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Classical binary computer

The classical binary computers, on which a Hartree-Fock (or density functional theory) calculation can be executed, already allows seeing a limit in their progress, mainly in an “intensive character” of their progress that’s the size of semiconductor device, which cannot be less than the size of one atom of silicon — according to the logic of things. The processor architecture optimization for special mathematical operations and program code optimization to increase the speed of calculation can be included into the latter as well. Contrary the latter, there is an “extensive character” of progress of the classical binary computers that’s increasing the number of processors to parallelize a calculation as much as possible to get a calculation speed acceleration. I guess it obvious that the “extensive character” will have remained even since the “intensive one” will be over.

Quantum computer

There is a new (old) hope of the quantum computer development. But as far as I understand that the problem is to make it in materials such a way that to have possible to operate with a sufficient number of qubits (electrons or photons or other quantum particles) as well as a problem to get a sufficient number of entangled particles (qubits). I guess what sizes of quantum gates are important as well because a sufficient number of them allows to make a quantum circuit more functional. Besides that, at running a quantum computer for some calculation inevitably some deviations from a true result can be obtained that can be related to a quality of materials of which a quantum computer is made (correct my assertions if they are wrong). It’s probably all of these while causes difficulties to implement a quantum circuit for solving the Schrödinger, Hartree-Fock, or Kohn-Sham equation on a real quantum computer made of materials. Although there are tries for moving forward, for instance in this and this. (It's interesting is there in the internet any databases or handbooks or encyclopedias with free quantum algorithms for finding eigenvalues and eigenvectors for the simple molecules $\ce{H2}$, $\ce{O2}$, $\ce{N2}$, $\ce{H2O}$, $\ce{CO2}$, $\ce{NH3}$, $\ce{CH4}$, and so on?). So it’s okay, the quantum computers are a matter of efforts and a time.

Classical ternary computer

The Hartree-Fock (DFT) algorithm, as many other algorithms, supposes in itself a lot of making summing (for instance at computing integrals) that must be executed in 1.58 (that’s taken from $\log_{2}3$) times faster with using ternary computers than if it would be done by binary ones. I suppose that binary computers get more spreading due to their high noise immunity in that “old” time. It’s probably modern materials of today would be able to make ternary computer with sufficient level of noise immunity. Who knows…

Classical electronic analog computer

In the XIX century there were classical analog computers to be fully mechanical and to execute highly specialized tasks. In the XX century, analog computers were transformed with the advent of electronics and became more compact and functional. So electronic analog computers appeared. In general, classical analog computer can represent arrays of integrators, differentiators, adders, comparators, splitters and other devices among which programmatically connections can be set, in order to be as universal device to solve computational tasks of different kinds. Programmatic managing can be implemented with using an old way that's the classical binary computer as variant, see schematic and rough picture below. DAC and ADC as input/output systems must be included as well. — In our time it’s possible to use a hybrid: a classical binary computer in conjunction with an analog one. Logically, a classic electronic analog computer must make calculation very fast, but also give some calculation error. Temperature, parasitic capacitance, parasitic inductance, background noise of sub-devices can affect this error. Maybe there is a way that’s able to stabilize work of the devices and due to that to minimize this error. For example, if a problem is due to warming internal devices (integrators, differentiators and so on), so knowing a dependency a current going through the ones of temperature automatically to be adjusting a needed value in correspondent with a right value. So this procedure will allow to exclude the factor decreasing calculation accuracy. (There is a paper, to put it shortly, about one of tries of solving a Schrödinger equation to be of special kind, being not hard not quite about a matter modeling but though.)

Very rough scheme of a classical analog electronic computer

Main question

Thus, classical binary computers have a limit, quantum computers are in a stage of development, classical ternary computers are not enough spreaded. Nevertheless, despite the fact that the above things can be sufficiently developed in the future, let’s consider analog computers though. It can seem that there is no enough motivation to think about it and it’s probably really so. One thought has been finding this problem as interested, another thought dissuades from its consideration. It's curious how could a circuit of binds of analog components look like that allows solving (finding eigenvalues and eigenvectors) the Schrödinger or Hartree-Fock equation for one of the simple molecules $\ce{H2}$, $\ce{O2}$, $\ce{H2O}$, $\ce{N2}$, $\ce{CO2}$, $\ce{CH4}$, $\ce{NH3}$? What would the thing be from which you have begun? It is not understood how to represent a wave function as a signal or a composition of signals. For example, a wave function for $\ce{H2}$ is a function of 4 variables: $\psi(\mathbf{r_1},\sigma_1,\mathbf{r_2},\sigma_2)$, where $\mathbf{r_1}$ and $\mathbf{r_2}$ — the spatial variables of the first and second hydrogen atom, $\sigma_1$ and $\sigma_2$ — the spinal variables analogically. Maybe an electric signal as a basis of an analog computer in this case is not quite suitable and another basis is needed.

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    $\begingroup$ Related question: mattermodeling.stackexchange.com/questions/1556/… $\endgroup$
    – Tyberius
    Apr 25, 2023 at 0:14
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    $\begingroup$ I gave my +1 long ago, but I am finding this question hard to follow. Why do you want to know anything about using analog computers for solving the HF or KS equations? Don't you think that people would be doing it right now if it was useful? Also, your question seems to be how could a circuit of binds of analog components look like that allows solving (finding eigenvalues and eigenvectors) the Schrödinger or Hartree-Fock equation for one of the simple molecules, but this either seems to be asking people to do a research project and publish the answer here, or for a reference. $\endgroup$ Nov 5, 2023 at 14:59
  • $\begingroup$ @NikeDattani Currently, a pivotal thing of quantum calculation is the speed of processing computations. I suppose this circumstance constraints our evolution in this scope, especially problems including analytical-synthesis approach (in a methodological sense without going into details), evolutionary algorithms as well as some other things. Recently, pathos has appeared in analog computers apparently not being a long-forgotten old thing. $\endgroup$
    – SFriendly
    Nov 11, 2023 at 6:08
  • $\begingroup$ @NikeDattani Also on analog electronic computers there are hopes to implement neural networks that one of them can be useful for good initial guess in HF and KS but this is another problem. Regarding the usefulness, if there is not enough facts to state definitely of its being so I guess hoping must be kept. $\endgroup$
    – SFriendly
    Nov 11, 2023 at 6:09
  • $\begingroup$ @NikeDattani You are right, the question is you've highlighted above. $\endgroup$
    – SFriendly
    Nov 11, 2023 at 6:32

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