Why Is Analog Quantum Computing Needed?

Quantum computing is motivated by a perception that simulating atomic physics described mathematically by the Schrödinger Equation SE of Quantum Mechanics QM, is exponentially hard and so is impossible. This is because SE for a system with $N$ electrons involves $3N$ spatial dimensions with computational work increasing exponentially with $N$.

In other words, digital simulation of QM is viewed to be so computationally demanding that the alternative of analog simulation must be explored. This is the idea analog quantum computing launched by Richard Feynman 50 years ago:

• Simulate a real quantum system by a controllable laboratory quantum system. 

This is the same idea as testing a physical model of a real airplane in a wind tunnel under controllable conditions. Or building a toy model of a bridge and testing its bearing capacity. No mathematics is needed, just craftsman skill.

The basic idea is thus to give up building mathematical models of realities in terms of Cartesian geometry based on numbers with digital representation, as the scientific method behind the evolution of the modern industrial/digital society. 

Such a step can be seen as a step back to a more primitive science based on analog modeling without mathematics. 

In any case, massive investment is now going into creating quantum computers as controllable analog quantum systems. The design work has to cope with the perceived impossibility to test different designs using mathematical digital modeling, and so has to rely on tricky experimental testing. The time frame for a useful analog quantum computer appears to be decades rather than years.

With this perspective it is natural to ask if the exponential computational complexity of the microscopics of quantum mechanics is written in stone. Macroscopics of continuum mechanics rarely comes with exponential complexity, because evolving a macroscopic system, like the weather, one time step involves only local connections in 3 space dimensions which has polynomial complexity. 

If macroscopics has polynomial complexity, then microscopics on smaller scales should have as well. RealQM offers a version of quantum mechanics of polynomial complexity. If nothing else, it can be used to test different designs of an analog quantum computer. Want to try RealQM?

Another mission of analog quantum computing put forward to motivate investors, is improved potential of factorisation of large natural numbers with promise to break cryptography codes. But analog computation about properties of numbers instead of digital appears far-fetched.

PS Recall that at each clock cycle

• a digital computer operates on $n$ factual states
• a quantum computer operates on $2^n$ possible states  

with simplistic promise of an enormous increase of capacity from linear to exponential. Is it too good to be true?

Quantum Computing Without Mathematics

Schrödinger's Equation SE for the Hydrogen atom with one electron formulated in 1926 by the Austrian physicist Erwin Schrödinger as a model of a negative electron charge density subject to Coulomb attraction from a positive kernel,  was generalised to atoms with many electrons by a formal mathematical procedure adding a new independent 3d spatial Euclidean space for each electron into a linear multi-dimensional SE with $3N$ spatial dimensions for an atom with $N$ electrons, to form the foundation of the modern physics of Quantum Mechanics. 

The mathematics of the multi-d SE was quickly formalised by the mathematician von Neumann into highly abstract functional analysis in Hilbert spaces as a triumph of symbolic abstract mathematics. Physics of real atoms was thus hijacked by mathematicians, but the task of making physical sense of the abstraction was left to physicists without the mathematical training required to make real sense of von Neumann's functional analysis. The problem of physical interpretation remains unresolved today, which is behind the present manifest crisis of a modern quantum physics hijacked by mathematics.

The multi-d SE showed to harbour a serious problem when confronted with mathematical computation. Because of the many dimensions the computational complexity showed to be exponential, which made SE uncomputable on digital computers, and so in effect useless. 

Abstract mathematics had created a model of real physics, which showed to be an uncomputable monster, which was not useful except as an exercise of functional analysis.

Quantum computing is a new form of computing fundamentally different from digital computing as mathematical computing with numbers. The idea was launched in the 1970s by the physicist Richard Feynman as a new approach to tackle the uncomputability of QM. The radical idea was to replace uncomputable functional analysis by a form of analog quantum computation, where a real atomic quantum systems is modeled in a laboratory by another real quantum system acting as analog quantum computer.

Recall that the option of replacing a mathematical model by an analog model is also used classically, when a model of an airplane is studied in a wind tunnel instead of solving the Navier-Stokes equations deemed to be impossible.

The success of mathematisation of quantum physics into functional analysis in Hilbert spaces hundred years ago carried its own destruction by coming with exponential complexity, which could not be met within mathematical computation. 

Heavy investment in is now being directed into building a quantum computer supposed to function according to a mathematical formalism, which is being replaced by analog quantum computing.

Does this appear to be strange? To build on mathematical quantum mechanics which is replaced by analog quantum computing based on what was replaced? What would Wittgenstein say about something like that? In any case the investment in quantum computing is high risk. 

RealQM offers a way out of this mess, in the form of a different Schrödinger equation which is computable as digital mathematics and thus does not have to be replaced by analog quantum computing.  Why not give it a try?