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Quantuminspired Evolutionary Algorithm for a Class of Combinatorial Optimization
 IEEE TRANS. EVOLUTIONARY COMPUTATION
, 2002
"... This paper proposes a novel evolutionary algorithm inspired by quantum computing, called a quantuminspired evolutionary algorithm (QEA), which is based on the concept and principles of quantum computing, such as a quantum bit and superposition of states. Like other evolutionary algorithms, QEA is a ..."
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Cited by 110 (7 self)
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This paper proposes a novel evolutionary algorithm inspired by quantum computing, called a quantuminspired evolutionary algorithm (QEA), which is based on the concept and principles of quantum computing, such as a quantum bit and superposition of states. Like other evolutionary algorithms, QEA is also characterized by the representation of the individual, the evaluation function, and the population dynamics. However, instead of binary, numeric, or symbolic representation, QEA uses a Qbit, defined as the smallest unit of information, for the probabilistic representation and a Qbit individual as a string of Qbits. A Qgate is introduced as a variation operator to drive the individuals toward better solutions. To demonstrate its effectiveness and applicability, experiments are carried out on the knapsack problem, which is a wellknown combinatorial optimization problem. The results show that QEA performs well, even with a small population, without premature convergence as compared to the conventional genetic algorithm.
PolynomialTime Quantum Algorithms for Pell's Equation and the Principal Ideal Problem
 in Proceedings of the 34th ACM Symposium on Theory of Computing
, 2001
"... Besides Shor's polynomialtime quantum algorithms for factoring and discrete log, all progress in understanding when quantum algorithms have an exponential advantage over classical algorithms has been through oracle problems. Here we give efficient quantum algorithms for two more nonoracle pro ..."
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Cited by 108 (7 self)
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Besides Shor's polynomialtime quantum algorithms for factoring and discrete log, all progress in understanding when quantum algorithms have an exponential advantage over classical algorithms has been through oracle problems. Here we give efficient quantum algorithms for two more nonoracle problems. The first is Pell's equation. Given a positive nonsquare integer d, Pell's equation is x&sup2;  dy&sup2; = 1 and the goal is to find its integer solutions. Factoring integers reduces to finding integer solutions of Pell's equation, but a reduction in the other direction is not known and appears more difficult. The second problem is the principal ideal problem in real quadratic number fields. Solving this problem is at least as hard as solving Pell's equation, and is the basis of a cryptosystem which is broken by our algorithm. We also state some related open problems from the area of computational algebraic number theory.
A transformation based algorithm for reversible logic synthesis
 in Design Automation Conf
"... A digital combinational logic circuit is reversible if it maps each input pattern to a unique output pattern. Such circuits are of interest in quantum computing, optical computing, nanotechnology and lowpower CMOS design. Synthesis approaches are not well developed for reversible circuits even for ..."
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Cited by 102 (23 self)
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A digital combinational logic circuit is reversible if it maps each input pattern to a unique output pattern. Such circuits are of interest in quantum computing, optical computing, nanotechnology and lowpower CMOS design. Synthesis approaches are not well developed for reversible circuits even for small numbers of inputs and outputs. In this paper, a transformation based algorithm for the synthesis of such a reversible circuit in terms of n n Toffoli gates is presented. Initially, a circuit is constructed by a single pass through the specication with minimal lookahead and no backtracking. Reduction rules are then applied by simple template matching. The method produces nearoptimal results for 3input circuits and also produces very good results for larger problems.
Resilient quantum computation
 Science
, 1998
"... This article is a short introduction to and review of the clusterstate model of quantum computation, in which coherent quantum information processing is accomplished via a sequence of singlequbit measurements applied to a fixed quantum state known as a cluster state. We also discuss a few novel pr ..."
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Cited by 93 (3 self)
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This article is a short introduction to and review of the clusterstate model of quantum computation, in which coherent quantum information processing is accomplished via a sequence of singlequbit measurements applied to a fixed quantum state known as a cluster state. We also discuss a few novel properties of the model, including a proof that the cluster state cannot occur as the exact ground state of any naturally occurring physical system, and a proof that measurements on any quantum state which is linearly prepared in one dimension can be efficiently simulated on a classical computer, and thus are not candidates for use as a substrate for quantum computation. Key words: quantum computation, cluster states, oneway quantum computer 1.
Quantum algorithms for the triangle problem
 PROCEEDINGS OF SODA’05
, 2005
"... We present two new quantum algorithms that either find a triangle (a copy of K3) in an undirected graph G on n nodes, or reject if G is triangle free. The first algorithm uses combinatorial ideas with Grover Search and makes Õ(n10/7) queries. The second algorithm uses Õ(n13/10) queries, and it is b ..."
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Cited by 93 (10 self)
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We present two new quantum algorithms that either find a triangle (a copy of K3) in an undirected graph G on n nodes, or reject if G is triangle free. The first algorithm uses combinatorial ideas with Grover Search and makes Õ(n10/7) queries. The second algorithm uses Õ(n13/10) queries, and it is based on a design concept of Ambainis [6] that incorporates the benefits of quantum walks into Grover search [18]. The first algorithm uses only O(log n) qubits in its quantum subroutines, whereas the second one uses O(n) qubits. The Triangle Problem was first treated in [12], where an algorithm with O(n + √ nm) query complexity was presented, where m is the number of edges of G.
Quantum search of spatial regions
 THEORY OF COMPUTING
, 2005
"... Can Grover’s algorithm speed up search of a physical region—for example a 2D grid of size √ n × √ n? The problem is that √ n time seems to be needed for each query, just to move amplitude across the grid. Here we show that this problem can be surmounted, refuting a claim to the contrary by Beniof ..."
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Cited by 85 (8 self)
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Can Grover’s algorithm speed up search of a physical region—for example a 2D grid of size √ n × √ n? The problem is that √ n time seems to be needed for each query, just to move amplitude across the grid. Here we show that this problem can be surmounted, refuting a claim to the contrary by Benioff. In particular, we show how to search a ddimensional hypercube in time O ( √ n) for d ≥ 3, or O ( √ nlog 5/2 n) for d = 2. More generally, we introduce a model of quantum query complexity on graphs, motivated by fundamental physical limits on information storage, particularly the holographic principle from black hole thermodynamics. Our results in this model include almosttight upper and lower bounds for many search tasks; a generalized algorithm that works for any graph with good expansion properties, not just hypercubes; and relationships among several notions of ‘locality’ for unitary matrices acting on graphs. As an application of our results, we give an O (√ n)qubit communication protocol for the disjointness problem, which improves an upper bound of Høyer and de Wolf and matches a lower bound of Razborov.
Adiabatic quantum computation is equivalent to standard quantum computation
 SIAM Journal on Computing
"... Adiabatic quantum computation has recently attracted attention in the physics and computer science communities, but its computational power was unknown. We describe an efficient adiabatic simulation of any given quantum algorithm, which implies that the adiabatic computation model and the convention ..."
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Cited by 80 (12 self)
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Adiabatic quantum computation has recently attracted attention in the physics and computer science communities, but its computational power was unknown. We describe an efficient adiabatic simulation of any given quantum algorithm, which implies that the adiabatic computation model and the conventional quantum computation model are polynomially equivalent. Our result can be extended to the physically realistic setting of particles arranged on a twodimensional grid with nearest neighbor interactions. The equivalence between the models provides a new vantage point from which to tackle the central issues in quantum computation, namely designing new quantum algorithms and constructing fault tolerant quantum computers. In particular, by translating the main open questions in the area of quantum algorithms to the language of spectral gaps of sparse matrices, the result makes these questions accessible to a wider scientific audience, acquainted with mathematical physics, expander theory and rapidly mixing Markov chains. 1
The Complexity of the Local Hamiltonian Problem
 In Proc. of 24th FSTTCS
, 2004
"... The kLOCAL HAMILTONIAN problem is a natural complete problem for the complexity class QMA, the quantum analog of NP. It is similar in spirit to MAXkSAT, which is NPcomplete for k ≥ 2. It was known that the problem is QMAcomplete for any k ≥ 3. On the other hand 1LOCAL HAMILTONIAN is in P, and h ..."
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Cited by 80 (6 self)
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The kLOCAL HAMILTONIAN problem is a natural complete problem for the complexity class QMA, the quantum analog of NP. It is similar in spirit to MAXkSAT, which is NPcomplete for k ≥ 2. It was known that the problem is QMAcomplete for any k ≥ 3. On the other hand 1LOCAL HAMILTONIAN is in P, and hence not believed to be QMAcomplete. The complexity of the 2LOCAL HAMILTONIAN problem has long been outstanding. Here we settle the question and prove that it is QMAcomplete. One component in our proof is a powerful technique for analyzing the sum of two Hamiltonians; this technique is based on perturbation theory and we believe that it might prove useful elsewhere. Our proof also implies that adiabatic computation with twolocal interactions on qubits is equivalent to standard quantum computation. 1
Growth and generation in SL2(Z/pZ)
 ANN. OF MATH
, 2005
"... We show that every subset of SL2(Z/pZ) grows rapidly under multiplication. It follows readily that, for every set of generators A of SL2(Z/pZ), every element of SL2(Z/pZ) can be expressed as a product of at most O((log p) c) elements of A ∪ A −1, where c and the implied constant are absolute. ..."
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Cited by 79 (6 self)
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We show that every subset of SL2(Z/pZ) grows rapidly under multiplication. It follows readily that, for every set of generators A of SL2(Z/pZ), every element of SL2(Z/pZ) can be expressed as a product of at most O((log p) c) elements of A ∪ A −1, where c and the implied constant are absolute.
Parallel repetition: Simplifications and the nosignaling case
 In STOC’07
, 2007
"... Consider a game where a referee chooses (x,y) according to a publicly known distribution PXY, sends x to Alice, and y to Bob. Without communicating with each other, Alice responds with a value a and Bob responds with a value b. Alice and Bob jointly win if a publicly known predicate Q(x,y, a, b) hol ..."
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Cited by 76 (0 self)
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Consider a game where a referee chooses (x,y) according to a publicly known distribution PXY, sends x to Alice, and y to Bob. Without communicating with each other, Alice responds with a value a and Bob responds with a value b. Alice and Bob jointly win if a publicly known predicate Q(x,y, a, b) holds. Let such a game be given and assume that the maximum probability that Alice and Bob can win is v < 1. Raz (SIAM J. Comput. 27, 1998) shows that if the game is repeated n times in parallel, then the probability that Alice and Bob win all games simultaneously is at most ¯v log(s), where s is the maximal number of possible responses from Alice and Bob in the initial game, and ¯v < 1 is a constant depending only on v. In this work, we simplify Raz’s proof in various ways and thus shorten it significantly. Further we study the case where Alice and Bob are not restricted to local computations and can use any strategy which does not imply communication among them. 1