Quantum computing is incompatible with Born Rule and the related measurement postulates, the qubit reading capability is falsified experimentally, and the Feynman-Deutsch quantum computing theory is low-strict, low-testable and infeasible.
Abstract (1) Deutsch stated: it is shown that quantum theory and the ‘universal quantum computer’ are compatible with the principle, that is, it is shown the compatibility of quantum computing with quantum mechanics. Born Rule and Time Evolution Postulate are the important ingredients of quantum theory. Deutsch et al. did not provide a proof of the compatibility of quantum computing with Born Rule (and Time Evolution Postulate), avoided the extraction of unitary transformation results in their papers, and defaulted that the output probability density can map to the observation-space CS C and to be read. We attribute the proof of the incompatibility of quantum computing with quantum mechanics to the proof of the incompatibility of quantum computing with Born Rule. The basic form of Born Rule is that \({\left| \psi \right|^2}\) represents the probability density for particles; if the system is in a state \(\left| \psi \right\rangle\), then the probability that the eigenvalue λ i of Q is found when Q is measured is P(λ i ) =\({\left| {\left\langle {{\lambda _n}} \right|\left. \psi \right\rangle } \right|^2}\). We show that in the technical scheme of quantum computing, there is an imposed strong constraint on Born Rule, that is, double \({\left| \psi \right|^2}\) hypothesis (i.e., Feynman-Deutsch’s \({\left| \psi \right|^2}\)(double) reading hypothesis), which requires that there is such a mapping from the state-space CS M to the observation-space CS C : \({\left| {{\psi _M}} \right|^2}\)→\({\left| {{\psi _C}} \right|^2}\), where \({\left| {{\psi _M}} \right|^2}\)is the probability density of the wavefunction \({\text{a}}\left| 0 \right\rangle +b\left| 1 \right\rangle\) of the single-qubit, \({\left| {{\psi _C}} \right|^2}\) is the probability density of the corresponding outcomes within CS C , and \({\left| {{\psi _C}} \right|^2}\) is equivalent to \({\left| {{\psi _M}} \right|^2}\). The original Born Rule does not assert the mapping \({\left| {{\psi _M}} \right|^2}\)→\({\left| {{\psi _C}} \right|^2}\). We demonstrate that in the technical scheme of quantum computing it is assumed that the single-qubit carrier-symbol snapshot traceability is the basis for the realization of the double \({\left| \psi \right|^2}\). We, using the Stern-Gerlach experiment and the chip engineer's contraction mapping experiment, experimentally falsify the single-qubit carrier-symbol snapshot traceability, and thus experimentally falsify the Feynman-Deutsch’s \({\left| \psi \right|^2}\)(double) reading hypothesis. In this way, we show that quantum computing is incompatible with Born Rule and the related measurement postulates, and the Feynman-Deutsch quantum computing theory is not feasible. The principal error of the Feynman-Deutsch theory is the mapping \({\left| {{\psi _M}} \right|^2}\)→\({\left| {{\psi _C}} \right|^2}\); the root of all errors in quantum computing is the double probability density hidden in the technical scheme, which is the false Born Rule. (2) The Feynman-Deutsch theory is a low-strict, low-testable and non-transparent theory for the following reasons: postulate-level assumptions are hidden in the technical scheme; the testability (falsifiability) of various parts of quantum computing has never been publicized. It has long been believed that quantum computing centers on unitary transformation sequence. We show that qubit reading is the only falsifiable part of quantum computing. Because falsifiable issue takes precedence over unfalsifiable issue, and principle-level issue takes precedence over non-principle-level issue, the priority of quantum computing research is the inapplicability of Born Rule to quantum computing rather than the applicability of unitary transformation. i.e., the real heart of quantum computing is the existence of \({\left| \psi \right|^2}\) mapping from CS M to CS C . We prove the fact that although algorithm and the algorithm’s output within CS M are carried out in the name of quantum technology, the "double \({\left| \psi \right|^2}\)" is actually independent of Born Rule, and independent of quantum mechanics. The methodological error of the Feynman-Deutsch theory is to exclude the inapplicability of Born Rule from the research field. (3) We re-examined the effectiveness of three published quantum computing (Shor factoring) experiments, and show that they are invalid in the sense of falsifiability criterion and the traversal criterion. The high risk of the infeasibility of quantum computing is completely underestimated. (4) In the last sections, we examine a topic with a higher degree of generality: the traceability (direction non-constraint) of single-particle information process. Using the “reproducible effect” method suggested by Popper, we show that the realization of quantum computing violates human's existing knowledge about the contraction mapping experiments of two spaces. The proofs in this paper show that the compression mapping experiment in meso-micro scale and the physical version of Gödel's incompleteness theorem are more profound principles of quantum physics (including Born Rule and the related measurement postulates).
Abstract (1) Deutsch stated: it is shown that quantum theory and the ‘universal quantum computer’ are compatible with the principle, that is, it is shown the compatibility of quantum computing with quantum mechanics. Born Rule and Time Evolution Postulate are the important ingredients of quantum theory. Deutsch et al. did not provide a proof of the compatibility of quantum computing with Born Rule (and Time Evolution Postulate), avoided the extraction of unitary transformation results in their papers, and defaulted that the output probability density can map to the observation-space CS C and to be read. We attribute the proof of the incompatibility of quantum computing with quantum mechanics to the proof of the incompatibility of quantum computing with Born Rule. The basic form of Born Rule is that ({\left| \psi \right|^2}) represents the probability density for particles; if the system is in a state (\left| \psi \right\rangle), then the probability that the eigenvalue λ i of Q is found when Q is measured is P(λ i ) =({\left| {\left\langle {{\lambda _n}} \right|\left. \psi \right\rangle } \right|^2}). We show that in the technical scheme of quantum computing, there is an imposed strong constraint on Born Rule, that is, double ({\left| \psi \right|^2}) hypothesis (i.e., Feynman-Deutsch’s ({\left| \psi \right|^2})(double) reading hypothesis), which requires that there is such a mapping from the state-space CS M to the observation-space CS C : ({\left| {{\psi _M}} \right|^2})→({\left| {{\psi _C}} \right|^2}), where ({\left| {{\psi _M}} \right|^2})is the probability density of the wavefunction ({\text{a}}\left| 0 \right\rangle +b\left| 1 \right\rangle) of the single-qubit, ({\left| {{\psi _C}} \right|^2}) is the probability density of the corresponding outcomes within CS C , and ({\left| {{\psi _C}} \right|^2}) is equivalent to ({\left| {{\psi _M}} \right|^2}). The original Born Rule does not assert the mapping ({\left| {{\psi _M}} \right|^2})→({\left| {{\psi _C}} \right|^2}). We demonstrate that in the technical scheme of quantum computing it is assumed that the single-qubit carrier-symbol snapshot traceability is the basis for the realization of the double ({\left| \psi \right|^2}). We, using the Stern-Gerlach experiment and the chip engineer's contraction mapping experiment, experimentally falsify the single-qubit carrier-symbol snapshot traceability, and thus experimentally falsify the Feynman-Deutsch’s ({\left| \psi \right|^2})(double) reading hypothesis. In this way, we show that quantum computing is incompatible with Born Rule and the related measurement postulates, and the Feynman-Deutsch quantum computing theory is not feasible. The principal error of the Feynman-Deutsch theory is the mapping ({\left| {{\psi _M}} \right|^2})→({\left| {{\psi _C}} \right|^2}); the root of all errors in quantum computing is the double probability density hidden in the technical scheme, which is the false Born Rule. (2) The Feynman-Deutsch theory is a low-strict, low-testable and non-transparent theory for the following reasons: postulate-level assumptions are hidden in the technical scheme; the testability (falsifiability) of various parts of quantum computing has never been publicized. It has long been believed that quantum computing centers on unitary transformation sequence. We show that qubit reading is the only falsifiable part of quantum computing. Because falsifiable issue takes precedence over unfalsifiable issue, and principle-level issue takes precedence over non-principle-level issue, the priority of quantum computing research is the inapplicability of Born Rule to quantum computing rather than the applicability of unitary transformation. i.e., the real heart of quantum computing is the existence of ({\left| \psi \right|^2}) mapping from CS M to CS C . We prove the fact that although algorithm and the algorithm’s output within CS M are carried out in the name of quantum technology, the "double ({\left| \psi \right|^2})" is actually independent of Born Rule, and independent of quantum mechanics. The methodological error of the Feynman-Deutsch theory is to exclude the inapplicability of Born Rule from the research field. (3) We re-examined the effectiveness of three published quantum computing (Shor factoring) experiments, and show that they are invalid in the sense of falsifiability criterion and the traversal criterion. The high risk of the infeasibility of quantum computing is completely underestimated. (4) In the last sections, we examine a topic with a higher degree of generality: the traceability (direction non-constraint) of single-particle information process. Using the “reproducible effect” method suggested by Popper, we show that the realization of quantum computing violates human's existing knowledge about the contraction mapping experiments of two spaces. The proofs in this paper show that the compression mapping experiment in meso-micro scale and the physical version of Gödel's incompleteness theorem are more profound principles of quantum physics (including Born Rule and the related measurement postulates).
These notes introduce quantum computation and quantum error correction, emphasising the importance of stabilisers and the mathematical foundations in basic Lie theory. We begin by using the double cover map SU2→SO3(R) to illustrate the distinction between states and measurements for a single qubit. We then discuss entanglement and CNOT gates, the Deutsch--Jozsa Problem, and finally quantum error correction, using the Steane [[7,1,3]]-code as the main example. The necessary background physics of unitary evolution and Born rule measurements is developed as needed. The circuit model is used throughout.
This preprint presents a complete and fully operator-theoretic resolution of the quantum measurement problem. Including collapse, the Born rule, the uniqueness of outcomes, and the irrelevance of observersm derived entirely from a universal structural principle called the lawhood identity. Rather than adding new physics, modifying the Schrödinger equation, introducing hidden variables, invoking many-worlds branching, or appealing to consciousness, this work shows that quantum measurement is a direct, unavoidable consequence of a general fixed-point law that applies to all stable physical systems. When instantiated for quantum mechanics, this law forces every genuine superposition to fail the criteria for being a physically lawful state. Collapse is not an extra postulate: it is structurally mandatory. Key results include: • The No-Superposition Theorem: All superpositions are proven to be structurally unlawful; they cannot survive the measurement map.• Mandatory Collapse: Collapse emerges automatically from the composition of three physical operators (boundary, decoherence, and closure), without modifying quantum dynamics.• Derivation of the Born Rule: The familiar probabilities arise uniquely from the internal consistency of the measurement operators. No additional assumptions, decision theory, or symmetry arguments are required.• Observer Independence: Awareness and measurement are formally separated. Collapse is a physical, observer-independent process; knowledge acquisition is epistemic, not causal.• Uniqueness of Outcomes: The framework proves that there is only one stable classical outcome for any initial quantum state, directly resolving the many-worlds branching issue.• Experimental Signatures: The formulation predicts measurable relationships involving entropy, decoherence rates, and effective dimensionality. This paper shows that the puzzles of quantum collapse, superposition, outcome selection, probability, and the apparent role of observers, are not mysteries requiring new interpretations. They are the natural consequences of a deeper structural principle that governs all lawful physical systems. Quantum mechanics, under this lens, becomes a specific case of a universal operator identity that determines which states can persist as stable reality. Readers will find a complete, logically self-contained argument, with theorems and proofs written in a way that does not rely on philosophical interpretation or speculative assumptions. The result is a clear and testable framework that resolves a problem that has persisted since the birth of quantum theory. This work is part of a broader research program developing the universal Tier-0 lawhood framework and the “Everything Equation,” which unifies stability, collapse, and fixed-point structure across physics, computation, thermodynamics, and information. Relation to Tier-1 dynamical formulation: This work develops a law-level (structural) resolution of quantum measurement. A complementary Tier-1 dynamical realization of these structural principles, formulated entirely within a constrained operator framework and independent of law-level recursion language, is developed in: J. Rodgers, Saturation Geometry and the Structural Emergence of Measurement and the Born Rule in a Capacity-Constrained Dirac–Λ System, Zenodo (2026).https://doi.org/10.5281/zenodo.18704783 The Tier-1 formulation may be read independently and provides a concrete operator-theoretic instantiation of the structural mechanisms discussed here.
Traditionally, some version of Born’s rule is always considered to be an indispensable part of any interpretation of quantum mechanics, either as a postulate, or as a result derived-not always on the basis of convincing reasoning-from other postulates. In this section, we have a close look at the possible forms of Born’s rule and discuss the limits of its validity. All traditional foundations of quantum mechanics heavily depend on the concept of (hypothetical, idealized) experiments-far too heavily. This is one of the reasons why these foundations are still unsettled, over 90 years after the discovery of the basic equations for modern quantum mechanics. No other theory has such controversial foundations. The main reason is that the starting point of the usual interpretations is an idealization of the measurement process that is taken too seriously, namely as the indisputable truth about everything measured. But in reality, this idealization is only a didactical trick for the newcomer to make the formal definitions of quantum mechanics a bit easier to swallow. Except in a few very simple cases, it is too far removed from experimental practice to tell much about real measurements, and hence about how quantum physics is used in real applications. In experimental physics, measurement is a very complex thing, far more complex than Born’s rule (the usual starting point) suggests. To measure the distance between two galaxies, the mass of the top quark, or the Lamb shift, just to mention three basic examples, cannot be captured by the idealistic measurement concept used there, nor by any of the refinements of it discussed in the literature. In each of the three cases mentioned, one assembles a lot of auxiliary information and ultimately calculates the measurement result from a best fit of a model to the data. Clearly, the theory must already be in place in order to do that. We do not even know what a top quark should be, whose mass we are measuring, unless we have a theory that tells us this! The two most accurately determined observables in the history of quantum physics, namely the anomalous magnetic moment of the electron and Lamb shift, are not even q-observables! To present the stage for the criticism of Born’s rule in Section 14.3, we first need to clarify the meaning of the term “Born’s rule”. To distinguish different useful meanings, we look in Sections 14.1-14.2 at the early history of Born’s rule.
Few, if any, issues in physics have engendered as much discussion as the measurement problem in quantum mechanics. It is generally agreed that the `normal' dynamical evolution of the state vector in quantum mechanics is given by a unitary map. The linearity of this map implies that the state vector will, in general, be found in a superposition of eigenstates of a given observable (or, similarly, that the density matrix describing a subsystem will not correspond to a definite value of this observable). However, when we make a measurement of an observable, we always obtain a define value—although it is impossible to predict with certainty which value will be obtained. The traditional response to this issue is to postulate that when a measurement is made, the wavefunction `collapses' to an eigenstate of the observable being measured, perhaps due to the inherent classicality of the measuring apparatus (Bohr), or to the consciousness of the observer (Wigner), or possibly to some modification of quantum dynamics that occurs even when observations are not being made. The main motivation for the Everett (`many worlds') interpretation is to avoid introducing any such collapse postulate. This volume commemorates the 50th anniversary of the publication of Everett's paper in 1957 and contains 20 original articles as well as the transcripts of several discussions that took place at meetings devoted to the Everett interpretation at Oxford University and the Perimeter Institute. The attractiveness of the Everett interpretation is very succinctly summarized by a sentence from Vaidman's contribution (p 582): `The collapse, with its randomness, non-locality and the lack of a well-defined moment of occurrence, is such an ugly scar on quantum theory, that I, along with many others, am ready to follow Everett and deny its existence.' But the main drawback of the interpretation is then equally succinctly stated in the next sentence: `The price is the many worlds interpretation, i.e., the existence of numerous parallel worlds.' In other words, if the state vector evolves to have a non-zero amplitude for two different values of an observable that I `measure' (i.e., interact with in an appropriate way), each of these alternatives remains equally `real', i.e., there is a `branch' of the wavefunction in which the observable takes the first value and I believe that it has this first value and a branch of the wavefunction where it takes the second value and I believe that it has this second value. Each of these `me's' sees a definite value for the observable, thus nicely accounting for what I believe I experience without invoking collapse. But then, it seems that I have to believe that there are many other `me's' around—even if they are very difficult for this particular me to access—and no way I can predict which `me' I will be in the future. Indeed, Saunders writes on p 192, `If Alice is a person (of course she is) then we must say, even prior to branching at tj that there are many Alices present, atom for atom duplicates...Alice should be uncertain after all— each Alice should be uncertain—for each as of tj does not (and as a matter of principle, cannot) know which of these branching persons she is.' Thus, if taken seriously, the Everett viewpoint leads directly to questions like `What determines which ``me'' I am now and/or will feel that I am in the future?' The response `You are all of them' somehow doesn't seem satisfying, and further pursuit of these issues would seem to lead to questions as fruitless as questions like `Why am I ``me'' and not ``you''?' I believe that this accounts for why the Everett interpretation does not have universal appeal. But enough of `me' for now (more later); let's discuss the book. Unlike many edited volumes and conference proceedings in physics—which often consist of hastily slapped-together re-hashes of previous papers—this volume meets very high standards in the choice of topics and authors and in the quality of the writing. Although each of the articles is closely tied in with the Everett interpretation, there are many articles of considerable value and interest in their own right, such as the discussion of quasiclassical realms by Hartle and and the discussion of the de Broglie–Bohm interpretation by Valentini. There is also an excellent introduction by Saunders to the quantum measurement problem and the main issues that arise in the Everett interpretation. A central issue in the Everett interpretation is the status of the `Born rule', which asserts that, for state ψ, the probability of obtaining a particular outcome of a measurement is ||Pψ||2, where P is the projection operator onto the eigensubspace associated with the measurement outcome. In traditional interpretations, the Born rule is simply postulated as part of the collapse hypothesis. In the Everett interpretation, it is far from obvious that the Born rule even has any meaning—if all outcomes occur, how can one talk about the probability of a particular outcome? Given the importance of this issue, it is highly appropriate that four chapters of the book (by Saunders, Papineau, Wallace, and Greaves and Myrvold) are devoted to addressing probability and the Born rule from the Everett viewpoint, and three chapters (by Kent, Albert, and Price) are devoted to criticising these views. As indicated by the above quote from Saunders' article, uncertainty and probability can enter the Everett interpretation via each Alice being uncertain as to which Alice she is or will be. However, in view of the difficulties to which this kind of argumentation will lead, it is perhaps not surprising that in the 123 pages of combined discussion by Saunders, Papineau, Wallace, and Greaves and Myrvold, there are only fleeting references to the use of probability along these lines. Rather, the approach taken is reminiscent of that of adept politicians: if you can't answer a question, then answer a different question, and answer it with conviction and detail. Instead of considering probabilities of outcomes—a highly problematical notion when all of the possible outcomes actually occur—consider only decision theory issues. In particular, Wallace gives a precise and detailed mathematical proof that, under certain hypotheses, the only rational strategy that a quantum observer can follow in making decisions is to weight the possible outcomes by Born rule probabilities. To illustrate this more graphically, suppose tomorrow's weather were determined by some quantum mechanical, Stern–Gerlach-like process. I would like to know whether it is likely to rain tomorrow. However, when I ask an Everett follower (who knows the state vector and has done the Born rule calculation), all she will tell me in direct response to my question is that `it will rain and it won't rain.' Nevertheless, she is willing tell me whether or not it is rational for me to take an umbrella. I suppose that if she tells me that it would be highly irrational for me to take an umbrella, I can take the hint and deduce that it is very unlikely to rain. However, there does seem something wrong with the fact that she is not allowed to say this directly. In any case, if the conclusion of a mathematically correct argument is that rational decision strategies require the Born rule, then there must be quite a bit lying in the assumptions. The articles by Kent, Albert, and Price do a good job of fleshing out these assumptions and pointing out the weaknesses and flaws in the probability and decision theory discussions within the Everett framework. I hope it is clear from what I have written above that this is an outstanding volume, which should be of great interest to anyone interested in the foundations of quantum theory. Indeed, I feel quite fortunate to be the particular `me' that happens to exist in the same branch of the wavefunction of the universe in which this book was published.
Deutsch, Feynman, and Manin viewed quantum computing as a kind of universal physical simulation procedure. Much of the writing about quantum Turing machines has shown how these machines can simulate an arbitrary unitary transformation on a finite number of qubits. This interesting problem has been addressed most famously in a paper by Deutsch, and later by Bernstein and Vazirani. Quantum Turing machines form a class closely related to deterministic and probabilistic Turing machines and one might hope to find a universal machine in this class. A universal machine is the basis of a notion of programmability. The extent to which universality has in fact been established by the pioneers in the field is examined and a key notion in theoretical computer science (universality) is scrutinised. In a forthcoming paper, the authors will also consider universality in the quantum gate model.
This work provides a structural derivation of the quadratic form of the Born rule in quantum mechanics. Rather than postulating the probability rule as a foundational axiom, the paper proves that the exponent governing quantum probabilities is uniquely fixed by stability, contractivity, and duality requirements imposed on admissible physical laws. The analysis is carried out within a general operator-theoretic and law-space framework, in which physical laws are treated as fixed points of recursive generative dynamics. By combining strict contractive flow (Kappa Law), topological rigidity (Law of Endogenous Constraint), and algebra–geometry duality (Monad Duality), the paper demonstrates that only the Hilbert–Schmidt geometry is compatible with global stability of law evolution. As a consequence, the quadratic probability rule emerges as a rigid structural invariant rather than a free modeling choice. The result implies that any hypothetical modification of quantum theory based on non-quadratic probability rules would necessarily violate at least one of the fundamental structural requirements of admissible physical law, such as uniform contractivity, spectral stability, or duality consistency. Relation to Tier-1 dynamical formulation: This work develops a law-level (structural) resolution of the Born exponent. A complementary Tier-1 dynamical realization, formulated entirely within a constrained operator framework and independent of law-level recursion language, is developed in: J. Rodgers, Saturation Geometry and the Structural Emergence of Measurement and the Born Rule in a Capacity-Constrained Dirac–Λ System, Zenodo (2026).https://doi.org/10.5281/zenodo.18704783 The Tier-1 formulation may be read independently and derives the quadratic Born rule as a consequence of modular implementability and saturation geometry within the coupled Dirac–Λ system.