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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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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.
Limits on the Power of Quantum Statistical ZeroKnowledge
, 2003
"... In this paper we propose a definition for honest verifier quantum statistical zeroknowledge interactive proof systems and study the resulting complexity class, which we denote QSZK ..."
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Cited by 39 (4 self)
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In this paper we propose a definition for honest verifier quantum statistical zeroknowledge interactive proof systems and study the resulting complexity class, which we denote QSZK
Lower Bounds for Local Search by Quantum Arguments
"... The problem of finding a local minimum of a blackbox function is central for understanding local search as well as quantum adiabatic algorithms. For functions on the Boolean hypercube {0,1} n (, we show a lower bound of Ω 2 n/4) /n on the number of queries needed by a quantum computer to solve this ..."
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Cited by 32 (2 self)
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The problem of finding a local minimum of a blackbox function is central for understanding local search as well as quantum adiabatic algorithms. For functions on the Boolean hypercube {0,1} n (, we show a lower bound of Ω 2 n/4) /n on the number of queries needed by a quantum computer to solve this problem. More surprisingly, our approach, based on Ambainis’s quantum ( adversary method, also yields a lower bound of Ω 2 n/2 /n 2 on the problem’s classical randomized query complexity. This improves and simplifies a 1983 result of Aldous. Finally, in both the randomized and quantum cases, we give the first nontrivial lower bounds for finding local minima on grids of constant dimension d ≥ 3. 1.
Quantum complexities of ordered searching, sorting, and element distinctness
, 2001
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Quantum query complexity and semidefinite programming
 In Proceedings of the 18th IEEE Annual Conference on Computational Complexity
, 2003
"... We reformulate quantum query complexity in terms of inequalities and equations for a set of positive semidefinite matrices. Using the new formulation we: 1. show that the workspace of a quantum computer can be limited to at most n + k qubits (where n and k are the number of input and output bits res ..."
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We reformulate quantum query complexity in terms of inequalities and equations for a set of positive semidefinite matrices. Using the new formulation we: 1. show that the workspace of a quantum computer can be limited to at most n + k qubits (where n and k are the number of input and output bits respectively) without reducing the computational power of the model. 2. give an algorithm that on input the truth table of a partial boolean function and an integer t runs in time polynomial in the size of the truth table and estimates, to any desired accuracy, the minimum probability of error that can be attained by a quantum query algorithm attempts to evaluate f in t queries. 3. use semidefinite programming duality to formulate a dual SDP ˆ P (f,t,ɛ) that is feasible if and only if f can not be evaluated within error ɛ by a tstep quantum query algorithm Using this SDP we derive a general lower bound for quantum query complexity that encompasses a lower bound method of Ambainis and its generalizations. 4. Give an interpretation of a generalized form of branching in quantum computation.
Quantum Search Algorithms
, 2005
"... We review some of quantum algorithms for search problems: Grover’s search algorithm, its generalization to amplitude amplification, the applications of amplitude amplification to various problems and the recent quantum algorithms based on quantum walks. 1 ..."
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We review some of quantum algorithms for search problems: Grover’s search algorithm, its generalization to amplitude amplification, the applications of amplitude amplification to various problems and the recent quantum algorithms based on quantum walks. 1
On the power of Ambainis’s lower bounds
 Theoretical Computer Science, 339(23):241– 256, 2005. Earlier version in ICALP’04. 569 Copyright © by SIAM. Unauthorized
"... The polynomial method and Ambainis’s lower bound method are two main quantum lower bound techniques. Recently Ambainis showed that the polynomial method is not tight. The present paper aims at studying the limitation of Ambainis’s lower bounds. We first give a generalization of the three known Ambai ..."
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The polynomial method and Ambainis’s lower bound method are two main quantum lower bound techniques. Recently Ambainis showed that the polynomial method is not tight. The present paper aims at studying the limitation of Ambainis’s lower bounds. We first give a generalization of the three known Ambainis’s lower bound theorems. Then it is shown that all these four Ambainis’s lower bounds have an upper bound, which is in terms of certificate complexity. This implies that for some problems such as TRIANGLE, kCLIQUE, and BIPARTITE/GRAPH MATCHING whose quantum query complexities are still open, the best known lower bounds cannot be further improved by using Ambainis’s techniques. Another consequence is that all the Ambainis’s lower bounds are not tight. Finally, we show that for total functions, this upper bound for Ambainis’s lower bounds can be further improved. This also implies limitation of Ambainis’s method on some specific problems such as ANDOR TREE, whose precise quantum complexity is still unknown. 1
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