Quantum Mechanics and Consciousness

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 Mefiante (September 10, 2009, 16:55:17 PM):
In what way would a quantum computer essentially differ from [a hybrid computer?]
As the linked-to article describes, a hybrid computer uses an analogue “computer” to generate a reasonably good initial value (or set of values) that forms the starting point for digital processing by a normal computer in cases where problem-solving requires iterative techniques. The reason for doing this is basically to reduce the solve time on the digital computer: the better the initial guess, the fewer iterations are needed to satisfy some predefined tolerance or accuracy criterion.

An analogue “computer” must not be thought of as comparable to a digital computer because it doesn’t per se perform any calculations. Instead, it simulates one physical process by another that is mathematically similar – hence the “analogue” descriptor. Something as simple as an electronic Inductance-Resistance-Capacitance (LRC) circuit built with variable resistors, capacitors and/or inductors can qualify as an analogue “computer.” Such a circuit can, for example, be used to simulate forced damped vibration behaviour in materials (or mechanical wave propagation in them) by subjecting the circuit to AC current of an appropriate frequency and measuring certain electrical responses in the circuit. Another example is to simulate stress/strain state changes in materials subjected to impulsive forces by measuring transient electrical behaviour in the circuit. Such problems have quite complex formulations that are of the same or a very similar mathematical form to that of the analogue that is used to simulate them, usually a set of linked non-linear partial differential equations.

Put briefly, an analogue computer is a clever way of initialising the analysis of a problem in order to reduce the computational effort that a digital computer alone would need and thereby reduce the processing time. Analogue computers are not general computing machines like a digital computer is. They are limited in their applicability to a small set of problems and are purpose-built for specific problems, which is perhaps the main reason why they and also hybrid computers are less favoured.

So much for the background.

It should be clear from the above that a hybrid computer is fully deterministic because the digital aspect of it greatly refines the approximate answer provided by the analogue component. In contrast and as described in an earlier post, a quantum computer is not deterministic, and that is the essential difference between the two types. One possible way of thinking about a quantum computer is to picture it as an array of bits (i.e. binary digits) that have a curious property: until each bit is actually examined, it exists in a superposed state of both 0 and 1, and it becomes definitely 1 or 0 only once it is examined. Such special bits are called “qubits.” Moreover, the state that each qubit will assume upon examination depends on its neighbours and what operations have been performed on the whole collection of them and in what order (this only works if the qubits are entangled, and this is the main technical obstacle in the way of the “quamputer”). These operations and their sequence can be thought of as the algorithm.

Here’s a very simple example for the sake of illustration: Suppose you have a four-qubit computer. It has 16 (= 24) possible states. Suppose further that an algorithm is loaded that forces the third qubit always to be the complement (not = negation) of the exclusive-or (xor) result of the first two, and the fourth qubit is the logical-and (and) result of the second and third qubits. This algorithm has two degrees of freedom because the third and fourth qubits are fixed by the value of the first two. The algorithm also predisposes the first qubit to come up high (= 1) 80% of the time, and low (= 0) for the remaining 20%, while the second qubit comes up high 33% of the time and low over the remaining 67%. Because this is a trivial problem, it’s not hard to work out the probability of each of the four possible states but it should be clear that the complexity of the algorithm and the quantum computer can be vastly increased in theory to address more meaningful problems. While the quantum computer can be used to implement such a simulation directly, a deterministic computer must either analyse the problem symbolically or use a source of randomness (or, more usually, pseudorandomness) to compute the likelihood of the possible outcomes.

'Luthon64
 Peter Grant (September 13, 2009, 19:03:29 PM):
It should be clear from the above that a hybrid computer is fully deterministic because the digital aspect of it greatly refines the approximate answer provided by the analogue component. In contrast and as described in an earlier post, a quantum computer is not deterministic, and that is the essential difference between the two types.

Hmm, I guess I just assumed that a quantum computer would also have a deterministic, digital component which we would use to interface with the quantum part. I'm more interested in the the analogue part at the moment though. Would it be deterministic? If not, is it non-deterministic for a very different reason than the quantum computer?

This is the part which really grabbed my attention:

Quote
Consider that the nervous system in animals is a form of hybrid computer. Signals pass across the synapses from one nerve cell to the next as discrete (digital) packets of chemicals, which are then summed within the nerve cell in an analog fashion by building an electro-chemical potential until its threshold is reached, whereupon it discharges and sends out a series of digital packets to the next nerve cell. The advantages are at least threefold: noise within the system is minimized (and tends not to be additive), no common grounding system is required, and there is minimal degradation of the signal even if there are substantial differences in activity of the cells along a path (only the signal delays tend to vary). The individual nerve cells are analogous to analog computers; the synapses are analogous to digital computers.

Put briefly, an analogue computer is a clever way of initialising the analysis of a problem in order to reduce the computational effort that a digital computer alone would need and thereby reduce the processing time. Analogue computers are not general computing machines like a digital computer is. They are limited in their applicability to a small set of problems and are purpose-built for specific problems, which is perhaps the main reason why they and also hybrid computers are less favoured.

But wouldn't a programmable analogue computer be seriously cool? Imagine being able to write programs which ran simulations that were actually real! (At least in the numerical sense)

NEC just started miniaturisation in June this year:

http://www.necel.com/news/en/archive/0906/1802.html

Also check out Factorizing RSA Keys, An Improved Analogue Solution:

 Mefiante (September 14, 2009, 14:27:38 PM):
Apologies for the delayed reply – I have been indisposed these past few days.

Hmm, I guess I just assumed that a quantum computer would also have a deterministic, digital component which we would use to interface with the quantum part.
Well, yes, probably there would be such but it would be a component merely serving an input-output (IO) function. Its presence would most assuredly not suddenly change a quantum computer into a deterministic machine. That would be a bit like saying that the presence or absence of a speedometer in your car changes the type of fuel it needs between petrol and diesel.

Would [the analogue part of a hybrid computer] be deterministic?
Ideally, yes – or as close to it as macroscopic (i.e. non-quantum) models will allow in theory. In actuality, all real analogue devices are subject to certain inaccuracies, however small they may be. These inaccuracies are for all practical purposes random, if not entirely unknowable, and they could be the result of any number of environmental factors or conditions. The errors may be tiny but no instrument can measure with 100% precision the real quantity it was designed to measure, and it is furthermore highly doubtful whether perfect accuracy is even achievable.

But wouldn't a programmable analogue computer be seriously cool?
Sure it would, but it is hard to see how one might go about constructing a general-purpose programmable analogue machine. As outlined earlier, an analogue machine is one that makes use of a process that is mathematically similar to the one of interest, whereas a digital computer (usually) treats the mathematics itself of the process of interest (which is why a digital computer isn’t inherently limited to a rather narrow range of simulations or physical problems). It is not apparent how one might find an analogue of sufficient generality (short of reality itself, which clearly isn’t an analogue) to cover all of the requirements adequately for a super analogue computer to be built.

'Luthon64
 Peter Grant (September 14, 2009, 19:24:15 PM):
Apologies for the delayed reply – I have been indisposed these past few days.

No worries, gave me time to do some more reading. ;D

Hmm, I guess I just assumed that a quantum computer would also have a deterministic, digital component which we would use to interface with the quantum part.
Well, yes, probably there would be such but it would be a component merely serving an input-output (IO) function. Its presence would most assuredly not suddenly change a quantum computer into a deterministic machine. That would be a bit like saying that the presence or absence of a speedometer in your car changes the type of fuel it needs between petrol and diesel.

Agreed, in the same way that, with this quantum consciousness theory, the digital parts of our nervous system do not suddenly change our brains into deterministic machines. The same could be said for an analogue/digital hybrid consciousness theory.

Would [the analogue part of a hybrid computer] be deterministic?
Ideally, yes – or as close to it as macroscopic (i.e. non-quantum) models will allow in theory. In actuality, all real analogue devices are subject to certain inaccuracies, however small they may be. These inaccuracies are for all practical purposes random, if not entirely unknowable, and they could be the result of any number of environmental factors or conditions. The errors may be tiny but no instrument can measure with 100% precision the real quantity it was designed to measure, and it is furthermore highly doubtful whether perfect accuracy is even achievable.

Are you sure? This quote from Wikipedia seems to say the opposite:

Quote
Although digital computer simulation of electronic circuits is very successful and routinely used in design and development, there is one category of analog circuit that cannot be simulated digitally, and that is an (analog) circuit made to exhibit chaotic behavior. Because everything in the analog circuit is essentially simultaneous, but a digital simulation is sequential, simulation a chaotic circuit fails.
http://en.wikipedia.org/wiki/Analog_computer

But wouldn't a programmable analogue computer be seriously cool?
Sure it would, but it is hard to see how one might go about constructing a general-purpose programmable analogue machine. As outlined earlier, an analogue machine is one that makes use of a process that is mathematically similar to the one of interest, whereas a digital computer (usually) treats the mathematics itself of the process of interest (which is why a digital computer isn’t inherently limited to a rather narrow range of simulations or physical problems). It is not apparent how one might find an analogue of sufficient generality (short of reality itself, which clearly isn’t an analogue) to cover all of the requirements adequately for a super analogue computer to be built.

'Luthon64

They have designed programmable analogue computers, but they still use punch cards! I'm talking about miniaturization and bringing the complexity up to that of today's digital computers. As to finding analogues of problems, as with digital computing, one breaks them down into components. Analogue computers are great at:

* summation
* integration with respect to time
* inversion
* multiplication
* exponentiation
* logarithm
* division, although multiplication is much preferred

BTW did you get a chance to check out that Factorizing RSA Keys, An Improved Analogue Solution:

Isn't this one of those BQP problems?
 Mefiante (September 15, 2009, 11:13:13 AM):
Are you sure? This quote from Wikipedia seems to say the opposite:…
Yes I am, and not really, respectively. The first thing to realise is that the terms “chaos” and “chaotic behaviour” have precise mathematical meaning. Second, “chaos” does not imply indeterminism or non-computability or some such. Third, the phrasing of the cited Wikipedia excerpt is a little misleading because true chaotic behaviour can always be digitally simulated to arbitrary precision. If it were not so, then, for example, the Mandelbrot set would come out looking differently every time it is computed. It’s just a question of how long you wish to wait for answers, as well as of the capability of the resources at your disposal.

That things happen essentially simultaneously in an analogue circuit of a certain kind also does not in principle preclude digital simulation. It is not a good reason at all. For example, in brittle failure modes (which are mathematically chaotic), things like stress redistributions and strain-energy releases also happen essentially simultaneously, yet we can model such scenarios quite satisfactorily on powerful digital machines. I think that what the article means to say is really just what I wrote earlier, namely that while analogues are fully deterministic in theory, it is an enormously difficult task to predict or model certain types’ behaviour in practice – so much so that it is fair to call them intractable or even infeasible. The important difference to bear in mind here is the distinction between “practically undoable” and “impossible even in principle.”

They have designed programmable analogue computers, but they still use punch cards! I'm talking about miniaturization and bringing the complexity up to that of today's digital computers. As to finding analogues of problems, as with digital computing, one breaks them down into components.
Yes, true enough all around. However, the observation that your suggestions haven’t been happening much should tell you a few important things: for reasons of physics, analogues do not generally lend themselves well to miniaturisation; the individual components are very limited in their capabilities and ranges of application; assembling a solution involves a hands-on approach to arranging the components in a particular way according to some design (sort of like using bits and pieces from a programming library, except that the analogue constituents are palpable); constructing a general-purpose machine that automatically assembles an analogue and runs it according to some schematic concept merely shifts the problem back by one level; and so on. In short, the limitations of analogues and the practical difficulties of implementing and using them are daunting. That is not to say, however, that they do not find good use in certain dedicated niches, nor that these difficulties are technically insurmountable.

BTW did you get a chance to check out that [paper]?

Isn't [composite integer factorisation] one of those BQP problems?
Yes, I’ve read the analogue factorisation paper – thank you for locating it. It’s a very interesting theoretical exercise that has as much to say about complexity theory as about integer factorisation. As for what complexity class the problem of general integer factorisation is, it’s still unknown. Most number theorists are reasonably sure that it is squarely NP. Certainly, all known algorithms place it there, but definitive proof is still lacking. If it is indeed NP, it probably falls outside the scope of the BQP class (because it is also not entirely clear just how far the BQP class extends).

'Luthon64