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.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.

ACM Symposium on Theory of Computing (STOC)Jul 6, 200178 citations

Introduction to Special Section on Quantum Computation

Umesh V. Vazirani

The rapid evolution of computers in the half century since their invention has resulted in dramatically smaller and faster computers. However, from a computational point of view, all these computers look alike; for example, they are built out of simple logic gates. A fundamental thesis of computer science---the modern form of the Church--Turing thesis---asserts that this is inevitable in a deep sense. Any computer can be simulated with at most a polynomial factor slowdown by a probabilistic Turing machine. Quantum computation poses the first credible challenge to this thesis. It goes back to a suggestion by Feynman [4], who pointed out that there appears to be no efficient way of simulating a quantum mechanical system on a computer, and suggested that, perhaps, a computer based on quantum physical principles might be able to carry out the simulation efficiently. Two formal models for quantum computers---the quantum Turing machine [2] and quantum computational networks [3]---were defined by Deutsch. The first three papers in this issue describe efficient quantum algorithms for computational tasks that we do not know how to solve classically. In "Quantum Complexity Theory," Bernstein and Vazirani give the first formal evidence that quantum computers violate the modern form of the Church--Turing thesis. They show that a certain problem---the recursive Fourier sampling problem---can be solved in polynomial time on a quantum Turing machine, but relative to an oracle, requires superpolynomial time on a classical probabilistic Turing machine. Simon, in the paper "On the Power of Quantum Computation" introduces a fundamental projection technique and uses it to design an efficient quantum algorithm to determine whether a certain type of function is 2-1 or 1-1. He further shows that, relative to an oracle, this problem requires exponential time on a classical probabilistic Turing machine. In the paper "Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer," Shor gives remarkable polynomial time quantum algorithms for two of the most famous problems in computer science: factoring and discrete log. Since the computational hardness of these problems is the basis of several famous cryptosystems, Shor's paper very dramatically underlines the power of quantum computers. To understand the computational power of quantum computers, it is helpful to consider a quantum mechanical system of n particles, each of which can be in one of two states, labeled $|0\rangle$ and $|1\rangle$ . If this were a classical system, then its instantaneous state could be described by n bits. However, in quantum physics, the system is allowed to be in a linear superposition of configurations, and indeed the instantaneous state of the system is described by a unit vector in the 2n dimensional vector space, whose basis vectors correspond to all the2n classical configurations. Therefore, to describe the instantaneous state of the system, we must specify 2n complex numbers. Nature must update 2n complex numbers at each instant to evolve the system in time. This is an extraordinary amount of effort, since even for n = 200, 2n is larger than estimates of the number of elementary particles in the visible universe. Nonetheless, there are limits to the power of quantum computers. In "Strengths and Weaknesses of Quantum Computing," Bennett, Bernstein, Brassard, and Vazirani show that, relative to a random oracle, with probability 1, the class NP cannot be solved on a quantum Turing machine in time o(2n/2). This bound is tight, since recent work of Grover [5] has shown how to accept any language in NP in time O(2n/2) on a quantum Turing machine. Quantum computers are necessarily time reversible. Indeed, Bennett's work [1] on reversible computation inspired early work on quantum computation that preceded Feynman's paper [4]. The reversibility requirement makes it quite complex to implement even basic computational primitives such as looping or composition. In "Quantum Complexity Theory," Bernstein and Vazirani show how to implement quantum programming primitives and give a construction for an efficient universal quantum Turing machine. The structure of the universal quantum Turing machine is quite simple: it consists of a deterministic Turing machine with a single "quantum coin flip." In "Quantum Computability," Adleman, DeMarrais, and Huang greatly simplify this further by showing that a very simple type of coin flip is sufficient---a rotation by an angle $\theta$ such that ${\rm sin} \theta = 3/5$ . Making quantum computers robust against noise and decoherence is an important and challenging problem. In "Stabilization of Quantum Computations by Symmetrization," Barenco, Berthiaume, Deutsch, Ekert, Jozsa, and Macchiavello show how to use the quantum watchdog effect to stabilize a quantum computation against noise. Their method is based on running several copies of the quantum computer in parallel and projecting its state into the symmetric subspace at frequent intervals. They show that the quantum watchdog effect results in the suppression of errors that lie outside the symmetric subspace. Quantum computation touches upon the foundations of both computer science and quantum physics. It is not unlikely that the issues raised by quantum computation will stimulate further research into the foundations of quantum physics. I wish to express my gratitude to several people who made this special section possible. Oded Goldreich acted as editor for two of the papers in the issue and dealt with them with his characteristic efficiency and judgment. The editorial staff at SIAM, most notably Lisa Dougherty, Beth Gallagher, Deidre Wunderlich, and Sam Young, were extremely helpful, patient, and resourceful. Finally, I would like to thank a number of referees whose careful and timely reviews were critical to putting together this issue. Umesh VaziraniThe rapid evolution of computers in the half century since their invention has resulted in dramatically smaller and faster computers. However, from a computational point of view, all these computers look alike; for example, they are built out of simple logic gates. A fundamental thesis of computer science---the modern form of the Church--Turing thesis---asserts that this is inevitable in a deep sense. Any computer can be simulated with at most a polynomial factor slowdown by a probabilistic Turing machine. Quantum computation poses the first credible challenge to this thesis. It goes back to a suggestion by Feynman [4], who pointed out that there appears to be no efficient way of simulating a quantum mechanical system on a computer, and suggested that, perhaps, a computer based on quantum physical principles might be able to carry out the simulation efficiently. Two formal models for quantum computers---the quantum Turing machine [2] and quantum computational networks [3]---were defined by Deutsch. The first three papers in this issue describe efficient quantum algorithms for computational tasks that we do not know how to solve classically. In "Quantum Complexity Theory," Bernstein and Vazirani give the first formal evidence that quantum computers violate the modern form of the Church--Turing thesis. They show that a certain problem---the recursive Fourier sampling problem---can be solved in polynomial time on a quantum Turing machine, but relative to an oracle, requires superpolynomial time on a classical probabilistic Turing machine. Simon, in the paper "On the Power of Quantum Computation" introduces a fundamental projection technique and uses it to design an efficient quantum algorithm to determine whether a certain type of function is 2-1 or 1-1. He further shows that, relative to an oracle, this problem requires exponential time on a classical probabilistic Turing machine. In the paper "Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer," Shor gives remarkable polynomial time quantum algorithms for two of the most famous problems in computer science: factoring and discrete log. Since the computational hardness of these problems is the basis of several famous cryptosystems, Shor's paper very dramatically underlines the power of quantum computers. To understand the computational power of quantum computers, it is helpful to consider a quantum mechanical system of n particles, each of which can be in one of two states, labeled $|0\rangle$ and $|1\rangle$. If this were a classical system, then its instantaneous state could be described by n bits. However, in quantum physics, the system is allowed to be in a linear superposition of configurations, and indeed the instantaneous state of the system is described by a unit vector in the 2n dimensional vector space, whose basis vectors correspond to all the2n classical configurations. Therefore, to describe the instantaneous state of the system, we must specify 2n complex numbers. Nature must update 2n complex numbers at each instant to evolve the system in time. This is an extraordinary amount of effort, since even for n = 200, 2n is larger than estimates of the number of elementary particles in the visible universe. Nonetheless, there are limits to the power of quantum computers. In "Strengths and Weaknesses of Quantum Computing," Bennett, Bernstein, Brassard, and Vazirani show that, relative to a random oracle, with probability 1, the class NP cannot be solved on a quantum Turing machine in time o(2n/2). This bound is tight, since recent work of Grover [5] has shown how to accept any language in NP in time O(2n/2) on a quantum Turing machine. Quantum computers are necessarily time reversible. Indeed, Bennett's work [1] on reversible computation inspired early work on quantum computation that preceded Feynman's paper [4]. The reversibility requirement makes it quite complex to implement even basic computational primitives such as looping or composition. In "Quantum Complexity Theory," Bernstein and Vazirani show how to implement quantum programming primitives and give a construction for an efficient universal quantum Turing machine. The structure of the universal quantum Turing machine is quite simple: it consists of a deterministic Turing machine with a single "quantum coin flip." In "Quantum Computability," Adleman, DeMarrais, and Huang greatly simplify this further by showing that a very simple type of coin flip is sufficient---a rotation by an angle $\theta$ such that ${\rm sin} \theta = 3/5$. Making quantum computers robust against noise and decoherence is an important and challenging problem. In "Stabilization of Quantum Computations by Symmetrization," Barenco, Berthiaume, Deutsch, Ekert, Jozsa, and Macchiavello show how to use the quantum watchdog effect to stabilize a quantum computation against noise. Their method is based on running several copies of the quantum computer in parallel and projecting its state into the symmetric subspace at frequent intervals. They show that the quantum watchdog effect results in the suppression of errors that lie outside the symmetric subspace. Quantum computation touches upon the foundations of both computer science and quantum physics. It is not unlikely that the issues raised by quantum computation will stimulate further research into the foundations of quantum physics. I wish to express my gratitude to several people who made this special section possible. Oded Goldreich acted as editor for two of the papers in the issue and dealt with them with his characteristic efficiency and judgment. The editorial staff at SIAM, most notably Lisa Dougherty, Beth Gallagher, Deidre Wunderlich, and Sam Young, were extremely helpful, patient, and resourceful. Finally, I would like to thank a number of referees whose careful and timely reviews were critical to putting together this issue. Umesh Vazirani

SIAM Journal on ComputingOct 1, 19974 citations

Quantum Complexity Theory

Ethan Bernstein, Umesh V. Vazirani

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 $O(\log T)$ 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 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 $O(\log T)$ 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.

SIAM Journal on ComputingOct 1, 19971,500 citations

Quantum complexity theory

Ethan Bernstein, Umesh V. Vazirani

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.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.

ACM Symposium on Theory of Computing (STOC)Jan 1, 19931,279 citations

Quantum circuit complexity

Andrew Chi-Chih Yao

We propose a complexity model of quantum circuits analogous to the standard (acyclic) Boolean circuit model. It is shown that any function computable in polynomial time by a quantum Turing machine has a polynomial-size quantum circuit. This result also enables us to construct a universal quantum computer which can simulate, with a polynomial factor slowdown, a broader class of quantum machines than that considered by E. Bernstein and U. Vazirani (1993), thus answering an open question raised by them. We also develop a theory of quantum communication complexity, and use it as a tool to prove that the majority function does not have a linear-size quantum formula.>We propose a complexity model of quantum circuits analogous to the standard (acyclic) Boolean circuit model. It is shown that any function computable in polynomial time by a quantum Turing machine has a polynomial-size quantum circuit. This result also enables us to construct a universal quantum computer which can simulate, with a polynomial factor slowdown, a broader class of quantum machines than that considered by E. Bernstein and U. Vazirani (1993), thus answering an open question raised by them. We also develop a theory of quantum communication complexity, and use it as a tool to prove that the majority function does not have a linear-size quantum formula.>

Annual Symposium on Foundations of Computer ScienceDec 30, 2002629 citations

Computational Power of the Quantum Turing Automata

Sina Jafarpour, Mohammad Ghodsi, Keyvan Sadri +1 more

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 futureLots 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, 20070 citations

|0\rangle

and

|1\rangle

. If this were a classical system, then its instantaneous state could be described by n bits. However, in quantum physics, the system is allowed to be in a linear superposition of configurations, and indeed the instantaneous state of the system is described by a unit vector in the 2n dimensional vector space, whose basis vectors correspond to all the2n classical configurations. Therefore, to describe the instantaneous state of the system, we must specify 2n complex numbers. Nature must update 2n complex numbers at each instant to evolve the system in time. This is an extraordinary amount of effort, since even for n = 200, 2n is larger than estimates of the number of elementary particles in the visible universe. Nonetheless, there are limits to the power of quantum computers. In "Strengths and Weaknesses of Quantum Computing," Bennett, Bernstein, Brassard, and Vazirani show that, relative to a random oracle, with probability 1, the class NP cannot be solved on a quantum Turing machine in time o(2n/2). This bound is tight, since recent work of Grover [5] has shown how to accept any language in NP in time O(2n/2) on a quantum Turing machine. Quantum computers are necessarily time reversible. Indeed, Bennett's work [1] on reversible computation inspired early work on quantum computation that preceded Feynman's paper [4]. The reversibility requirement makes it quite complex to implement even basic computational primitives such as looping or composition. In "Quantum Complexity Theory," Bernstein and Vazirani show how to implement quantum programming primitives and give a construction for an efficient universal quantum Turing machine. The structure of the universal quantum Turing machine is quite simple: it consists of a deterministic Turing machine with a single "quantum coin flip." In "Quantum Computability," Adleman, DeMarrais, and Huang greatly simplify this further by showing that a very simple type of coin flip is sufficient---a rotation by an angle

\theta

such that

{\rm sin} \theta = 3/5

. Making quantum computers robust against noise and decoherence is an important and challenging problem. In "Stabilization of Quantum Computations by Symmetrization," Barenco, Berthiaume, Deutsch, Ekert, Jozsa, and Macchiavello show how to use the quantum watchdog effect to stabilize a quantum computation against noise. Their method is based on running several copies of the quantum computer in parallel and projecting its state into the symmetric subspace at frequent intervals. They show that the quantum watchdog effect results in the suppression of errors that lie outside the symmetric subspace. Quantum computation touches upon the foundations of both computer science and quantum physics. It is not unlikely that the issues raised by quantum computation will stimulate further research into the foundations of quantum physics. I wish to express my gratitude to several people who made this special section possible. Oded Goldreich acted as editor for two of the papers in the issue and dealt with them with his characteristic efficiency and judgment. The editorial staff at SIAM, most notably Lisa Dougherty, Beth Gallagher, Deidre Wunderlich, and Sam Young, were extremely helpful, patient, and resourceful. Finally, I would like to thank a number of referees whose careful and timely reviews were critical to putting together this issue. Umesh Vazirani

O(\log T)

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.