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Polynomial degree vs. quantum query complexity
 Proceedings of FOCS’03
"... The degree of a polynomial representing (or approximating) a function f is a lower bound for the quantum query complexity of f. This observation has been a source of many lower bounds on quantum algorithms. It has been an open problem whether this lower bound is tight. We exhibit a function with pol ..."
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Cited by 81 (14 self)
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The degree of a polynomial representing (or approximating) a function f is a lower bound for the quantum query complexity of f. This observation has been a source of many lower bounds on quantum algorithms. It has been an open problem whether this lower bound is tight. We exhibit a function with polynomial degree M and quantum query complexity Ω(M 1.321...). This is the first superlinear separation between polynomial degree and quantum query complexity. The lower bound is shown by a new, more general version of quantum adversary method. 1
Search via quantum walk
 LOGIC PROGRAMMING, PROC. OF THE 1994 INT. SYMP
, 2007
"... We propose a new method for designing quantum search algorithms for finding a “marked ” element in the state space of a classical Markov chain. The algorithm is based on a quantum walk à la Szegedy [24] that is defined in terms of the Markov chain. The main new idea is to apply quantum phase estimat ..."
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Cited by 56 (8 self)
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We propose a new method for designing quantum search algorithms for finding a “marked ” element in the state space of a classical Markov chain. The algorithm is based on a quantum walk à la Szegedy [24] that is defined in terms of the Markov chain. The main new idea is to apply quantum phase estimation to the quantum walk in order to implement an approximate reflection operator. This operator is then used in an amplitude amplification scheme. As a result we considerably expand the scope of the previous approaches of Ambainis [6] and Szegedy [24]. Our algorithm combines the benefits of these approaches in terms of being able to find marked elements, incurring the smaller cost of the two, and being applicable to a larger class of Markov chain. In addition, it is conceptually simple, avoids several technical difficulties in the previous analyses, and leads to improvements in various aspects of several algorithms based on quantum walk.
All quantum adversary methods are equivalent
 THEORY OF COMPUTING
, 2006
"... The quantum adversary method is one of the most versatile lowerbound methods for quantum algorithms. We show that all known variants of this method are equivalent: spectral adversary (Barnum, Saks, and Szegedy, 2003), weighted adversary (Ambainis, 2003), strong weighted adversary (Zhang, 2005), an ..."
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Cited by 50 (4 self)
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The quantum adversary method is one of the most versatile lowerbound methods for quantum algorithms. We show that all known variants of this method are equivalent: spectral adversary (Barnum, Saks, and Szegedy, 2003), weighted adversary (Ambainis, 2003), strong weighted adversary (Zhang, 2005), and the Kolmogorov complexity adversary (Laplante and Magniez, 2004). We also present a few new equivalent formulations of the method. This shows that there is essentially one quantum adversary method. From our approach, all known limitations of these versions of the quantum adversary method easily follow.
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
Quantum walk based search algorithms
 In Proceedings of the 5th Conference on Theory and Applications of Models of Computation
, 2008
"... Abstract. In this survey paper we give an intuitive treatment of the discrete time quantization of classical Markov chains. Grover search and the quantum walk based search algorithms of Ambainis, Szegedy and Magniez et al. will be stated as quantum analogues of classical search procedures. We presen ..."
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Cited by 37 (1 self)
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Abstract. In this survey paper we give an intuitive treatment of the discrete time quantization of classical Markov chains. Grover search and the quantum walk based search algorithms of Ambainis, Szegedy and Magniez et al. will be stated as quantum analogues of classical search procedures. We present a rather detailed description of a somewhat simplified version of the MNRS algorithm. Finally, in the query complexity model, we show how quantum walks can be applied to the following
Any ANDOR formula of size N can be evaluated in time N1/2+o(1) on a quantum computer
 in Proceedings of the 48th IEEE FOCS
"... Abstract. Consider the problem of evaluating an ANDOR formula on an Nbit blackbox input. We present a boundederror quantum algorithm that solves this problem in time N1/2+o(1). In particular, approximately balanced formulas can be evaluated in O( N) queries, which is optimal. The idea of the alg ..."
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Cited by 35 (10 self)
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Abstract. Consider the problem of evaluating an ANDOR formula on an Nbit blackbox input. We present a boundederror quantum algorithm that solves this problem in time N1/2+o(1). In particular, approximately balanced formulas can be evaluated in O( N) queries, which is optimal. The idea of the algorithm is to apply phase estimation to a discretetime quantum walk on a weighted tree whose spectrum encodes the value of the formula.
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.
Quantum information processing in continuous time
, 2004
"... Quantum mechanical computers can solve certain problems asymptotically faster than any classical computing device. Several fast quantum algorithms are known, but the nature of quantum speedup is not well understood, and inventing new quantum algorithms seems to be difficult. In this thesis, we explo ..."
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Cited by 20 (6 self)
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Quantum mechanical computers can solve certain problems asymptotically faster than any classical computing device. Several fast quantum algorithms are known, but the nature of quantum speedup is not well understood, and inventing new quantum algorithms seems to be difficult. In this thesis, we explore two approaches to designing quantum algorithms based on continuoustime Hamiltonian dynamics. In quantum computation by adiabatic evolution, the computer is prepared in the known ground state of a simple Hamiltonian, which is slowly modified so that its ground state encodes the solution to a problem. We argue that this approach should be inherently robust against lowtemperature thermal noise and certain control errors, and we support this claim using simulations. We then show that any adiabatic algorithm can be implemented in a different way, using only a sequence of measurements of the Hamiltonian. We illustrate how this approach can achieve quadratic speedup for the unstructured search problem. We also demonstrate two examples of quantum speedup by quantum walk, a quantum mechanical analog of random walk. First, we consider the problem of searching a region