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arXiv:1604.07517 · Advanced Quantum Technologies

Quantifiable Simulation of Quantum Computation beyond Stochastic Ensemble Computation

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

Published Aug 24, 2018• 4 citations
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Abstract

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.

DOI
10.1002/qute.201800037

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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.
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.
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.
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].
The quantum model of computation is a model, analogous to the probabilistic Turing machine (PTM), in which the normal laws of chance are replaced by those obeyed by particles on a quantum mechanical scale, rather than the rules familiar to us from the macroscopic world. We present here a problem of distinguishing between two fairly natural classes of functions, which can provably be solved exponentially faster in the quantum model than in the classical probabilistic one, when the function is given as an oracle drawn equiprobably from the uniform distribution on either class. We thus offer compelling evidence that the quantum model may have significantly more complexity theoretic power than the PTM. In fact, drawing on this work, Shor has recently developed remarkable new quantum polynomial-time algorithms for the discrete logarithm and integer factoring problems.
A model of quantum computation based on unitary matrix operations was introduced by Feynman and Deutsch. It has been asked whether the power of this model exceeds that of classical Turing machines. We show here that a significant class of these quantum computations can be simulated classically in polynomial time. In particular we show that two-bit operations characterized by 4 \times 4 matrices in which the sixteen entries obey a set of five polynomial relations can be composed according to certain rules to yield a class of circuits that can be simulated classically in polynomial time. This contrasts with the known universality of two-bit operations, and demonstrates that efficient quantum computation of restricted classes is reconcilable with the Polynomial Time Turing Hypothesis. In other words it is possible that quantum phenomena can be used in a scalable fashion to make computers but that they do not have superpolynomial speedups compared to Turing machines for any problem. The techniques introduced bring the quantum computational model within the realm of algebraic complexity theory. In a manner consistent will one view of quantum physics, the wave function is simulated deterministically, and randomization arises only in the course of making measurements. The results generalize the quantum model in that they do not require the matrices to be unitary. In a different direction these techniques also yield deterministic polynomial time algorithms for the decision and parity problems for certain classes of read-twice Boolean formulae. All our results are based on the use of gates that are defined in terms of their graph matching properties.
In this paper we study quantum computation from a complexity theoretic viewpoint. Our first result is the existence of an efficient 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 specified. We prove that O(log⁡T)O(\log T)O(logT) bits of precision suffice to support a T step computation. This justifies 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 first 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 efficiently decidable (with small error-probability) on a quantum Turing machine satisfies $\BPP \subseteq \BQP \subseteq \Ptime^{\SP}$. 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.