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
The Church-Turing thesis says that a digital computer is a universal computational device; that is, it is able to simulate any physically realizable computational device. It has generally been believed that this simulation can be made efficient so that it entails at most a polynomial increase in computation time. This may not be true if quantum mechanics is taken into consideration. A quantum computer is a hypothetical machine based on quantum mechanics. We explain quantum computing, and give an algorithm for prime factorization on a quantum computer that runs asymptotically much faster than the best known algorithm on a digital computer. It is not clear whether it will ever be possible to build large-scale quantum computers. One of the main difficulties is in manipulating coherent quantum states without introducing errors or losing coherence. We discuss quantum error-correcting codes and fault-tolerant quantum computing, which can guarantee highly reliable quantum computation, given only moderately reliable quantum computing hardware.
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(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.
Quantum computing is a rapidly developing field with the potential to revolutionize our ability to solve certain computational problems . While classical computers operate using classical bits, quantum computers use quantum bits (qubits), which can exist in a superposition of states, allowing for parallel computations. The existence of quantum algorithms, which can solve certain problems exponentially faster than classical algorithms, is one of the most exciting aspects of quantum computing. These exotic algorithms are designed to take advantage of quantum mechanical properties such as entanglement and interference, and have the potential to transform a range of fields, including cryptography, optimization, and machine learning.
Quantum computing is a rapidly developing field with the potential to revolutionize our ability to solve certain computational problems . While classical computers operate using classical bits, quantum computers use quantum bits (qubits), which can exist in a superposition of states, allowing for parallel computations. The existence of quantum algorithms, which can solve certain problems exponentially faster than classical algorithms, is one of the most exciting aspects of quantum computing. These exotic algorithms are designed to take advantage of quantum mechanical properties such as entanglement and interference, and have the potential to transform a range of fields, including cryptography, optimization, and machine learning.
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].