Introductory Theory of Computer Science

Lecture Notes in Computer Science

2003

Self Cite

This paper studies the problems involved in the speed-up of the classical computational algorithms using the quantum computational paradigm. In particular, we relate the primitive recursive function approach used in computability theory with the harmonic oscillator basis used in quantum physics. Also, we raise some basic issues concerning quantum computational paradigm: these include failures in programmability and scalability, limitation on the size of the decoherence-free space available and lack of methods for proving quantum programs correct. In computer science, time is discrete and has a well-founded structure. But in physics, time is a real number, continuous and is infinitely divisible; also time can have a fractal dimension. As a result, the time complexity measures for conventional and quantum computation are incomparable. Proving properties of programs and termination rest heavily on the well-founded properties, and the transfinite induction principle. Hence transfinite induction is not applicable to reason about quantum programs.

“…Two approaches for modelling computability are [11,13,14]: (i) Turing machine and (ii) Recursive function.…”

Section: Primitive Recursion and Harmonic Oscillator Functionsmentioning

confidence: 99%

Section: Primitive Recursion and Harmonic Oscillator Functionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Problems and Prospects for Quantum Computational Speed-up

Lecture Notes in Computer Science

2003

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“…[1][2][3] The formal definition of primitive/partial recursive functions consists of four rules: (i) initial functions rule, (ii) composition rule, (iii) primitive recursion rule, and (iv) minimalization rule. We define the above four rules inductively.…”

Section: Recursive Functionsmentioning

confidence: 99%

Quantum Field Theory and Computational Paradigms

2001

Int. J. Mod. Phys. C

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We introduce the basic theory of quantization of radiation field in quantum physics and explain how it relates to the theory of recursive functions in computer science. We outline the basic differences between quantum mechanics (QM) and quantum field theory (QFT) and explain why QFT is better suited for a computational paradigm -based on algorithmic requirement, countably infinite degrees of freedom and the creation of macroscopic output objects. The quanta of the radiation field correspond to the non-negative integers and the harmonic oscillator spectra correspond to the recursive computationwith the creation and annihilation operators, respectively, playing the same role as the successor and predecessor in computability theory. Accordingly, this approach relates the classical computational model and the quantum physical model more directly than the Turing machine approach used earlier. Also, the application of Lambda calculus formalism and the associated denotational semantics (that is widely used in the classical computational paradigm involving recursive functions) for applications to computational paradigm based on quantum field theory is described. Finally, we explain where QFT and conventional paradigm depart from each other, and examine the concept of fixed points, phase transitions, programmability, emergent computation and related open problems.

“…From a pragmatic point of view, we are not even sure whether this study will lead us to the design of a quantum computer in the near future or almost never! Although, quantum version of Boolean logic circuits [1][2][3][4] have been shown to be feasible, the machine hierarchy (namely, finite state machines, push-down stack machine and Turing machine) and the corresponding Chomskian grammatical and lingusitic hierarchy, [5][6][7] and their connection to algorithms and related data structures have not been established in the quantum computational logic, leaving a wide gap in understanding. Yet, some problems have been shown to be amenable for quantum speed-up.…”

Section: Introductionmentioning

confidence: 99%

Mapping, Programmability and Scalability of Problems for Quantum Speed-Up

2003

Int. J. Mod. Phys. C

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This paper explores the reasons as to why the quantum paradigm is not so easy to extend to all of the classical computational algorithms. We also explain the failure of programmability and scalability in quantum speed-up. Due to the presence of quantum entropy, quantum algorithm cannot obviate the curse of dimensionality encountered in solving many complex numerical and optimization problems. Finally, the stringent condition that quantum computers have to be interaction-free, leave them with little versatility and practical utility.