Quantum Logic and Quantum Logic Gates: The beauty behind the mind and thinking process of a Quantum Computer

Once upon a time in our understanding of mathematics, complex number (square root of minus one) were considered to be mystical, evil, “imaginary” and possess supernatural capabilities; and the best way to deal with them was to avoid or simply ignore them. The lessons we learn from this sort of story is that our beliefs of the past can vary vastly, and be highly intense at the time; but these lessons, nevertheless, always looms around, lurking in the shadows. The ghost of imaginary numbers now hunts our physical conception of quantum phenomenon, and once again we are faced with hard lessons to learn, or remain in epistemological oblivion. If you are reading this article, congratulations! You are amongst those who has decided to face the hard truth about quantum logic and its gates, so let’s start from the beginning.

Classical computational system uses combinations of “yes” or “no” logical units to make complex automated decisions, this is generally regarded as a Boolean logic. A proposition is either true or false, and nothing in-between, thus they are regarded as propositional logic. Come to think about it, it is the natural way we are designed to think; up and down, backwards and forwards, North and South, light and darkness, left and right, and the list goes on. When at the junction of a bivalent logical proposition, based on some certain conditions, we are forced to chose one option: right or left. The conditions that determine what option we chose or eventually take is regarded as logic gate(s): thus, the result of several bivalent propositional logic can be composed to form a truth table. When the word quantum is used in correspondence with logic and gate, things get a lot more interesting and confusing.

The tale of quantum logic is not the tale of a promising idea gone bad; it is rather the tale of the unrelenting pursuit of a bad idea. —?Maudlin, Hilary Putnam.

In trying to understand quantum logic note that its essence, unlike with the Boolean logic, isn’t in the black and white, but in the grey area: that is where the magic happens. Note also that no matter how you cut it, quantum logic doesn’t provide us with propositions whose logic are bivalent; on the contrary, all the various interpretations suggest that we would require an entirely new structure and formalism for describing it. The formal approach that has been generally adopted is that quantum logic is bounded within an orthocomplemented lattice with least element 0 and greatest element 1. What does this even mean?

When complex numbers were finally accepted by mathematicians as valid mathematical objects, the question arose as to where in the number line these numbers should fit. Since it cannot fit into our already neatly ordered number system, the mathematicians had to invent a new branch/layer/dimension for them just besides our regular number system. What does this new mathematical object mean, and what does it describe? At the time, nobody knows, so it was just a mathematical abstraction. The mathematicians played with this new object for years, developing theories and formalisms for it, until suddenly, the physicist found physical phenomenon that fits its descriptions perfectly. Amongst these phenomena, complex numbers formed the core foundations for the mathematical description of quantum mechanics and quantum mechanical processes.

Contrary to other physical theories, such as general relativity, the defining axioms of quantum theory are quite ad hoc, with no obvious physical intuition. While they lead to the right experimental predictions, they do not come with a mental picture of the world where they fit.

The phenomenal question upon which quantum logic was built was, how do we translate the description of a quantum state (built with a combination of complex numbers) into measurements that we can quantify in our ordered number system? In order theory, orthocomplementation on order-reversing self-inversing functions are used to map elements to their complement. In phenomenological terms, particulate objects that we interact with are described in regular space-time (x, y, z, t) which are composed of our usual ordered numbers, while quantum objects are described in what we call Hilbert Space (a space for all numbers, including complex numbers).

With this description, we understand that quantum logic tries to make sense of objects (quantum states) by the process of quantum measurement. While the basic unit of a Boolean logic is the bit (represented by 0s and 1s), the basic unit of the quantum logic is a qubit (a superposition of quantum states). Think of the qubit as a magic dragon ball in a Hilbert Space (a space described by complex number): until you get a hold of it (measure it) it is neither 0 nor 1. But once you get a hold of it (measure it), it collapses into either a 0 or a 1. Forgive my over simplification. Note that the qubit is a combination of two orthonormal basis states, |0> and |1> in a Hilbert Space. It was observed that some physical observables could be described with the provided quantum proposition, but these descriptions were probabilistic due to inconsistencies encountered with quantum measurement.

Furthermore, the fact that nature violates the statistical conditions known as Bell inequalities indicates that the unpredictability of quantum measurement results cannot be explained away as due to ignorance about "local hidden variables" within quantum systems.

There was the idea that the principles of logic might be susceptible to revision on empirical grounds: consequently, not everyone was satisfied with the formal description of quantum logic. A solution to the problem was a logic of properties with a three-valued semantics; each property could have one of three possible truth-values: true, false, or indeterminate. Some believe that there must be some flaw in our understanding of quantum logic that led to the quantum measurement problem. Some posited that there were hidden variables within the computations, other used a method called wave particle collapse. Some even questioned whether quantum logic was actually logic in the sense of it, or just some clever language to describe quantum interactions. Others thought it was just some mathematical categorification.

Tim Maudlin writes that quantum "logic 'solves' the (measurement) problem by making the problem impossible to state."

The question has been asked whether the formal approach to quantum logic is strong enough to host quantum computational capabilities from a rigorous mathematical foundation. Do we need to design new and better form of logic and/or extend the mathematical foundations for the feasibility of quantum computations? These thoughts gave birth to ideas such as modal logic, linear logic, and many more logical systems designed for quantum computational tendencies. During our fight to understand and achieve control over the idea of using quantum logic for computations, we have come to ask ourselves, how can a machine use quantum logic to make decisions? The Boolean logic makes it easy for our classical computers to make decisions using classical propositional logic and logic gates. But how can we use quantum logic to make our quantum computers make similarly observable, translatable and understandable decisions? We would need a quantum logic gate, with which we can construct quantum logic circuit for quantum computations. How are these logic gates composed? Are they found wanting like the quantum logical foundations upon which they were designed?

To be Continued…

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