Classical Mental Models are Incommensurate with the Quantum Mindset

The Classical and Quantum Mindset

My blog today is to draw the reader into questioning one's own mindset in attempting to conceive and imagineer new quantum inspired or quantum computing models.  I will get back to the probability density polytope shortly from the previous post.

I would like to start out with a quote from a really good book on Quantum Computing:

"Algorithm design for quantum computers is hard because designers face two difficult problems not faced in the construction of algorithms for classical computers. First, our human intuition is rooted in the classical world. If we use that intuition as an aid to the construction of algorithms, then the algorithmic ideas we come up with will be classical ideas. To design good quantum algorithms one must ‘turn off’ one’s classical intuition for at least part of the design process, using truly quantum effects to achieve the desired algorithmic end. Second, to be truly interesting it is not enough to design an algorithm that is merely quantum mechanical. The algorithm must be better than any existing classical algorithm! Thus, it is possible that one may find an algorithm which makes use of truly quantum aspects of quantum mechanics, that is nevertheless not of widespread interest because classical algorithms with comparable performance characteristics exist. The combination of these two problems makes the construction of new quantum algorithms a challenging problem for the future."   Nielsen, Michael A.; Chuang, Isaac L. (2010-12-09). Quantum Computation and Quantum Information. Cambridge University Press. Kindle Edition. 

In simple terms, this hits to the heart of the Church-Turing Thesis (which has never been well challenged, disproved nor proved) - so it would seem that it will be very hard to build efficient true quantum algorithms,  even harder to build a quasi-quantum, pseudo-quantum algorithm, and plausibly moderately hard to build a quantum-inspired algorithm.

Ultimately, and at the foundation of current systems, it must hold from whichever variety of Church-Turing Thesis that you would like, that a conception of any programming language must be a Turing machine:  if, however, the foundation is changed, then conceptually we are in a place where the current history of science has not yet ventured and I have no idea about what that might look like - I admit it can exist, but, that the foundations must be radically not as they are today (in the conception of theories in modern physics).

However, in attempting to get outside the box of classical thinking, it does serve some purpose to expound upon a simple concept: the Qubit and the Bit.

In reviewing the traditional understanding of a Qubit, it is stated often that a qubit is just like a classical bit only that it can probabilistically flip between a state of "0" and a state of "1".   Therefore, the question oft comes up why we can not just use a statistical (hidden) variable with a probability distribution function that represents either "0" or "1" by a certain probability.

Here is the answer: because in the classical case, we have already chosen how the probability will outcome while in the quantum case we are forced to measure and interact.

This, in a deep way, implies that there is no way to describe a qubit classically. This point goes back to the essence of the quote from Nielsen and Chuang's text on Quantum Computation and Quantum Information.

Another difference is that a bit is a point-instance object while a qubit is a system: it is a quantum-mechanical system and can hold more information than just one bit. The outcome of this is that you cannot redefine a qubit as an interpolation between a function that outputs "0" and "1" in the same way that you can describe, say, a fair coin toss.  They completely not related.

However, my personal view is that while we cannot yet work effectively and efficiently at the level of individual qubits (though research is making progress), what about an ensemble?  An ensemble will exhibit statistical properties just like any ensemble and the relevance of individual qubits becomes subsumed within the ensemble, so that, at least from the algorithmic viewpoint, we have indirect access to qubits as a result of the statistics.  Is that possible?

The implication of that last statement is that ensembles, as long as they are made up of quantum systems, store individual qubits but can be indirectly manipulated via classical techniques at the level of the ensemble.  In the real world, for example, even though we do not have access to the individual quantum systems of a Superconductor, nonetheless, we have access to the ensemble effects of the quantum systems (i.e. its superconductivity properties) and indeed companies like D-Wave have taken advantage of this.

What about the research on individual qubits? Qubits are idealized mathematical objects and in nature, discoveries on how to implement the qubit abstraction have resulted in, for example:  the states of electrons orbiting an atom, or the nuclear spin in magnetic fields or as the polarizations of a photon.

And in between the pure ensemble model and individual qubit model is the model of the topological qubit that can be implemented in terms of states of matter (either via an ensemble or via a unit), such as for example, in  Graphene.  I believe, the future, that the novel functionally graded doped versions of Graphene will result in topological qubits (a personal belief of mine).

What does all this tell us about the Classical versus the Quantum mental models or mindsets that we use.  I claim that it tells us the following:  that we need to base our thinking on mathematical abstractions first, and then seek to discover design models for these abstractions in terms of real world physics, or equivalently, for us computer geeks, in terms of Turing Machine models.

Until Next Time!