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Quantum Circuit Complexity
, 1993
"... We study 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 polynomialsize quantum circuit. This result also enables us to construct a universal quantum compu ..."
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Cited by 320 (1 self)
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We study 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 polynomialsize 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 Bernstein and Vazirani [BV93], thus answering an open question raised in [BV93]. 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 linearsize quantum formula. Keywords. Boolean circuit complexity, communication complexity, quantum communication complexity, quantum computation AMS subject classifications. 68Q05, 68Q15 1 This research was supported in part by the National Science Foundation under grant CCR9301430. 1 Introduction One of the most intriguing questions in computation theroy ...
Oracle quantum computing
 Brassard & U.Vazirani, Strengths and weaknesses of quantum computing
, 1994
"... \Because nature isn't classical, dammit..." ..."
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Cited by 115 (8 self)
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\Because nature isn't classical, dammit..."
The Quantum Challenge to Structural Complexity Theory
, 1992
"... This is a nontechnical survey paper of recent quantummechanical discoveries that challenge generally accepted complexitytheoretic versions of the ChurchTuring thesis. In particular, building on pionering work of David Deutsch and Richard Jozsa, we construct an oracle relative to which there exi ..."
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Cited by 53 (4 self)
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This is a nontechnical survey paper of recent quantummechanical discoveries that challenge generally accepted complexitytheoretic versions of the ChurchTuring thesis. In particular, building on pionering work of David Deutsch and Richard Jozsa, we construct an oracle relative to which there exists a set that can be recognized in Quantum Polynomial Time (QP), yet any Turing machine that recognizes it would require exponential time even if allowed to be probabilistic, provided that errors are not tolerated. In particular, QP 6` ZPP relative to this oracle. Furthermore, there are cryptographic tasks that are demonstrably impossible to implement with unlimited computing power probabilistic interactive Turing machines, yet they can be implemented even in practice by quantum mechanical apparatus. 1 Deutsch's Quantum Computer In a bold paper published in the Proceedings of the Royal Society, David Deutsch put forth in 1985 the quantum computer [7] (see also [8]). Even though this may c...
Finding a BetterthanClassical Quantum AND/OR Algorithm using Genetic Programming
, 1999
"... This paper documents the discovery of a new, betterthanclassical quantum algorithm for the depthtwo AND/OR tree problem. We describe the genetic programming system that was constructed specifically for this work, the quantum computer simulator that is used to evaluate the fitness of evolving quant ..."
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Cited by 36 (2 self)
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This paper documents the discovery of a new, betterthanclassical quantum algorithm for the depthtwo AND/OR tree problem. We describe the genetic programming system that was constructed specifically for this work, the quantum computer simulator that is used to evaluate the fitness of evolving quantum algorithms, and the newly discovered algorithm. 1 Introduction Quantum computers use the dynamics of atomicscale objects to store and manipulate information. The behavior of atomicscale objects is governed by quantum mechanics rather than by classical physics, and the quantum mechanical properties of these systems can be harnessed to compute certain functions more efficiently than is possible on any classical computer [1]. For example, Shor's quantum factoring algorithm finds the prime factors of an ndigit number in time O(n 2 log(n) log log(n)) [2], while the best known classical factoring algorithms require time O(2 n 1 3 log(n) 2 3 ) and many researchers doubt the existence...
Quantum algorithms for weighing matrices and quadratic residues
 Algorithmica
, 2002
"... In this article we investigate how we can employ the structure of combinatorial objects like Hadamard matrices and weighing matrices to device new quantum algorithms. We show how the properties of a weighing matrix can be used to construct a problem for which the quantum query complexity is signific ..."
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Cited by 20 (2 self)
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In this article we investigate how we can employ the structure of combinatorial objects like Hadamard matrices and weighing matrices to device new quantum algorithms. We show how the properties of a weighing matrix can be used to construct a problem for which the quantum query complexity is significantly lower than the classical one. It is pointed out that this scheme captures both Bernstein & Vazirani’s innerproduct protocol, as well as Grover’s search algorithm. In the second part of the article we consider Paley’s construction of Hadamard matrices to design a more specific problem that uses the Legendre symbol χ (which indicates if an element of a finite field GF(p k) is a quadratic residue or not). It is shown how for a shifted Legendre function fs(x) = χ(x+s), the unknown s ∈ GF(p k) can be obtained exactly with only two quantum calls to fs. This is in sharp contrast with the observation that any classical, probabilistic procedure requires at least k log p queries to solve the same problem. 1
A Decision Procedure for WellFormed Linear Quantum Cellular Automata
, 1996
"... In this paper we introduce a new quantum computation model, the linear quantum cellular automaton. Wellformedness is an essential property for any quantum computing device since it enables us to dene the probability of a conguration in an observation as the squared magnitude of its amplitude. We gi ..."
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Cited by 17 (1 self)
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In this paper we introduce a new quantum computation model, the linear quantum cellular automaton. Wellformedness is an essential property for any quantum computing device since it enables us to dene the probability of a conguration in an observation as the squared magnitude of its amplitude. We give an ecient algorithm which decides if a linear quantum cellular automaton is wellformed. The complexity of the algorithm is O(n 2 ) in the algebraic model of computation if the input automaton has continuous neighborhood. key words: quantum computation, cellular automata, de Bruijn graphs 1 Introduction In order to analyze the complexity of algorithms, computer scientists usually choose some computational model, implement the algorithm on it and count the number of steps as a function of the size of the input. Dierent models, such as Turing machines (TM), random access machines, circuits, or cellular automata can be used. They are all universal in the sense that they can simulate e...
Quantum Domain Theory  Definitions and Applications
 Proceedings of CCA’03
, 2003
"... Domain theory is a branch of classical computer science. It has proven to be a rigourous mathematical structure to describe denotational semantics for programming languages and to study the computability of partial functions. In this paper, we study the extension of domain theory to the quantum sett ..."
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Cited by 8 (0 self)
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Domain theory is a branch of classical computer science. It has proven to be a rigourous mathematical structure to describe denotational semantics for programming languages and to study the computability of partial functions. In this paper, we study the extension of domain theory to the quantum setting. By defining a quantum domain we introduce a rigourous definition of quantum computability for quantum states and operators. Furthermore we show that the denotational semantics of quantum computation has the same structure as the denotational semantics of classical probabilistic computation introduced by Kozen [23]. Finally, we briefly review a recent result on the application of quantum domain theory to quantum information processing. 1
Artificial immune systems and the grand challenge for nonclassical computation
 Proceedings of the 2003 International Conference on Artificial Immune Systems, LNCS 2787
, 2003
"... Abstract. The UK Grand Challenges for Computing Research is an initiative to map out certain key areas that could be used to help drive research over the next 10–15 years. One of the identified Grand Challenges is NonClassical Computation, which examines many of the fundamental assumptions of Compu ..."
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Cited by 4 (0 self)
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Abstract. The UK Grand Challenges for Computing Research is an initiative to map out certain key areas that could be used to help drive research over the next 10–15 years. One of the identified Grand Challenges is NonClassical Computation, which examines many of the fundamental assumptions of Computer Science, and asks what would result if they were systematically broken. In this discussion paper, we explain how the subdiscipline of Artificial Immune Systems sits squarely in the province of this particular Grand Challenge, and we identify certain key questions.
Quantum computation and quantum information
 International Journal of Parallel, Emergent and Distributed Systems
, 2006
"... The paper is intended to be a survey of all the important aspects and results that have shaped the eld of quantum computation and quantum information. The reader is rst familiarized with those features and principles of quantum mechanics providing a more e cient and secure information processing. Th ..."
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Cited by 3 (3 self)
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The paper is intended to be a survey of all the important aspects and results that have shaped the eld of quantum computation and quantum information. The reader is rst familiarized with those features and principles of quantum mechanics providing a more e cient and secure information processing. Their applications to the general theory of information, cryptography, algorithms, computational complexity and errorcorrection are then discussed. Prospects for building a practical quantum computer are also analyzed. 1 Introduction and
Quantum Algorithms in group theory
, 2003
"... We present a survey of quantum algorithms, primarily for an intended audience of pure mathematicians. We place an emphasis on algorithms involving group theory. ..."
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Cited by 2 (0 self)
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We present a survey of quantum algorithms, primarily for an intended audience of pure mathematicians. We place an emphasis on algorithms involving group theory.