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SIAM Journal on Computing

Quantum Computability

Leonard M. Adleman, Jonathan DeMarrais, Ming-Deh A. Huang

Published Oct 1, 1997• 260 citations
Publisher

Abstract

In this paper some theoretical and (potentially) practical aspects of quantum computing are considered. Using the tools of transcendental number theory it is demonstrated that quantum Turing machines (QTM) with rational amplitudes are sufficient to define the class of bounded error quantum polynomial time (BQP) introduced by Bernstein and Vazirani [Proc. 25th ACM Symposium on Theory of Computation, 1993, pp. 11--20, SIAM J. Comput., 26 (1997), pp. 1277--1339]. On the other hand, if quantum Turing machines are allowed unrestricted amplitudes (i.e., arbitrary complex amplitudes), then the corresponding BQP class has uncountable cardinality and contains sets of all Turing degrees. In contrast, allowing unrestricted amplitudes does not increase the power of computation for error-free quantum polynomial time (EQP). Moreover, with unrestricted amplitudes, BQP is not equal to EQP. The relationship between quantum complexity classes and classical complexity classes is also investigated. It is shown that when quantum Turing machines are restricted to have transition amplitudes which are algebraic numbers, BQP, EQP, and nondeterministic quantum polynomial time (NQP) are all contained in PP, hence in ${\rm P}^{#{\rm P}}$ and PSPACE. A potentially practical issue of designing "machine independent" quantum programs is also addressed. A single ("almost universal") quantum algorithm based on Shor's method for factoring integers is developed which would run correctly on almost all quantum computers, even if the underlying unitary transformations are unknown to the programmer and the device builder.

DOI
10.1137/s0097539795293639

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Lots of efforts in the last decades have been done to prove or disprove whether the set of polynomially bounded problems is equal to the set of polynomially verifiable problems. This paper will present an overview of the current beliefs of quantum complexity theorists and discussion detail the impacts these beliefs may have on the future of the field. We introduce a new form of Turing machine based on the concepts of quantum mechanics and investigate the domain of this novel definition for computation. The main object is to show the domain of these computation machines and the new definition of algorithms using these automata. In section one we give a brief fundamental overview of classical complexity theory, Turing machine and all the fundamental concepts required for the next chapters. In section two we describe various classes of classical complexity automata. In next chapter we introduce quantum complexity featuring quantum Turing machines and discuss its relationship to classical complexity. Section four presents a relationship between quantum complexity classes based on the quantum Turing machines and quantum oracles, and their relationship with corresponding classical complexity classes. And section five presents a discussion on the practical phenomena in problem solving using quantum computers. Finally in section five we speculate briefly on the direction we believe the field to be headed, and what might reasonably be expected in the future

•Apr 1, 2007•0 citations
In this paper some theoretical and (potentially) practical aspects of quantum computing are considered. Using the tools of transcendental number theory it is demonstrated that quantum Turing machines (QTM) with rational amplitudes are sufficient to define the class of bounded error quantum polynomial time (BQP) introduced by Bernstein and Vazirani [Proc. 25th ACM Symposium on Theory of Computation, 1993, pp. 11--20, SIAM J. Comput., 26 (1997), pp. 1277--1339]. On the other hand, if quantum Turing machines are allowed unrestricted amplitudes (i.e., arbitrary complex amplitudes), then the corresponding BQP class has uncountable cardinality and contains sets of all Turing degrees. In contrast, allowing unrestricted amplitudes does not increase the power of computation for error-free quantum polynomial time (EQP). Moreover, with unrestricted amplitudes, BQP is not equal to EQP. The relationship between quantum complexity classes and classical complexity classes is also investigated. It is shown that when quantum Turing machines are restricted to have transition amplitudes which are algebraic numbers, BQP, EQP, and nondeterministic quantum polynomial time (NQP) are all contained in PP, hence in {\rm P}^{#{\rm P}} and PSPACE. A potentially practical issue of designing "machine independent" quantum programs is also addressed. A single ("almost universal") quantum algorithm based on Shor's method for factoring integers is developed which would run correctly on almost all quantum computers, even if the underlying unitary transformations are unknown to the programmer and the device builder.
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.
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.
In 1985, D. Deutsch introduced quantum Turing machines (QTMs for short) as Turing machines which can perform so called quantum parallel computations. Then, in 1994, P. Shor showed that QTM can factor integers with arbitrary small error probability in polynomial time. The quantum complexity theory is the computational complexity theory based on QTMs. In this paper, we first review the definitions of the QTM and major quantum complexity classes EQP, BQP, ZQP, etc. Then, we present the known results and major open questions in this field.
In this paper, we explore the power of quantum computers with restricted transition amplitudes. In 1997 Adleman, DeMarrais, and Huang showed that quantum Turing machines (QTMs) with the amplitudes from [Formula: see text] are computationally equivalent to ones with the polynomial-time computable amplitudes as machines implementing bounded-error polynomial-time algorithms. We show that QTMs with the amplitudes from [Formula: see text] is polynomial-time equivalent to deterministic Turing machines as machines implementing exact algorithms, i.e., algorithms that output correct answers with certainty. By extending this result, it is shown that exact quantum computers with rational biased coins are equivalent to classical computers. Moreover, we discuss the computational power of exact quantum computers with multiple types of coins. We also show that, from the viewpoint of zero-error polynomial-time algorithms, [Formula: see text] is not more powerful than [Formula: see text] as the set of amplitudes taken by QTMs; however, it is sufficient to solve the factoring problem.
Contrary to the classical case, the relation between quantum programming languages and quantum Turing Machines (QTM) has not been fully investigated. In particular, there are features of QTMs that have not been exploited, a notable example being the intrinsic infinite nature of any quantum computation. In this paper, we propose a definition of QTM, which extends and unifies the notions of Deutsch and Bernstein & Vazirani. In particular, we allow both arbitrary quantum input, and meaningful superpositions of computations, where some of them are “terminated” with an “output”, while others are not. For some infinite computations an “output” is obtained as a limit of finite portions of the computation. We propose a natural and robust observation protocol for our QTMs, which does not modify the probability of the possible outcomes of the machines. Finally, we use QTMs to define a class of quantum computable functions—any such function is a mapping from a general quantum state to a probability distribution of natural numbers. We expect that our class of functions, when restricted to classical input-output, will not be different from the set of the recursive functions.
Lots of efforts in the last decades have been done to prove or disprove whether the set of polynomially bounded problems is equal to the set of polynomially verifiable problems. This paper will present an overview of the current beliefs of quantum complexity theorists and discussion detail the impacts these beliefs may have on the future of the field. We introduce a new form of Turing machine based on the concepts of quantum mechanics and investigate the domain of this novel definition for computation. The main object is to show the domain of these computation machines and the new definition of algorithms using these automata. In section one we give a brief fundamental overview of classical complexity theory, Turing machine and all the fundamental concepts required for the next chapters. In section two we describe various classes of classical complexity automata. In next chapter we introduce quantum complexity featuring quantum Turing machines and discuss its relationship to classical complexity. Section four presents a relationship between quantum complexity classes based on the quantum Turing machines and quantum oracles, and their relationship with corresponding classical complexity classes. And section five presents a discussion on the practical phenomena in problem solving using quantum computers. Finally in section five we speculate briefly on the direction we believe the field to be headed, and what might reasonably be expected in the future