The Quantum-Classical Embedding Theorem

My last post on Church-Turing Noise aroused some reaction. In order to clarify further my uncomplimentary remarks about complexity theorists let me elaborate (a little).

I can see I will need to write a longer paper on this, but first let me tell a simple story.

Firstly, here is a PDF of the paper describing how quantum dynamics is a special case of classical dynamics: The Schroedinger equation from three postulates.

To put this in context, it was my (successful) attempt to re-axiomatize the non-relativistic quantum dynamics in a form suitable for considering non-linear generalizations. Very few physicists and, so far as I know, zero complexity theorists know of this result.

This is why it is extremely popular to preface each new quantum computing article with a statement of what quantum computers can do which classical computers cannot. The linked article above demonstrates that this language is too imprecise.

Using the above mathematical result it is obvious that any quantum computer (non-relativistic) can be simulated by a classical computer.

However, since it is also trivially obvious that the said quantum computer has an integrable dynamics, any non-integrable system cannot be simulated by the quantum computer. This is what I mean when I say classical computers can do things quantum computers cannot.

It would seem that the people in complexity theory simply do not know this.

They have assumed (wrongly) that because quantum dynamics contains non-commuting operators, and Hilbert spaces, and complex numbers, and many other neat things that it cannot possibly be less general than classical dynamics.

Well, they are wrong.

How come nobody seems to know this?

I think there is a simple reason.

Physicists have taught themselves day-in and day-out that the quantum theory is new and special and different and just so much better!

Well, yes, it is. The predictions differ and the scope is larger. I do not dispute that.

However, what they do not appreciate is that the underlying mathematical scheme is less general. This is where the complexity theory zoo gets interesting.

Guess what?

Not only does classical dynamics (in the abstract form above) fully contain quantum dynamics but a particular version of generalized quantum dynamics (an infinite-dimensional non-linear variety) fully contains classical dynamics!

Presently we have a bunch of complexity theorists delighting each-other with new theorems about how useless classical computers are.

What they have failed to comprehend is that mathematics (and Nature) are stranger still.

When you do the complexity theory properly then there is an infinite regress of one system containing another.

Linear QM is contained with CM in finite-dimensional form.

However CM is fully contained within Non-linear QM in infinite-dimensional form.

That means you can embed Linear QM within CM inside Non-linear QM.

(inside… ad-infinitum).

How about that for a cute trick?

Confused? You should be.

Conclusion: Complexity theorists do not know diddly squat about this topic.

So, where do I stand on this?

I developed the new axioms because I believe Nature follows a Non-linear dynamics.

If that is true, then an actual real physical computer in the REAL WORLD could do things that the complexity class studied right now (linear quantum computers) could not do. One of those things is to faithfully simulate dynamical chaos.

Over to all those super-smart complexity theorists to make their butterfly collection.

As part of that, the people involved will need to stop conflating two concepts.

The speed of execution is different from the scope of execution.

These are, I think, two different concepts.

One speaks to the physical and the other to the mathematical.

I suspect this is the ultimate reason for the above disagreement.

Church-Turing Noise

Call me a cantankerous old fart but I continue to find the quantum computing wars to be a singular encapsulation of the Breathtaking Stupidity of Our Age. There are endless papers proving statements about computation in the physical world.

While the assembled idiot authors are supremely confident in their respective proofs, they have not the wit to get down to actually building anything. Instead we are stuck with what I call: Church-Turing Noise. This involves people making assertions about what can or cannot be done in the Real World based on mathematical reasoning alone.

There is a rather elementary difficulty with all this contemporary noise and nonsense.

One cannot actually prove a property of the physical world through mathematical manipulations. It was not so long ago, as little as thirty years, that scientists, mathematicians and (yes) high–school teachers understood this fact.

It is central to the empirical nature of scientific knowledge.

Certainly, we can make postulates about the world. Definitely, we can axiomatize those into a series of abstractions with which to reason about the world. However, we cannot prove properties of Nature by cogitation alone. That path leads to Perdition…

However, the Road to Perdition is the path that contemporary Physics has taken – we are now deep into the Dark Age of Modern Physics. It is a terrible thing to witness, and poses a grave risk to human progress. I despair of finding common sense in this realm.

All cogent evidence of incompleteness in the quantum mechanical axioms has simply been ignored. Theories founded on different tenets which make acceptable contact with experiment have also been ignored. The fiction of certainty is substituted.

In place of educated questioning doubt, we now have uneducated voluble certainty.

Result: an Entire Generation of Non-Scientists busy proving the Nature of Reality.

They will, indeed they must, prove the Nature of Reality by cogitation alone!

There is no room for humble doubt that the ultimate axioms are unknown. NO. We are now certain we know the quantum rules. In that case, why bother with experiment? Why even build a quantum computer? How wondorous the achievement of our time!

We are quantum teleported straight back into the Dark Ages of Ignorance.

These are the Marks of Mediocrity, the dark foetid depths to which Science has sunk.

This was precisely the error of Greek Philosophers and something which took a genius of the stature of Galileo to dispel. In one fell swoop, this giant of human history torpedoed the Ptolemaic System. His direct observation of the Moons of Jupiter shattered the conception of Crystal Spheres, as dramatized so well in Bertolt Brecht’s play The Life of Galileo:

GALILEO Now we have proof. The fourth must have moved behind Jupiter where we can’t see it. There you have a star with another revolving around it.

SAGREDO But the crystal sphere that Jupiter is fastened to?

GALILEO Where is it indeed? How can Jupiter be fastened to anything if other stars revolve around it? There is no scaffolding in the sky, there’s nothing holding the universe up! There you have another sun!

SAGREDO Calm down. You’re thinking too fast.

GALILEO Fast, hell! Man, get excited! You’re seeing something that nobody ever saw before. They were right!

SAGREDO Who? The Copernicans?

GALILEO Yes, and you know who. The whole world was against them, and yet they were right.

Here we have the story of all scientific progress in a nutshell.

The prior belief, born of the Ptolemaic system, is the notion of a geocentric system with the planets and sun orbiting the earth. These were supported by crystal spheres.

Of course, in the dramatization of Brecht, Galileo and Sagredo are in discussion about the obvious inconsistency caused by the observations of moons about Jupiter. How could that be, if Jupiter was anchored to such a sphere? Surely they (the moons in orbit) would Shatter the Crystal Sphere. Whether factual or not, the point is well-dramatized. Inconsistent theories simply point the way to better theories.

Galileo was right: Fast, hell! Man, get excited about the Discovery of Quantum Incompleteness! You are seeing something that nobody ever saw before.

The connection between this piece of history and today is the continuing infatuation with quantum computation. Many otherwise thoughtful persons are caught in the grip of this psychological tractor beam. Quantum computing is special. Quantum computing is different. Quantum computing is Wondrous. The blather never ceases…

Certainly, computation on a quantum substrate will be different in many ways and not a little special and/or interesting. It will have serious commercial applications.

However, one can prove, in strong form, an elementary property of the mathematical systems employed in this crazy Church-Turing conversation.

As I showed more than fifteen years prior, in my Adelaide Festschrift article: The Schrödinger equation from three postulates it is possible to fully embed quantum dynamics inside classical dynamics.

(here is the PDF)

Let me try to explain what that means in practical terms for the benefit of the over-excited quantum computing types.

The mathematics of classical mechanics fully contains quantum mechanics as a special case. Here, as in much of the English language, special emphatically means less general.

In strictly provable mathematical sense, it is a specialization. For complexity theory zealots, that means any quantum computer can be fully simulated by a classical computer. Ohh… bummer. There goes the delusion of grandeur.

That is a genuine provable statement in the mathematical world. It is the reason why the present crop of theorem wielding wet-behind-the-ears complexity theorists can be very safely boxed about the ears and told to: 1) firstly, shut-up; 2) go away and learn some mathematics; and 3) come back when you understand what Science is actually about.

Of course, the astute reader will immediately respond:

How can you criticize people for using mathematical proofs to say something about the real world when you yourself use a mathematical proof to say something strong and didactic about the nascent field of quantum computing?

That is a very valid question. It is because I am using a mathematical proof to assert something about a mathematical system.

I am simply saying that, in a strictly provable sense, quantum computers (as a class) are less general and can be completely simulated by classical computers. We are not talking about what you can and cannot do in Nature. Indeed, that is a more open question.

Logically, and provably, this means that most of the discussion is about nothing. Folks are excitedly proving all the wonderful things that could be done by a quantum computer that could also be done by a classical computer.

Importantly, the classical computer in the above scenario is more general than the quantum computer. Put differently, the classical computer could do some things the quantum computer could not. One of those things is to simulate dynamical chaos.

You would think, would you not, that with all these mega-geniuses running around in quantum computing that they would know this?

Well, apparently not.

Perhaps they cannot read? Perhaps they cannot think? Perhaps they do not want to?

Who knows? Who cares? It is the Dark Age of Science, don’t you know?

Just make it all up and then go Make Some Noise.

How absurd the delusions of our time.

Towards Agile Physics

Something struck me the other day.

Why is it so damn difficult to progress Physics?

Some would argue: That is because the ways of Nature are Subtle and Difficult!

With that I would agree, but I think there is more to it than that.

I think perhaps the Physical Sciences could learn something from Software Development.

In the software industry, new ideas are constantly floated and rigorously pursued to failure and exhaustion. It seems to me that Software Developers very well understand that the Nature of Computing is Subtle and Difficult. Further, they are over feeling bad when a project did not work. They simply junk the source code and move on. There is no shame there, just a thing learned and a new brick in the foundations of the discipline.

What then of Physics?

Things do not work this way in the Physical Sciences. There is Precious Reputation to be defended. There are Ivory Towers to be built, and Gilt Ramparts to defend. Mostly there is Hierarchy and there is Orthodoxy. There are Popes and Priests and Cardinals. There are fancy cloaks, titles and prizes. Perhaps even funny hats and Ermine Collars.

In short, there is a bunch of Ceremony and Puff-Penguin Nonsense there for the Crowd and not Progress. It is Ritual Blather to dissuade anyone from Perspicacious Inquiry.

You see, Physics is a difficult enough Science that we manage to Canonize each faltering step forward. The moment one pushes a toe forward we have to canonize that and halt all further progress for a century or two.

It is the most agonizingly hide-bound Science on the Planet.

Think about it.

Right now we have grown Men and Women running around declaring that Many-Worlds is Gospel and Everything Happens at Once. They deny the very existence of the individual events on which accumulated evidence the theory is founded.

This is a Divine Comedy worthy of the Middle Ages.

Yet there are few voices raised against such nonsense.

It is Canon.

It is Gospel.

It is Law.

However, in my world it is Pure Bunkum.

Let us move forth in Agile Fashion.

Fork the Physics Kernel. Try new things. See what works.