Building on the work of Deutsch and Jozsa, we construct oracles relative to which (1) there is a decision problem that can be solved with certainty in worst-case polynomial time on the quantum computer, yet it cannot be solved classically in probabilistic expected polynomial time if errors are not tolerated, nor even in nondeterministic polynomial time, and (2) there is a decision problem that can be solved in exponential time on the quantum computer, which requires double exponential time on all but finitely many instances on any classical deterministic computer.
This paper continues the study of the power of oracles to separate quantum com.plexity classes from classical (including probabilistic and nondeterministic) complexity classes, which we initiated in an earlier paper [5]. The new results are that, under appropriate oracles, (1) there is a decision problem that can be solved in polynomial time on the quantum computer, which would classically require exponential time in the worst case even on a nondeterministic computer (in particular QP g NP relative to this oracle), and (2) there is a decision problem that can be solved in exponential time on the quantum computer for which any deterministic classical computer would require double exponential time on all but finitely many instances.
Recently a great deal of attention has been focused on quantum computation following a sequence of results [Bernstein and Vazirani, in Proc. 25th Annual ACM Symposium Theory Comput., 1993, pp. 11--20, SIAM J. Comput., 26 (1997), pp. 1277--1339], [Simon, in Proc. 35th Annual IEEE Symposium Foundations Comput. Sci., 1994, pp. 116--123, SIAM J. Comput., 26 (1997), pp. 1340--1349], [Shor, in Proc. 35th Annual IEEE Symposium Foundations Comput. Sci., 1994, pp. 124--134] suggesting that quantum computers are more powerful than classical probabilistic computers. Following Shor's result that factoring and the extraction of discrete logarithms are both solvable in quantum polynomial time, it is natural to ask whether all of $\NP$ can be efficiently solved in quantum polynomial time. In this paper, we address this question by proving that relative to an oracle chosen uniformly at random with probability 1 the class $\NP$ cannot be solved on a quantum Turing machine (QTM) in time
o(2n/2). We also show that relative to a permutation oracle chosen uniformly at random with probability 1 the class $\NP \cap \coNP$ cannot be solved on a QTM in time
o(2n/3). The former bound is tight since recent work of Grover [in {\it Proc.\
28th Annual ACM Symposium Theory Comput.}, 1996] shows how to accept the class $\NP$ relative to any oracle on a quantum computer in time
O(2n/2).
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⟩ and
∣1⟩. 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
θ such that
sinθ=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
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.