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arXiv:1310.8457 · Cambridge University Press eBooks

Critique of fault-tolerant quantum information processing

Robert Alicki

Published Sep 5, 2013• 4 citations
arXivHTMLPDF

Abstract

Introduction The idea of fault-tolerant quantum computation presents a bold challenge to the rather well established principles called the strong Church–Turing thesis and the Bohr correspondence principle. One of the variations of the strong Church–Turing thesis (SCTT), which is not due to Church or Turing, but rather gradually developed in the field of computational complexity theory, is the following [BV97]: Any “reasonable” model of computation can be efficiently simulated on a probabilistic Turing machine. This can be expressed even less formally but more practically as: No computer can be more efficient than a digital one equipped with a random number generator. Here, a computer A is “more efficient” than a computer B if A can solve, in polynomial time, a problem that cannot be solved in a polynomial time by computer B. On the other hand the Bohr correspondence principle (BCP) demands that: Classical physics and quantum physics give the same answer when the systems become large. A more rigorous form of the BCP is the following: For large systems the experimental data are consistent with classical probabilistic models . Here, by a classical probabilistic model we mean a model in which all observables are described by real functions, and all states by probability distributions on a certain set (“phase-space”).

DOI
10.1017/cbo9781139034807.028

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Introduction The idea of fault-tolerant quantum computation presents a bold challenge to the rather well established principles called the strong Church–Turing thesis and the Bohr correspondence principle. One of the variations of the strong Church–Turing thesis (SCTT), which is not due to Church or Turing, but rather gradually developed in the field of computational complexity theory, is the following [BV97]: Any “reasonable” model of computation can be efficiently simulated on a probabilistic Turing machine. This can be expressed even less formally but more practically as: No computer can be more efficient than a digital one equipped with a random number generator. Here, a computer A is “more efficient” than a computer B if A can solve, in polynomial time, a problem that cannot be solved in a polynomial time by computer B. On the other hand the Bohr correspondence principle (BCP) demands that: Classical physics and quantum physics give the same answer when the systems become large. A more rigorous form of the BCP is the following: For large systems the experimental data are consistent with classical probabilistic models . Here, by a classical probabilistic model we mean a model in which all observables are described by real functions, and all states by probability distributions on a certain set (“phase-space”).
There is strong evidence that computers based upon the principles of quantum physics represent an inherently new and more powerful model of computation. Such computers violate the modern form of the Church-Turing thesis (which lies at the foundations of computer science). This thesis can be informally summarized as follows: all physical implementations of computing devices can be described by the same abstract model of computation-the probabilistic Turing Machine or the equivalent random access machine. In this paper, we will describe two formal models for quantum computers: quantum circuits and quantum Turing Machines, introduced by Deutsch [1985].
Turning to quantum mechanics … secret, secret, close the doors! we always have had a great deal of difficulty in understanding the world view that quantum mechanics represents. … It has not yet become obvious to me that there's no real problem. I cannot define the real problem, therefore I suspect there's no real problem, but I'm not sure there's no real problem. So that's why I like to investigate things. – Richard Feynman, 1964 The only difference between a probabilistic classical world and the equations of the quantum world is that somehow or other it appears as if the probabilities would have to go negative. – Richard Feynman, in “Simulating Physics with Computers,” 1982 Our first result is the existence of an efficient universal quantum Turning machine in Deutsch's model. … We give the first formal evidence that quantum Turning machines violate the modern complexity theoretic formulation of the Church Turning thesis. We show the existence of a problem relative to an oracle that can be solved in polynomial time on a quantum Turning machine but require super polynomial time on a bounded error probabilistic Turning machine. – E. Bernstein and U. Vazirani, “Quantum Complexity Theory,” 1997 Quantum computing is a new computational model that may be physically realizable and may provide an exponential advantage over “classical” computational models such as probabilistic and deterministic Turing machines.
. We revisit the notion of a quantum Turing-machine , whose design is based on the laws of quantum mechanics. It turns out that such a machine is not more powerful, in the sense of computability, than the machine originally constructed by Turing. Quantum Turing-machines do not violate the Church–Turing thesis. The benefit of quantum computing lies in efficiency . Quantum computers appear to be more efficient, in time, than classical Turing-machines, however its exact additional computational power is unclear, as this question ties in with deep open problems in complexity theory. We will sketch where BQP, the quantum analogue of the complexity class P, resides in the realm of complexity classes. §1. Introduction . A decade before Turing developed his theory of computing, physicists struggled with the advent of quantum mechanics. During the famous 5th Solvay Conference in 1927 it was clear that a new era of physics had surfaced. Its strange features like superposition and entanglement still lead to heated discussions and much confusion. However strange and counter-intuitive, the theory has never been refuted by experiments that are performed daily and in great numbers throughout laboratories around the world. Time after time the predictions of quantum mechanics are in full agreement with experiment. Shortly after the advent of quantum mechanics, Church, Turing and Post developed the notion of computability [Chu36, Tur36, Pos36]. Less than 10 years later these formal ideas would be put to practice resulting in the ENIAC, the first general purpose machine.
It is possible in principle to construct quantum mechanical observables and unitary operators which, if implemented in physical systems as measurements and dynamical evolution, would contradict the Church-Turing thesis, which lies at the foundation of computer science. Elsewhere we have argued that the quantum measurement problem implies a finite, computational model of the measurement and evolution of quantum states. If correct, this approach helps to identify the key feature that can reconcile quantum mechanics with the Church-Turing thesis: finitude of the degree of fine-graining of Hilbert space. This suggests that the Church-Turing thesis constrains the physical universe and thereby highlights a surprising connection between purely logical and algorithmic considerations on the one hand and physical reality on the other.
In the paper we interpret the Church Turing thesis of Computation Algorithm and how it differs from the strong Church Turing thesis in the field of computation. Church Turing thesis can be classified as Strong Church Turing thesis and Normal Church Turing thesis. Church Turing hypothesis says “that a computing problem can be solved in any computer if and only if it can be solved on a very simple ‘machine’, named TURING MACHINE”. We introspect the difference in the model of computers that are developed on these two competitive algorithms. We will also be studying the physical realization of bits in a quantum computer with special mention to the commercial quantum computer systems offered at the moment. Our objective is to find out the relation of a quantum bit and the Strong Church Turing Thesis converging the gap between theoretical model and physical structure.
The extended Church-Turing thesis posits that any computable function can be calculated efficiently by a probabilistic Turing machine. If this thesis held true, the global effort to build quantum computers might ultimately be unnecessary. The thesis would however be strongly contradicted by a physical device that efficiently performs a task believed to be intractable for classical computers. BOSONSAMPLING—the sampling from a distribution of n photons undergoing some linear-optical process—is a recently developed, and experimentally accessible example of such a task [1].