In modern view on quantum mechanics wavefuntion collapse is no longer a ‘mystical out of physics’ phenomena, but is seen as a thermodynamical result of interaction with the environment (‘einselection’) –there is still some concrete unitary evolution behind.So there should exist ‘Hamiltonian of the Universe’ describing evolution of everything.

We have similar situation in (classical!) field theories: for Euler-Lagrange equations (like Klein-Gordon: d_tt psi = laplacian psi – m^2 psi ) the evolution operator is self-adjoint – can be diagonalized (spectral theorem). The evolution on the (lambda) coordinate is: d_tt x = lambda x.

So this operator should be non-positive, because otherwise some coordinates would explode.

For negative eigenvalues, we get unitary evolution –like in quantum mechanics, we can imagine it as superposition of different eigenfunctions, ‘rotating’ with different speeds. And so such hyperbolic PDE are called wave-like.

We have limited knowledge: cannot fully trace these unitary evolutions – from our perspective they 'loose their coherence':

- we don’t/can’t know precise parameters, like initial conditions,

- we cannot fully trace complicated motion (chaos),

- thermodynamically stable state usually have own dynamics, like atomic orbitals or quantum phases.

If we model such our lack of knowledge withproper statistical ensemble among possible scenarios- maximize uncertainty not locally like in Brownian motion, but globally - we get thermodynamical going to quantum mechanical ground state probability density. These new models also show why to translate from amplitude we are working on to the probability, we should take ‘the square’ ( http://arxiv.org/abs/0910.2724 ).

To understand the strength of quantum computers, it should be enough to focus on models with constant (fixed) number of particles, for what classical field theory is enough.

We can observe analogue of Young experiment on a surface of water - it can be described in eigenbase of evolution operator, like in standard view of quantum mechanics - but simulation in this case requires considering all scenarios/trajectories - time complexity grows exponentially with the number of slits/qbits ... but we can also simulate it as differential equation/action optimization - for which time complexity should rater grow polynomially.

What is nonituitive about them is that natural picture for such Lagrangian mechanics is ‘static 4D’ – particles are no just ‘moving points’, but rather their trajectories in the spacetime ... let’s look what gives it us for computational capabilities.

Quantum algorithm usually look like:

initialize qbits,

use Hadamar gates to get superposition of all possible inputs,

calculate classical function of the input,

extract some information from the superposition of results,

look at theclassical function calculation–it has to use reversible gates, like

(x,y,z)->(x,y,z XOR f(x,y) )

they are also reversible classically, so we can easily reverse the whole function calculation on standard computer.

Unfortunately it’s not so simple: there is a problem about it – such reversible calculations usually requires quite large number of auxiliary (q)bits, which had been initialized (to zero).

While taking classical reverse of such function, we rather cannot control that these auxiliary (q)bits are zeros – they would usually be just random – so we hadn’t really calculated what we wished.

If we could for example calculate square of a number modulo N or multiplicate of two numbers using ‘small’ number of auxiliary bits, we could guess their final value (e.g. randomly) and in a small number of trials we would be able to reverse such function (getting all zeros), what would allow to factorize N – so probably simple multiplication requires linear number of auxiliary bits.

The strength of quantum computers is that they can ‘mount qbits trajectories’ in both past and future – simultaneously initialize auxiliary qbits and using measurement focus only on scenarios having the same final value (the measured one).

In Shor’s algorithm case, we wouldn’t even need to know all the scenarios to make Fourier transform – knowing two would be already enough: if these powers gives the same value modulo N, their difference gives 1 modulo N.

On the 18th page of my presentation is diagram for Shor’s algorithm: https://docs.google.com/fileview?id=...MWVkOTJk&hl=en

For physics it’s natural to find global minimum of action, but simulating such computer in classical field theory, even after simplifications probably still would be difficult, but anyway it suggests thatto attack algorithmically difficult problems, it could be useful to translate them into continuous ones.

For example in 3SAT problem we have to valuate variables to fulfill all alternatives of triples of these variables or their negations – look that x OR y can be changed into optimizing

((x-1)^2+y^2)((x-1)^2+(y-1)^2)(x^2+(y-1)^2)

and analogously seven terms for alternative of three variables. Finding global minimum of sum of such polynomials for all terms, would solve our problem.

I’ve just found information that it looks like it is successfully done for a few years – enforcing that there is only one minimum, so local gradient would show the way to the solution: http://en.wikipedia.org/wiki/Cooperative_optimization

What do you think about it?