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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 94 (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.
Lower bounds for randomized and quantum query complexity using Kolmogorov arguments
 in Proc. of the 19th IEEE Conference on Computational Complexity
, 2004
"... Abstract. We prove a very general lower bound technique for quantum and randomized query complexity that is easy to prove as well as to apply. To achieve this, we introduce the use of Kolmogorov complexity to query complexity. Our technique generalizes the weighted and unweighted methods of Ambainis ..."
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Cited by 50 (3 self)
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Abstract. We prove a very general lower bound technique for quantum and randomized query complexity that is easy to prove as well as to apply. To achieve this, we introduce the use of Kolmogorov complexity to query complexity. Our technique generalizes the weighted and unweighted methods of Ambainis and the spectral method of Barnum, Saks, and Szegedy. As an immediate consequence of our main theorem, it can be shown that adversary methods can only prove lower bounds for Boolean functions f in O(min ( √ nC0(f), √ nC1(f))), where C0,C1 is the certificate complexity and n is the size of the input.
2006, Quantum verification of matrix products
 Proceedings of the 17th ACMSIAM Symposium on Discrete Algorithms
"... We present a quantum algorithm that verifies a product of two n×n matrices over any integral domain with bounded error in worstcase time O(n 5/3) and expected time O(n 5/3 / min(w, √ n) 1/3), where w is the number of wrong entries. This improves the previous best algorithm [ABH + 02] that runs in ..."
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Cited by 48 (0 self)
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We present a quantum algorithm that verifies a product of two n×n matrices over any integral domain with bounded error in worstcase time O(n 5/3) and expected time O(n 5/3 / min(w, √ n) 1/3), where w is the number of wrong entries. This improves the previous best algorithm [ABH + 02] that runs in time O(n 7/4). We also present a quantum matrix multiplication algorithm that is efficient when the result has few nonzero entries. 1
Spanprogrambased quantum algorithm for evaluating formulas
, 2008
"... We give a quantum algorithm for evaluating formulas over an extended gate set, including all two and threebit binary gates (e.g., NAND, 3majority). The algorithm is optimal on readonce formulas for which each gate’s inputs are balanced in a certain sense. The main new tool is a correspondence be ..."
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Cited by 34 (6 self)
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We give a quantum algorithm for evaluating formulas over an extended gate set, including all two and threebit binary gates (e.g., NAND, 3majority). The algorithm is optimal on readonce formulas for which each gate’s inputs are balanced in a certain sense. The main new tool is a correspondence between a classical linearalgebraic model of computation, “span programs,” and weighted bipartite graphs. A span program’s evaluation corresponds to an eigenvaluezero eigenvector of the associated graph. A quantum computer can therefore evaluate the span program by applying spectral estimation to the graph. For example, the classical complexity of evaluating the balanced ternary majority formula is unknown, and the natural generalization of randomized alphabeta pruning is known to be suboptimal. In contrast, our algorithm generalizes the optimal quantum ANDOR formula evaluation algorithm and is optimal for evaluating the balanced ternary majority formula.
Span programs and quantum query complexity: The general adversary bound is nearly tight for every boolean function
"... The general adversary bound is a semidefinite program (SDP) that lowerbounds the quantum query complexity of a function. We turn this lower bound into an upper bound, by giving a quantum walk algorithm based on the dual SDP that has query complexity at most the general adversary bound, up to a log ..."
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Cited by 25 (5 self)
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The general adversary bound is a semidefinite program (SDP) that lowerbounds the quantum query complexity of a function. We turn this lower bound into an upper bound, by giving a quantum walk algorithm based on the dual SDP that has query complexity at most the general adversary bound, up to a logarithmic factor. In more detail, the proof has two steps, each based on “span programs,” a certain linearalgebraic model of computation. First, we give an SDP that outputs for any boolean function a span program computing it that has optimal “witness size. ” The optimal witness size is shown to coincide with the general adversary lower bound. Second, we give a quantum algorithm for evaluating span programs with only a logarithmic query overhead on the witness size. The first result is motivated by a quantum algorithm for evaluating composed span programs. The algorithm
A new quantum lower bound method, with an application to strong direct product theorem for quantum search
, 2005
"... We give a new version of the adversary method for proving lower bounds on quantum query algorithms. The new method is based on analyzing the eigenspace structure of the problem at hand. We use it to prove a new and optimal strong direct product theorem for 2sided error quantum algorithms computing ..."
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Cited by 24 (3 self)
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We give a new version of the adversary method for proving lower bounds on quantum query algorithms. The new method is based on analyzing the eigenspace structure of the problem at hand. We use it to prove a new and optimal strong direct product theorem for 2sided error quantum algorithms computing k independent instances of a symmetric Boolean function: if the algorithm uses significantly less than k times the number of queries needed for one instance of the function, then its success probability is exponentially small in k. We also use the polynomial method to prove a direct product theorem for 1sided error algorithms for k threshold functions with a stronger bound on the success probability. Finally, we present a quantum algorithm for evaluating solutions to systems of linear inequalities, and use our direct product theorems to show that the timespace tradeoff of this algorithm is close to optimal. Categories and Subject Descriptors F.1.2 [Computation by Abstract Devices]: Modes of Computation; F.1.3 [Computation by Abstract Devices]: Complexity Measures and Classes—Relations among complexity
Lower bounds on quantum query complexity
 EATCS BULLETIN
, 2005
"... Shor’s and Grover’s famous quantum algorithms for factoring and searching show that quantum computers can solve certain computational problems significantly faster than any classical computer. We discuss here what quantum computers cannot do, and specifically how to prove limits on their computation ..."
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Cited by 23 (2 self)
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Shor’s and Grover’s famous quantum algorithms for factoring and searching show that quantum computers can solve certain computational problems significantly faster than any classical computer. We discuss here what quantum computers cannot do, and specifically how to prove limits on their computational power. We cover the main known techniques for proving lower bounds, and exemplify and compare the methods.
On the power of Ambainis lower bounds
, 2005
"... The polynomial method and the Ambainis lower bound (or Alb, for short) method are two main quantum lower bound techniques. While recently Ambainis showed that the polynomial method is not tight, the present paper aims at studying the power and limitation of Alb’s. We first use known Alb’s to derive( ..."
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Cited by 17 (4 self)
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The polynomial method and the Ambainis lower bound (or Alb, for short) method are two main quantum lower bound techniques. While recently Ambainis showed that the polynomial method is not tight, the present paper aims at studying the power and limitation of Alb’s. We first use known Alb’s to derive(n1.5) lower bounds for BIPARTITENESS, BIPARTITENESS MATCHING and GRAPH MATCHING, in which the lower bound for BIPARTITENESS improves the previous(n) one.We then show that all the three known Ambainis lower bounds have a limitation N min{C0(f), C1(f)}, where C0(f) and C1(f) are the 0 and 1certificate complexities, respectively. This implies that for many problems such asTRIANGLE, kCLIQUE, BIPARTITENESS andBIPARTITE/GRAPHMATCHING which draw wide interest and whose quantum query complexities are still open, the best known lower bounds cannot be further improved by using Ambainis techniques. Another consequence is that all theAmbainis lower bounds are not tight. For total functions, this upper bound for Alb’s can be further improved to min{√C0(f)C1(f), N · CI(f)}, where CI(f) is the size of max intersection of a 0 and a 1certificate set. Again this implies that Alb’s cannot improve the best known lower bound for some specific problems such as ANDOR TREE, whose precise quantum query complexity is still open. Finally, we generalize the three known Alb’s and give a new Alb style lower bound method, which may be easier to use for some problems.