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16
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 40 (5 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.
Any ANDOR formula of size n can be evaluated in time N 1/2+o(1) on a quantum computer
 In Proceedings of 48th IEEE FOCS
, 2007
"... For any ANDOR formula of size N, there exists a boundederror N 1 2 +o(1)time quantum algorithm, based on a discretetime quantum walk, that evaluates this formula on a blackbox input. Balanced, or “approximately balanced,” formulas can be evaluated in O ( √ N) queries, which is optimal. It foll ..."
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Cited by 23 (12 self)
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For any ANDOR formula of size N, there exists a boundederror N 1 2 +o(1)time quantum algorithm, based on a discretetime quantum walk, that evaluates this formula on a blackbox input. Balanced, or “approximately balanced,” formulas can be evaluated in O ( √ N) queries, which is optimal. It follows that the (2 − o(1))th power of the quantum query complexity is a lower bound on the formula size, almost solving in the positive an open problem posed by Laplante, Lee and Szegedy. 1
Negative weights make adversaries stronger. To appear in STOC’07
 Algorithmica
, 2002
"... The quantum adversary method is one of the most successful techniques for proving lower bounds on quantum query complexity. It gives optimal lower bounds for many problems, has application to classical complexity in formula size lower bounds, and is versatile with equivalent formulations in terms of ..."
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Cited by 19 (3 self)
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The quantum adversary method is one of the most successful techniques for proving lower bounds on quantum query complexity. It gives optimal lower bounds for many problems, has application to classical complexity in formula size lower bounds, and is versatile with equivalent formulations in terms of weight schemes, eigenvalues, and Kolmogorov complexity. All these formulations are informationtheoretic and rely on the principle that if an algorithm successfully computes a function then, in particular, it is able to distinguish between inputs which map to different values. We present a stronger version of the adversary method which goes beyond this principle to make explicit use of the existence of a measurement in a successful algorithm which gives the correct answer, with high probability. We show that this new method, which we call ADV ±, has all the advantages of the old: it is a lower bound on boundederror quantum query complexity, its square is a lower bound on formula size, and it behaves well with respect to function composition. Moreover ADV ± is always at least as large as the adversary method ADV, and we show an example of a monotone function for which ADV ± (f) = Ω(ADV(f) 1.098). We also give examples showing that ADV ± does not face limitations of ADV such as the certificate complexity barrier and the property testing barrier. 1
Spalek, Spanprogrambased quantum algorithm for evaluating formulas
 Proc. 40th ACM Symposium on Theory of Computing
, 2008
"... We present a timeefficient and queryoptimal quantum algorithm for evaluating adversaryboundbalanced formulas on an extended gate set. The allowed gates include arbitrary two and threebit gates, as well as bounded fanin AND, OR, PARITY and EQUAL gates. The technique behind the formula evaluatio ..."
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Cited by 10 (3 self)
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We present a timeefficient and queryoptimal quantum algorithm for evaluating adversaryboundbalanced formulas on an extended gate set. The allowed gates include arbitrary two and threebit gates, as well as bounded fanin AND, OR, PARITY and EQUAL gates. The technique behind the formula evaluation algorithm is a new framework for quantum algorithms based on span programs. For example, the classical complexity of evaluating the balanced ternary majority formula is unknown, and the natural generalization of the standard balanced ANDOR formula evaluation algorithm is known to be suboptimal. In contrast, a generalization of the optimal quantum {AND, OR, NOT} formula evaluation algorithm is optimal for evaluating the balanced ternary majority formula. 1
A note on quantum algorithms and the minimal degree of ɛerror polynomials for symmetric functions. Available at arXiv:0802.1816
, 2008
"... The degrees of polynomials representing or approximating Boolean functions are a prominent tool in various branches of complexity theory. Sherstov [She08a] recently characterized the minimal degree degε(f) among all polynomials (over R) that approximate a symmetric function f: {0, 1} n → {0, 1} up t ..."
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Cited by 9 (1 self)
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The degrees of polynomials representing or approximating Boolean functions are a prominent tool in various branches of complexity theory. Sherstov [She08a] recently characterized the minimal degree degε(f) among all polynomials (over R) that approximate a symmetric function f: {0, 1} n → {0, 1} up to worstcase error ε: degε(f) = � � Θ deg1/3(f) + � � n log(1/ε). In this note we show how a tighter version (without the logfactors hidden in the � Θnotation), can be derived quite easily using the close connection between polynomials and quantum algorithms. 1
Spalek. Tight adversary bounds for composite functions
"... The quantum adversary method is a very versatile method for proving lower bounds on quantum algorithms. It has many equivalent formulations, yields tight bounds for many computational problems, and has natural connections to classical lower bounds. One of its formulations is in terms of the spectral ..."
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Cited by 6 (3 self)
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The quantum adversary method is a very versatile method for proving lower bounds on quantum algorithms. It has many equivalent formulations, yields tight bounds for many computational problems, and has natural connections to classical lower bounds. One of its formulations is in terms of the spectral norm of some matrices. We define a weighted version of this spectral method and show that it possesses useful composition properties. The results generalize and unify previously known composition properties of adversary methods. 1
Product rules in semidefinite programming
 In Proc. of 16th International Symposium on Fundamentals of Computation Theory, LNCS 4638
, 2007
"... Abstract. In recent years we witness the proliferation of semidefinite programming bounds in combinatorial optimization [1,5,8], quantum computing [9,2,3,6,4] and even in complexity theory [7]. Examples to such bounds include the semidefinite relaxation for the maximal cut problem [5], and the quant ..."
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Cited by 6 (3 self)
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Abstract. In recent years we witness the proliferation of semidefinite programming bounds in combinatorial optimization [1,5,8], quantum computing [9,2,3,6,4] and even in complexity theory [7]. Examples to such bounds include the semidefinite relaxation for the maximal cut problem [5], and the quantum value of multiprover interactive games [3,4]. The first semidefinite programming bound, which gained fame, arose in the late seventies and was due to László Lovász [11], who used his theta number to compute the Shannon capacity of the five cycle graph. As in Lovász’s upper bound proof for the Shannon capacity and in other situations the key observation is often the fact that the new parameter in question is multiplicative with respect to the product of the problem instances. In a recent result R. Cleve, W. Slofstra, F. Unger and S. Upadhyay show that the quantum value of XOR games multiply under parallel composition [4]. This result together with [3] strengthens the parallel repetition theorem of Ran Raz [12] for XOR games. Our goal is to classify those semidefinite programming instances for which the optimum is multiplicative under a naturally defined product operation. The product operation we define generalizes the ones used in [11] and [4]. We find conditions under which the product rule always holds and give examples for cases when the product rule does not hold. 1
A new rank technique for formula size lower bounds
 Symposium on Theoretical Aspects of Computer Science, STACS 2007, volume 4393 of Lecture Notes in Computer Science
, 2007
"... We introduce a new technique for proving formula size lower bounds based on matrix rank. A simple form of this technique gives bounds at least as large as those given by the method of Khrapchenko, originally used to prove an��lower bound on the parity function. Applying our method to the parity func ..."
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Cited by 5 (0 self)
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We introduce a new technique for proving formula size lower bounds based on matrix rank. A simple form of this technique gives bounds at least as large as those given by the method of Khrapchenko, originally used to prove an��lower bound on the parity function. Applying our method to the parity function, we are able to give an exact expression for the formula size of parity: if������, where������, then the formula size of parity on �bits is exactly��������������� � ��. Such a bound cannot be proven by any of the lower bound techniques of Khrapchenko, Nečiporuk, Koutsoupias, or the quantum adversary method, which are limited by��. 1