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Advanced Quantum Technologies

Quantifiable Simulation of Quantum Computation beyond Stochastic Ensemble Computation (Adv. Quantum Technol. 2/2018)

Jeongho Bang, Junghee Ryu, Chang‐Woo Lee, Ki Hyuk Yee, Jinhyoung Lee, Wonmin Son

Published Oct 1, 2018• 0 citations
Publisher

Abstract

A model of quantum computation (QC) is compared with a generalized classical computation model, named stochastic ensemble computation (SEnC), in article number 1800037 by Jeongho Bang, Jinhyoung Lee, Wonmin Son and co-workers. SEnC runs in an ensemble consisting of finite copies of single probabilistic entities, and hence is a generalized probabilistic Turing machine. Impossibility of simulating a QC in the SEnC is proved, manifesting “QC is characterized beyond any classical probabilistic computation,” i.e., SEnC ⊆ QC.

DOI
10.1002/qute.201870025

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A model of quantum computation (QC) is compared with a generalized classical computation model, named stochastic ensemble computation (SEnC), in article number 1800037 by Jeongho Bang, Jinhyoung Lee, Wonmin Son and co-workers. SEnC runs in an ensemble consisting of finite copies of single probabilistic entities, and hence is a generalized probabilistic Turing machine. Impossibility of simulating a QC in the SEnC is proved, manifesting “QC is characterized beyond any classical probabilistic computation,” i.e., SEnC ⊆ QC.
Abstract In this study, a distinctive feature of quantum computation (QC) is characterized. To this end, a seemingly‐powerful classical computing model, called “stochastic ensemble machine (SEnM),” is considered. The SEnM runs with an ensemble consisting of finite copies of a single probabilistic machine, hence is as powerful as a probabilistic Turing machine (PTM). Then the hypothesis—that is, the SEnM can effectively simulate a general circuit model of QC—is tested by introducing an information‐theoretic inequality, named readout inequality. The inequality is satisfied by the SEnM and imposes a critical condition: if the hypothesis holds, the inequality should be satisfied by the probing model of QC. However, it is shown that the above hypothesis is not generally accepted with the inequality violation; namely, such a simulation necessarily fails, implying that PTM ⊆ QC.
In this article, we underline and compare different quantum and deterministic models of computation by means of a probabilistic point of view. The main models used for this are deterministic, probabilistic and quantum Turing machines (DTM, PTM and QTM, respectively). Then we suggest that extending PTM with only part of quantum features is not useful to implement real quantum supremacy if we don’t incorporate them fully. This is done reviewing some notorious quantum physical experiments and quantum algorithms, showing that generally PTM does not fit for them, also if in some particular cases it fits.Here we compare probabilistic and quantum models of computation to underline some common features and differences.
Quantum computing allows us to solve some problems much faster than existing classical algorithms. Yet, the quantum computer has been believed to be no more powerful than the most general computing model—the Turing machine. Undecidable problems, such as the halting problem, and unrecognizable inputs, such as the real numbers, are beyond the theoretical limit of the Turing machine. I suggest a model for a quantum computer, which is less general than the Turing machine, but may solve the halting problem for any task programmable on it. Moreover, inputs unrecognizable by the Turing machine can be recognized by the model, thus breaking the theoretical limit for a computational task. A quantum computer is not just a successful design of the Turing machine as it is widely perceived now, but is a different, less general but more powerful model for computing, the practical realization of which may need different strategies than those in use now.
The quantum model of computation not only offers entirely new ways to manipulate information, but also allows information processing tasks to be formulated in unconventional, genuine quantum mechanical terms. We show that the task of distinguishing among entangled quantum states combines entanglement and non-determinism in a way that makes the quantum solution impossible to simulate on any classical machine (even one equipped with the same measurement capabilities as the quantum computational device). A new class of information processing tasks is thus uncovered whose members are readily carried out by a quantum computer, yet are impossible to perform on any classical machine (whether deterministic or probabilistic). In the broad, unconventional context created by quantum mechanics, the computational power of a quantum computer is therefore strictly greater than that of a classical computer.
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.
Abstract. In this paper we study quantum computation from a complexity theoretic viewpoint. Our rst result is the existence of an ecient universal quantum Turing machine in Deutsch’s model of a quantum Turing machine (QTM) [Proc. Roy. Soc. London Ser. A, 400 (1985), pp. 97{117]. This construction is substantially more complicated than the corresponding construction for classical Turing machines (TMs); in fact, even simple primitives such as looping, branching, and composition are not straightforward in the context of quantum Turing machines. We establish how these familiar primitives can be implemented and introduce some new, purely quantum mechanical primitives, such as changing the computational basis and carrying out an arbitrary unitary transformation of polynomially bounded dimension. We also consider the precision to which the transition amplitudes of a quantum Turing machine need to be specied. We prove that O(log T) bits of precision suce to support a T step computation. This justies the claim that the quantum Turing machine model should be regarded as a discrete model of computation and not an analog one. We give the rst formal evidence that quantum Turing machines violate the modern (complexity theoretic) formulation of the Church{Turing thesis. We show the existence of a problem, relative to an oracle, that can be solved in polynomial time on a quantum Turing machine, but requires superpolynomial time on a bounded-error probabilistic Turing machine, and thus not in the class BPP. The class BQP of languages that are eciently decidable (with small error-probability) on a quantum Turing machine satises BPP ⊆ BQP ⊆ P]P. Therefore, there is no possibility of giving a mathematical proof that quantum Turing machines are more powerful than classical probabilistic Turing machines (in the unrelativized setting) unless there is a major breakthrough in complexity theory.