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PolynomialTime Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer
 SIAM J. on Computing
, 1997
"... A digital computer is generally believed to be an efficient universal computing device; that is, it is believed able to simulate any physical computing device with an increase in computation time by at most a polynomial factor. This may not be true when quantum mechanics is taken into consideration. ..."
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Cited by 1277 (4 self)
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A digital computer is generally believed to be an efficient universal computing device; that is, it is believed able to simulate any physical computing device with an increase in computation time by at most a polynomial factor. This may not be true when quantum mechanics is taken into consideration. This paper considers factoring integers and finding discrete logarithms, two problems which are generally thought to be hard on a classical computer and which have been used as the basis of several proposed cryptosystems. Efficient randomized algorithms are given for these two problems on a hypothetical quantum computer. These algorithms take a number of steps polynomial in the input size, e.g., the number of digits of the integer to be factored.
A Fast Quantum Mechanical Algorithm for Database Search
 ANNUAL ACM SYMPOSIUM ON THEORY OF COMPUTING
, 1996
"... Imagine a phone directory containing N names arranged in completely random order. In order to find someone's phone number with a probability of , any classical algorithm (whether deterministic or probabilistic)
will need to look at a minimum of names. Quantum mechanical systems can be in a supe ..."
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Cited by 1135 (10 self)
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Imagine a phone directory containing N names arranged in completely random order. In order to find someone's phone number with a probability of , any classical algorithm (whether deterministic or probabilistic)
will need to look at a minimum of names. Quantum mechanical systems can be in a superposition of states and simultaneously examine multiple names. By properly adjusting the phases of various operations, successful computations reinforce each other while others interfere randomly. As a result, the desired phone number can be obtained in only steps. The algorithm is within a small constant factor of the fastest possible quantum mechanical algorithm.
Quantum complexity theory
 in Proc. 25th Annual ACM Symposium on Theory of Computing, ACM
, 1993
"... Abstract. 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 constructi ..."
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Cited by 574 (5 self)
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Abstract. 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 boundederror probabilistic Turing machine, and thus not in the class BPP. The class BQP of languages that are efficiently decidable (with small errorprobability) on a quantum Turing machine satisfies 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.
Faulttolerant quantum computation
 In Proc. 37th FOCS
, 1996
"... It has recently been realized that use of the properties of quantum mechanics might speed up certain computations dramatically. Interest in quantum computation has since been growing. One of the main difficulties in realizing quantum computation is that decoherence tends to destroy the information i ..."
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Cited by 264 (5 self)
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It has recently been realized that use of the properties of quantum mechanics might speed up certain computations dramatically. Interest in quantum computation has since been growing. One of the main difficulties in realizing quantum computation is that decoherence tends to destroy the information in a superposition of states in a quantum computer, making long computations impossible. A further difficulty is that inaccuracies in quantum state transformations throughout the computation accumulate, rendering long computations unreliable. However, these obstacles may not be as formidable as originally believed. For any quantum computation with t gates, we show how to build a polynomial size quantum circuit that tolerates O(1 / log c t) amounts of inaccuracy and decoherence per gate, for some constant c; the previous bound was O(1 /t). We do this by showing that operations can be performed on quantum data encoded by quantum errorcorrecting codes without decoding this data. 1.
Quantum lower bounds by quantum arguments
 In Proceedings of the ACM Symposium on Theory of Computing
, 2000
"... We propose a new method for proving lower bounds on quantum query algorithms. Instead of a classical adversary that runs the algorithm with one input and then modifies the input, we use a quantum adversary that runs the algorithm with a superposition of inputs. If the algorithm works correctly, its ..."
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Cited by 193 (19 self)
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We propose a new method for proving lower bounds on quantum query algorithms. Instead of a classical adversary that runs the algorithm with one input and then modifies the input, we use a quantum adversary that runs the algorithm with a superposition of inputs. If the algorithm works correctly, its state becomes entangled with the superposition over inputs. We bound the number of queries needed to achieve a sufficient entanglement and this implies a lower bound on the number of queries for the computation. Using this method, we prove two new Ω ( √ N) lower bounds on computing AND of ORs and inverting a permutation and also provide more uniform proofs for several known lower bounds which have been previously proven via variety of different techniques. 1
TwoBit Gates Are Universal for Quantum Computation
, 1995
"... A proof is given, which relies on the commutator algebra of the unitary Lie groups, that quantum gates operating on just two bits at a time are sufficient to construct a general quantum circuit. The best previous result had shown the universality of threebit gates, by analogy to the universality of ..."
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Cited by 182 (9 self)
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A proof is given, which relies on the commutator algebra of the unitary Lie groups, that quantum gates operating on just two bits at a time are sufficient to construct a general quantum circuit. The best previous result had shown the universality of threebit gates, by analogy to the universality of the Toffoli threebit gate of classical reversible computing. Twobit quantum gates may be implemented by magnetic resonance operations applied to a pair of electronic or nuclear spins. A "gearbox quantum computer" proposed here, based on the principles of atomic force microscopy, would permit the operation of such twobit gates in a physical system with very long phase breaking (i.e., quantum phase coherence) times. Simpler versions of the gearbox computer could be used to do experiments on EinsteinPodolskyRosen states and related entangled quantum states.
Quantum walk algorithms for element distinctness
 In: 45th Annual IEEE Symposium on Foundations of Computer Science, OCT 1719, 2004. IEEE Computer Society Press, Los Alamitos, CA
, 2004
"... We use quantum walks to construct a new quantum algorithm for element distinctness and its generalization. For element distinctness (the problem of finding two equal items among N given items), we get an O(N 2/3) query quantum algorithm. This improves the previous O(N 3/4) quantum algorithm of Buhrm ..."
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Cited by 174 (13 self)
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We use quantum walks to construct a new quantum algorithm for element distinctness and its generalization. For element distinctness (the problem of finding two equal items among N given items), we get an O(N 2/3) query quantum algorithm. This improves the previous O(N 3/4) quantum algorithm of Buhrman et al. [11] and matches the lower bound by [1]. We also give an O(N k/(k+1) ) query quantum algorithm for the generalization of element distinctness in which we have to find k equal items among N items. 1
Quantum vs. classical communication and computation
 Proc. 30th Ann. ACM Symp. on Theory of Computing (STOC ’98
, 1998
"... We present a simple and general simulation technique that transforms any blackbox quantum algorithm (à la Grover’s database search algorithm) to a quantum communication protocol for a related problem, in a way that fully exploits the quantum parallelism. This allows us to obtain new positive and ne ..."
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Cited by 158 (14 self)
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We present a simple and general simulation technique that transforms any blackbox quantum algorithm (à la Grover’s database search algorithm) to a quantum communication protocol for a related problem, in a way that fully exploits the quantum parallelism. This allows us to obtain new positive and negative results. The positive results are novel quantum communication protocols that are built from nontrivial quantum algorithms via this simulation. These protocols, combined with (old and new) classical lower bounds, are shown to provide the first asymptotic separation results between the quantum and classical (probabilistic) twoparty communication complexity models. In particular, we obtain a quadratic separation for the boundederror model, and an exponential separation for the zeroerror model. The negative results transform known quantum communication lower bounds to computational lower bounds in the blackbox model. In particular, we show that the quadratic speedup achieved by Grover for the OR function is impossible for the PARITY function or the MAJORITY function in the boundederror model, nor is it possible for the OR function itself in the exact case. This dichotomy naturally suggests a study of boundeddepth predicates (i.e. those in the polynomial hierarchy) between OR and MAJORITY. We present blackbox algorithms that achieve near quadratic speed up for all such predicates.