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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Algorithmic approach to quantum theory 1: features of many particle quantum dynamics

Yuri Ozhigov, Igor Semenikhin

Published May 24, 2006• 0 citations
Publisher

Abstract

Algorithmic approach to quantum theory is considered. It is based on the supposition that every evolution of many particle system can be simulated by classical algorithms of polynomial complexity. This hypothesis agrees with all known experiments but it presumes the principle cut-off of quantum formalism because it excludes a scalable quantum computer. Algorithmic approach describes quantum evolution uniformly, without separation of measurements from the unitary dynamics; it is shown how Bohrn rule for quantum probability follows from the basic principles of this approach. The radical difference of algorithmic approach from the standard and its perspectives are discussed.

DOI
10.1117/12.683114

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Algorithmic approach is based on the assumption that any quantum evolution of many particle system can be simulated on a classical computer with the polynomial time and memory cost. Algorithms play the central role here but not the analysis, and a simulation gives a "film" which visualizes many particle quantum dynamics and is demonstrated to a user of the model. Restrictions following from the algorithm theory are considered on a level of fundamental physical laws. Born rule for the calculation of quantum probability as well as the decoherence is derived from the existence of a nonzero minimal value of amplitude module - a grain of amplitude. The limitation on the classical computational resources gives the unified description of quantum dynamics that is not divided to the unitary dynamics and measurements and does not depend on the existence of observer. It is proposed the description of states based on the nesting of particles in each other that permits to account the effects of all levels in the same model. Algorithmic approach admits the possibility of refutation, because it forbids the creation of a scalable quantum computer that is allowed in the conventional quantum formalism.

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An algorithmic approach to theoretical quantum physics is described postulating that any multiparticle quantum evolution can be simulated by classical algorithms of polynomial complexity. Its implementation is presented that is based on the concept of an amplitude quantum, i.e., the smallest carrier of amplitude. As a consequence, the Born rule for quantum probability is derived within the classical urn scheme. Decoherence is treated as stemming from the inherently limited amount of computational resources available for multiparticle simulation. The algorithmic approach denies the possibility of creating a scalable quantum computer. It is therefore refutable by construction of such a computer.

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Quantum simulation, which is generically the task to employ quantum computers to simulate quantum physical models, has been one of the most significant motivations of quantum computing. Quantum dynamics, unitary or nonunitary Markovian dynamics driven by local interactions, has been proved to be efficiently simulatable on quantum computers. Extending the underlying models from unitary to nonunitary evolution, and from continuous-time to discrete-time evolution is essential not only for quantum simulation of more general processes, e.g., dissipative processes with evident non-Markovian effects, but also for developing alternative quantum computing models and algorithms. In this thesis, we explore quantum simulation problems mainly from the following three themes. First, we extend quantum simulation framework of Hamiltonian-driven evolution to quantum simulation of quantum channels, combined with the scheme of algorithmic simulation that accepts a promised simulation accuracy, hence algorithmic quantum channel simulation. Our simulation scheme contains a classical preprocessing part, i.e. a classical algorithm for quantum-circuit design, and a quantum part that is the execution of a quantum circuit for the quantum channel simulator. Second, we employ simulation algorithms for arbitrary quantum channels beyond the dilation method. We explore channel decomposition in terms of convex combination of smaller channels, known as generalized extreme channels, which is known as a nontrivial open problem. To attack this problem, we develop an optimization algorithm for approximate decomposition into a convex sum of generalized extreme channels. We provide an ansatz that proves to be able to yield arbitrary generalized extreme channels and allows a precise quantum circuit description. Furthermore, our numerical simulation has demonstrated the validity of our optimization algorithm for low-dimensional quantum channels. Third, by considering quantum simulation problems beyond the rough distinction between digital and analog simulations, and beyond the quantum-state generation problem, we define uniform, strong, and weak quantum simulations from the point view of operator topology. Besides strong simulation of quantum channels, we define a general weak quantum simulation problem, which simulates observable effects instead of the effects on state generation. Also we study the channel simulation problem in the quantum query model, and provide the query complexity by employing uniform quantum simulation method.

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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.

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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).

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Algorithmic approach to quantum theory is considered. It is based on the supposition that every evolution of many particle system can be simulated by classical algorithms of polynomial complexity. This hypothesis agrees with all known experiments but it presumes the principle cut-off of quantum formalism because it excludes a scalable quantum computer. Algorithmic approach describes quantum evolution uniformly, without separation of measurements from the unitary dynamics; it is shown how Bohrn rule for quantum probability follows from the basic principles of this approach. The radical difference of algorithmic approach from the standard and its perspectives are discussed.
Algorithmic approach is based on the assumption that any quantum evolution of many particle system can be simulated on a classical computer with the polynomial time and memory cost. Algorithms play the central role here but not the analysis, and a simulation gives a "film" which visualizes many particle quantum dynamics and is demonstrated to a user of the model. Restrictions following from the algorithm theory are considered on a level of fundamental physical laws. Born rule for the calculation of quantum probability as well as the decoherence is derived from the existence of a nonzero minimal value of amplitude module - a grain of amplitude. The limitation on the classical computational resources gives the unified description of quantum dynamics that is not divided to the unitary dynamics and measurements and does not depend on the existence of observer. It is proposed the description of states based on the nesting of particles in each other that permits to account the effects of all levels in the same model. Algorithmic approach admits the possibility of refutation, because it forbids the creation of a scalable quantum computer that is allowed in the conventional quantum formalism.
An algorithmic approach to theoretical quantum physics is described postulating that any multiparticle quantum evolution can be simulated by classical algorithms of polynomial complexity. Its implementation is presented that is based on the concept of an amplitude quantum, i.e., the smallest carrier of amplitude. As a consequence, the Born rule for quantum probability is derived within the classical urn scheme. Decoherence is treated as stemming from the inherently limited amount of computational resources available for multiparticle simulation. The algorithmic approach denies the possibility of creating a scalable quantum computer. It is therefore refutable by construction of such a computer.
Quantum simulation, which is generically the task to employ quantum computers to simulate quantum physical models, has been one of the most significant motivations of quantum computing. Quantum dynamics, unitary or nonunitary Markovian dynamics driven by local interactions, has been proved to be efficiently simulatable on quantum computers. Extending the underlying models from unitary to nonunitary evolution, and from continuous-time to discrete-time evolution is essential not only for quantum simulation of more general processes, e.g., dissipative processes with evident non-Markovian effects, but also for developing alternative quantum computing models and algorithms. In this thesis, we explore quantum simulation problems mainly from the following three themes. First, we extend quantum simulation framework of Hamiltonian-driven evolution to quantum simulation of quantum channels, combined with the scheme of algorithmic simulation that accepts a promised simulation accuracy, hence algorithmic quantum channel simulation. Our simulation scheme contains a classical preprocessing part, i.e. a classical algorithm for quantum-circuit design, and a quantum part that is the execution of a quantum circuit for the quantum channel simulator. Second, we employ simulation algorithms for arbitrary quantum channels beyond the dilation method. We explore channel decomposition in terms of convex combination of smaller channels, known as generalized extreme channels, which is known as a nontrivial open problem. To attack this problem, we develop an optimization algorithm for approximate decomposition into a convex sum of generalized extreme channels. We provide an ansatz that proves to be able to yield arbitrary generalized extreme channels and allows a precise quantum circuit description. Furthermore, our numerical simulation has demonstrated the validity of our optimization algorithm for low-dimensional quantum channels. Third, by considering quantum simulation problems beyond the rough distinction between digital and analog simulations, and beyond the quantum-state generation problem, we define uniform, strong, and weak quantum simulations from the point view of operator topology. Besides strong simulation of quantum channels, we define a general weak quantum simulation problem, which simulates observable effects instead of the effects on state generation. Also we study the channel simulation problem in the quantum query model, and provide the query complexity by employing uniform quantum simulation method.
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 ∣ψ∣2{\left| \psi \right|^2}∣ψ∣2 represents the probability density for particles; if the system is in a state ∣ψ⟩\left| \psi \right\rangle∣ψ⟩, then the probability that the eigenvalue λ i of Q is found when Q is measured is P(λ i ) =∣⟨λn∣ψ⟩∣2{\left| {\left\langle {{\lambda _n}} \right|\left. \psi \right\rangle } \right|^2}∣⟨λn​∣ψ⟩∣2. We show that in the technical scheme of quantum computing, there is an imposed strong constraint on Born Rule, that is, double ∣ψ∣2{\left| \psi \right|^2}∣ψ∣2 hypothesis (i.e., Feynman-Deutsch’s ∣ψ∣2{\left| \psi \right|^2}∣ψ∣2(double) reading hypothesis), which requires that there is such a mapping from the state-space CS M to the observation-space CS C : ∣ψM∣2{\left| {{\psi _M}} \right|^2}∣ψM​∣2→∣ψC∣2{\left| {{\psi _C}} \right|^2}∣ψC​∣2, where ∣ψM∣2{\left| {{\psi _M}} \right|^2}∣ψM​∣2is the probability density of the wavefunction a∣0⟩+b∣1⟩{\text{a}}\left| 0 \right\rangle +b\left| 1 \right\ranglea∣0⟩+b∣1⟩ of the single-qubit, ∣ψC∣2{\left| {{\psi _C}} \right|^2}∣ψC​∣2 is the probability density of the corresponding outcomes within CS C , and ∣ψC∣2{\left| {{\psi _C}} \right|^2}∣ψC​∣2 is equivalent to ∣ψM∣2{\left| {{\psi _M}} \right|^2}∣ψM​∣2. The original Born Rule does not assert the mapping ∣ψM∣2{\left| {{\psi _M}} \right|^2}∣ψM​∣2→∣ψC∣2{\left| {{\psi _C}} \right|^2}∣ψC​∣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 ∣ψ∣2{\left| \psi \right|^2}∣ψ∣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 ∣ψ∣2{\left| \psi \right|^2}∣ψ∣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 ∣ψM∣2{\left| {{\psi _M}} \right|^2}∣ψM​∣2→∣ψC∣2{\left| {{\psi _C}} \right|^2}∣ψC​∣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 ∣ψ∣2{\left| \psi \right|^2}∣ψ∣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 ∣ψ∣2{\left| \psi \right|^2}∣ψ∣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).
At the time of this writing more than 60 (sixty) companies in the world are building quantum computers. These computers, based on quantum physics principles, are radically different from those that operate according to the more familiar principles of classical physics. A quantum algorithm takes a number of classical bits as its input, manipulates them so as to create a superposition of all their possible states, further manipulates this exponentially large superposition to obtain the final quantum result, and then measures the result to get (with the appropriate probability distribution) the same number of output bits as in its input. For the middle phase, there are elementary operations which count as one step and yet manipulate all the exponentially many amplitudes of the superposition. The natural language of these quantum gates is that of linear algebra in a complex (Hilbert) vector space. Since 2017 it is known that it is possible to replace the linear algebra with some string-rewriting rules which are no more complicated than the basic rules of arithmetic. The original system was introduced by Terry Rudolph and has been promoted and disseminated in large-scale outreach projects (among others) by Diana Franklin (University of Chicago) and Sofia Economou and Ed Barnes (Virginia Tech) as well as several other educators at the high-school level. In this paper we show how a slightly modified (though still very elementary) system can be used to communicate a visual and entirely operational understanding of key quantum computation concepts such as: superposition, probability, entanglement, phase, interference and unitary state evolution, as they occur in well-known quantum algorithms. We give concrete examples of proving properties for quantum gates and quantum circuits without resorting at all to complex numbers or matrix multiplication. Only simple, abacus-like operations are used-hence the title of the paper. The system we present allows a novice learner to actually trace a quantum algorithm as if it were a classical computation, which is a rare (and, frankly, borderline incredible) luxury in the area of quantum computation, where traditional debugging is impossible. Examples include the phase kickback phenomenon and the famous Deutsch-Josza algorithm. We end with a discussion (and more examples) of how this approach can create a genuine bridge to the mathematics of quantum computation, that is, of vector and tensor algebras in complex spaces for students who may have little or no proper mathematical background.
This paper explores the potential of algorithmic quantum simulation to accelerate the study and analysis of complex, multi-dimensional dynamical systems. We introduce a novel quantum algorithm designed to efficiently explore the state space of these systems, offering a significant advantage over classical simulation techniques, particularly for systems with intricate dynamics and high dimensionality. The core mechanism focuses on utilizing quantum properties to overcome computational bottlenecks and enable more rapid simulations. We outline the algorithm's architecture, key quantum operations, and potential applications within diverse fields such as physics, chemistry, and materials science. The paper concludes with a discussion of future research directions and the expected impact of this approach on the field of quantum computation.