Quantum Computer as Test of Standard Quantum Mechanics

Quantum computing is suddenly booming with many start-ups after 50 years of brooding. The main objective of quantum computing is to solve problems of quantum mechanics which are not tractable by digital computing because of exponential computational complexity. The prospect is that a quantum computer will deliver exponential computational capacity meeting exponential complexity. 

Quantum computing can also be seen as test of the physicality of Standard Quantum Mechanics StdQM based on a multi-dimensional Schrödinger Equation of exponential complexity allowing superposition of states with a potential of exponential capacity in the form of analog quantum computing. 

If a quantum computer based on StdQM can be constructed capable of computing/simulating real physical systems described by StdQM as the expectation of investors, this will give support to the validity of StdQM as a functional model of real physics. 

But there is no quantum computer yet and skeptics believe that controled superposition as key feature of StdQM will be impossible to realise because the physics is missing. 

So the quest for a quantum computer can be seen as the ultimate test of physicality of StdQM. 

What are the odds today? Will there by a quantum computer in 10 years, in 50 years or ever?

   

Church-Turing vs Quantum Computing Illusion

A natural system like the weather can be viewed to perform a form of analog computation as it evolves from one time instant to the next when molecules in the air interact with their neighbors. The computational complexity can be viewed to be polynomial in the size of the physical system. This is expressed in the Physical Church-Turing Thesis PCTT:

• Any physical process can be simulated by a Turing machine.  

Here a Turing machine is a model of a digital computer with polynomial computational capacity capable of simulating a physical process of polynomial computational complexity.

According to PCTT there is thus no physical process expressing exponential complexity, which would be beyond the capacity of digital computing. 

Quantum computing is a form of analog computation with promise of exponential capacity capable of meeting the needs of systems of exponential complexity. It is motivated by a view that quantum mechanics carries exponential complexity in the form of a multi-dimensional wave function and so cannot be simulated on a digital computer. 

We meet here a contradiction:  

• An analog quantum computer is realised in a physical process which according to PCTT is limited to polynomial complexity and so does not have exponential capacity.

We see that PCTT says that a quantum computer with exponential capacity cannot be constructed. No wonder that no such quantum computer has been constructed.

If PCTT is correct, it means that the evolution of a quantum system of atoms and molecules as a physical process, does not expresses polynomial complexity and so in principle can be simulated by digital computation with polynomial capacity. The multi-dimensionality of the wave function appearing to demand exponential capacity thus is an illusion. 

RealQM deconstructs the illusion, by offering simulation of systems of atoms and molecules by digital computation (of polynomial complexity).

PS1 Recall that an N-body simulation has a computational complexity between $N$ and $N^2$ depending on interaction between bodies. 

PS2 If macroscopics has polynomial complexity, then so has microscopics as the basis of macroscopics. If microscopics has exponential complexity, then so has macroscopics based on microscopics. 

The Black Hole of Quantum Computing

Standard Quantum Mechanics StdQM based on a multi-dimensional Schrödinger Equation SE is viewed to have exponential digital computational complexity effectively making SE uncomputable even on super-computers and thus useless for digital modeling of atoms/molecules in practice.

Quantum computing is an attempt to model atoms and molecules instead by analog computation performed on quantum computers capable of meeting exponential complexity. A quantum computer models a physical quantum system of atoms/molecules not by SE, but by another (simpler) physical quantum system which is controllable as being described by SE. 

A quantum computer thus offers a model/map of a real system which is itself a real system of the same form and type. The quantum computer model can be viewed as a 1:10 exact physical model of an airplane or ship with some details removed. The digital computational complexity of the original physical system is then irrelevant since no digital computation is performed, only a form of analog computation performed in a physical model of essentially the same computational complexity. 

Quantum computing thus represents a step back from the scientific revolution based on Calculus offering  mathematical models of reality of abstract symbolic from based on numbers which can be made alive by digital computing. 

Quantum computing throws away digital Calculus because it is found to be useless, and seeks a replacement which works in terms of analog computation. 

But is it possible that real quantum systems really perform analog computations of exponential complexity? If a digital computer does not have the capacity to meet exponential complexity, what says that a real system as it evolves somehow can express exponential complexity? 

We know that the exponential complexity of SE comes from the multi-dimensionality of the wave function representing possibilities rather than actualities. But real systems do not realise all possibilities only a few actualities, and so the computational complexity of SE as model is much bigger than the complexity of what is modeled. The model/picture totally swamps the origin and so asks for infinite digital computational power to no avail as concerns the origin.

RealQM is a version of the quantum mechanics of atoms and molecules, which is digitally computable and so does not need quantum computing.

Quantum Computing as Theoretical Fantasy

Here is a summary by chatGPT of Dyakonov's view on the realism of quantum computing:

Physicist Mikhail Dyakonov has been one of the most prominent skeptics of large-scale quantum computation. His main critique can be summarized as follows as of 2019:

1. Extreme Fragility of Quantum States – Quantum bits (qubits) are extraordinarily sensitive to any external disturbance. Maintaining coherent quantum states long enough for complex computations is, in his view, practically impossible.

2. Error Correction Limits – While quantum error correction is theoretically possible, Dyakonov argues that the real-world requirements (millions of physical qubits per logical qubit, perfectly calibrated control systems, etc.) make it infeasible.

3. Unrealistic Scaling – He believes scaling from a few dozen qubits to millions (as required for useful computation) involves an exponential explosion of complexity in control, synchronization, and noise management.

4. Overly Optimistic Theory vs. Practice – Dyakonov claims that most of quantum computing theory assumes idealized conditions that cannot be realized physically. The engineering challenges are, in his words, “beyond human capabilities.”

5. Skepticism Toward “Quantum Supremacy” Claims – He views demonstrations like Google’s 2019 experiment as contrived and not indicative of practical progress toward general-purpose quantum computing.

In essence, Dyakonov argues that large-scale quantum computers are a theoretical fantasy, comparable to perpetual motion machines — possible on paper, but not in reality.

end chatGPT

Let us compare with the latest hype:

• Google’s Willow Chip Achieves Historic Quantum Computing Breakthrough
• Willow Chip Powers Verifiable Quantum Advantage, Running Algorithm 13,000x Faster Than Supercomputers.

This what chatGPT says about the Willow Chip:

• Dyakonov’s scepticism remains valuable and largely justified — many of his fundamental concerns (noise, control, scale, assumption-validity) are not yet fully overcome.
• The Willow chip is important: it represents a tangible, hardware-driven step toward the goals of large-scale quantum computing. It shows we are making progress on the error-correction and scaling front.
• However, Willow does not yet fully invalidate Dyakonov’s concerns. It addresses some, but many of the major leaps (millions of qubits, full fault-tolerance, broad practical algorithmic supremacy) are still ahead.
• In other words: Willow moves the bar, but the “gap” that Dyakonov insisted on remains significant. The achievement doesn’t guarantee the remaining engineering challenges will vanish.
• If I were to pick one key takeaway: Willow demonstrates that at least some of the previously theoretical obstacles (error-correction scaling) can be approached in hardware — which means the sceptical “maybe impossible” thesis is being challenged. 
• But “practically useful, general-purpose quantum computers in the near-term” still remain uncertain.

It seems that quantum computing is still a theoretical fantasy, 50 years after the idea was launched. 

Recall that quantum computing is based on unitary evolution of quantum systems of thousands of qubits in superposition of possibilities as fantasy physics: Compare with the first proposition in Wittgenstein's Tractatus:

• The World is all that is the case.

It is clear that "to be the case" requires more than what is present in a quantum system of possibilities, which means that according to Wittgenstein a quantum computer does not belong to the World. But a quantum computer is an analog computer and as such must belong to the World. Wittgenstein would thus view the Willow chip with utter skepticism. And you?

Recall that the idea of a quantum computer is a model of an uncontrollable real/analog quantum system as part of the World in the form of a controllable real/analog quantum system as part of the World, with the caveat that the model is not "the case" because it plays with possibilities and not with realities.   

Notice that this contradiction does no appear with a digital computer because the computing is abstract mathematical and so does not need real analog computing.  

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?

 

Is Quantum Computing Possible?

Quantum superposition is the crucial component of Quantum Mechanics believed to open a new world of quantum computing on quantum computers.    

An $n$-qubit quantum computer with $n=2,3,..,$ is supposed to operate in parallel on $2^n$ states in superposition, to be compared with a classical digital computer operating on $n$ bits at a time, where a bit can take the value 0 or 1. Quantum computing thus gives promise of exponential speed-up vs digital computing.   

The idea of quantum computing was first presented by Richard Feynman in an attempt to get around the apparent exponential complexity of Quantum Mechanics QM based on Schrödinger's multi-dimensional wave equation making digital simulation impossible by demanding computational work scaling with $2^N$ for a system with $N$ electrons. The idea was to replace digital simulation with some form of analog quantum computation where the simulation of a quantum system would be performed on a quantum system. An intriguing idea, but could it work? The apparent exponential complexity would then be met with an exponential computational power making simulation possible. To meet a perceived difficulty by ramping up the capacity or simply to "fight fire with fire". 

Let us then consider the basic idea of a quantum computing, which is 

• Simultaneous operation on states in superposition.       

What then is "superposition"? The mathematical answer is the following: Consider a linear algebraic or differential equation with solutions $\psi_1$ and $\psi_2$. Then the algebraic sum $\psi =\psi_1+\psi_2$ is also a solution and $\psi$ is viewed to be the superposition $\psi_1$ and $\psi_2$ with the + sign signifying algebraic sum. 

As concerns classical physical wave mechanics, the algebraic superposition can take two forms with physicality of (i) the algebraic sum $\psi$ or (ii) of the individual terms $\psi_1$ and $\psi_2$. Case (i) represents the interference pattern seen of the surface of one pond, while (ii) represents the beat interference generated by two vibrating strings with nearby frequencies. 

As concerns QM the physicality is supposed to be displayed in the double-slit experiment: Let a single photon/electron pass a double-slit and then be detected on a screen behind a as a spot. Repeat the experiment many times and notice a fringe pattern appearing on the screen, which is the same interference pattern developed by a macroscopic wave passing through both slits. The photon/electron after passing the double split is representing the quantum superposition supposed to carry quantum computing. This is not a superposition of realities as in (ii) above, but a superposition of possibilities made real by repetition of the one photon/electron experiment which represents the physics of a 1-qubit: The single photon/electron is by passing through the double slits put into a superposition of passing the left slit and passing the right slit not as realities but as possibilities. It is here essential that the superposition only concerns one photon/electron in superposition of $2^1=2$ states. This is supposed to generalise to $n$ entangled photons/electrons in superposition of $2^n$ possible states.

The evidence that quantum computing is possible thus boils down to the double-slit experiment. If one photon/electron can be put in superposition of two states which appears to support interference with fringe pattern, then constructing of a $n$-qubit computer may be possible. If two photons/electrons are needed for two states then we are back to classical computing with bits.

The crucial question is now: Is it sure that because there is one click on the screen at a time, the input is one photon/electron at a time?

Think of this, and return to see a more precise analysis.

QM in its standard multi-dimensional form has exponential complexity, which requires exponential computing power. RealQM is an alternative with polynomial complexity which can be met by classical computing.  

Maybe quantum computing is neither possible nor needed? 

Quantum Computing Swindle

The previous post gave expression of a Quantum Hype Bubble About to Burst. We may compare with the hype of the 2022 Nobel Prize in Physics:

• The ineffable effects of quantum mechanics are starting to find applications. There is now a large field of research that includes quantum computers, quantum networks and secure quantum encrypted communication.
• It has become increasingly clear that a new kind of quantum technology is emerging.
• The fundamentals of quantum mechanics are not just a theoretical or philosophical issue. Intense research and development are underway to utilise the special properties of individual particle systems to construct quantum computers, improve measurements, build quantum networks and establish secure quantum encrypted communication.
• This year’s laureates have explored these entangled quantum states, and their experiments laid the foundation of the revolution currently underway in quantum technology.
• Being able to manipulate and manage quantum states and all their layers of properties gives us access to tools with unexpected potential. This is the basis for quantum computation, the transfer and storage of quantum information, and algorithms for quantum encryption.

We note that "ineffable" means too great or extreme to be expressed or described in words!!

The Physics Nobel Prize Committee is joined by Knut and Alice Wallenberg Foundation as the major Swedish private funding agency of research with a $100 million donation to Wallenberg Centre for Quantum Technology at Chalmers (my Alma Mater) based on the following hype:

• Through an extensive research programme, we aim at developing and securing Swedish expertise within the main areas of quantum technology: quantum computing and simulation, quantum communications and quantum sensing. Our main project is to develop a high-end quantum computer that can solve problems far beyond the reach of the best conventional supercomputers.

We see that Sweden has been totally carried away by the hype into seeking to take a leading role in finally liberating the ineffable potential of quantum mechanics after 100 years of hibernation. But a hype is a hype is a hype... 

Knut Wallenberg (1853-1938) was a clever Swedish banker who earned a fortune, which he locked into a fund for science to secure that it was not going be swindled away by less clever inheritors, apparently under the impression that science in his time could not be swindle. 

More on swindle (for investors): 

• Dyakonov The Case Against Quantum Computing
• Hoofnagel and Garfinkel Law and Policy for the Quantum Age
• Galitsky Quantum Computing Hype is Bad for Science

Is Quantum Computing Possible?

The possibility of quantum computing was suggested by the famous physicist Richard Feynman some 50 years ago to meet the need of excessive computing power supposed to required to simulate atomic quantum systems on a conventional digital computer. Today much effort is spent on seeking to build a quantum computer operating with qubits instead of the bits of a digital computer. 

The basic idea is that a qubit can represent a superposition of two states (both 0 and 1) instead of the single state of either 0 or 1 of a bit. For a system with $N$ qubits/bits this increases the number of logical states from $N$ with bits to $2^N$ with qubits, thus from linear to exponential. 

IBM claims to have built the most powerful quantum processor in the world – the Osprey, which boasts a massive 433 quantum bits (qubits). The physical type of qubit used by IBM is described as:

• a superconducting transmon qubit, which is made from superconducting materials such as niobium and aluminum, patterned on a silicon substrate. 
• Such systems are not natural qubits, but are instead formed by isolating two energy levels out of many to form our approximate qubit.
• For a superconducting qubit to behave as the abstract notion of the qubit, we must have the device at drastically low temperatures to minimize ambient noise or heat that could excite the superconducting qubit and increase the error probability. 
• Once a system has cooled to the target temperature, which takes several days, the qubit reaches equilibrium at the ground state 0.

Quantum computing boils down to transforming qubits from one state to another by logical gates. It is a tricky subject with standard measurement and randomness playing a central role. The Hadamard gate creates a superposition of 0 and 1 by starting in either the state 0 or 1,  which followed by standard measurement gives 0 or 1 with a probability of 0.5. Altogether this looks like flipping a coin, but the creation of superposition of 0 and 1 prior to measurement is a unique qubit feature.  

Let us compare with the discussion in the previous post on a radiating atom in superposition of a ground state 0 and an excited state 1 under certain exterior forcing with frequency matching the difference of excited and ground state energies. In this case it is the exterior forcing which creates a superposition of the ground state and matching excited state. By varying the frequency of the exterior forcing, the atomic spectrum can be determined, which can be seen as a form of deterministic spectrum measurement. In this setting the spectrum of the atom is observable, but not the underlying electronic structure.  

A bit can be realised as an atom with value 0 in non-radiating ground state and value 1 in radiating superposition of ground state and excited state, both deterministically observable/measurable.  A bit is in direct contact with exterior forcing determining its value to be 0 or 1. 

As a qubit a radiating atom would allow superposition of ground and excited state and thereby offer a richer playground of states for processing by certain logical gates performed with zero exterior forcing. But measuring the result would amount to exterior forcing destroying superposition to randomely deliver either ground or excited state. The richer processing would thus come with randomness and a net gain would seem to be possible only for very special problems including randomness.      

Summary so far: A bit is in direct contact with exterior forcing allowing deterministic processing and reading without destruction of the state. A qubit must be insulated from exterior forcing during processing, while reading destroys the state and delivers a random result. 

Main question to answer: Is superposition possible without radiation/exterior forcing? If No, quantum computing may be impossible.