the difference between right or wrong? Good and evil? The questions are unending, and so are the possibilities available for the future of quantum technology.

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Next week’s publication: Quantum Computations and the birth of Quantum Computers: What can they do, and what can they not do?