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Quantum lower bounds by quantum arguments
 In Proceedings of the ACM Symposium on Theory of Computing
, 2000
"... We propose a new method for proving lower bounds on quantum query algorithms. Instead of a classical adversary that runs the algorithm with one input and then modifies the input, we use a quantum adversary that runs the algorithm with a superposition of inputs. If the algorithm works correctly, its ..."
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Cited by 146 (15 self)
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We propose a new method for proving lower bounds on quantum query algorithms. Instead of a classical adversary that runs the algorithm with one input and then modifies the input, we use a quantum adversary that runs the algorithm with a superposition of inputs. If the algorithm works correctly, its state becomes entangled with the superposition over inputs. We bound the number of queries needed to achieve a sufficient entanglement and this implies a lower bound on the number of queries for the computation. Using this method, we prove two new Ω ( √ N) lower bounds on computing AND of ORs and inverting a permutation and also provide more uniform proofs for several known lower bounds which have been previously proven via variety of different techniques. 1
Exponential lower bound for 2query locally decodable codes via a quantum argument
 Journal of Computer and System Sciences
, 2003
"... Abstract A locally decodable code encodes nbit strings x in mbit codewords C(x) in such a way that one can recover any bit xi from a corrupted codeword by querying only a few bits of that word. We use a quantum argument to prove that LDCs with 2 classical queries require exponential length: m = 2 ..."
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Cited by 123 (18 self)
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Abstract A locally decodable code encodes nbit strings x in mbit codewords C(x) in such a way that one can recover any bit xi from a corrupted codeword by querying only a few bits of that word. We use a quantum argument to prove that LDCs with 2 classical queries require exponential length: m = 2 \Omega (n). Previously this was known only for linear codes (Goldreich et al. 02). The
Complexity Measures and Decision Tree Complexity: A Survey
 Theoretical Computer Science
, 2000
"... We discuss several complexity measures for Boolean functions: certificate complexity, sensitivity, block sensitivity, and the degree of a representing or approximating polynomial. We survey the relations and biggest gaps known between these measures, and show how they give bounds for the decision tr ..."
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Cited by 122 (15 self)
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We discuss several complexity measures for Boolean functions: certificate complexity, sensitivity, block sensitivity, and the degree of a representing or approximating polynomial. We survey the relations and biggest gaps known between these measures, and show how they give bounds for the decision tree complexity of Boolean functions on deterministic, randomized, and quantum computers. 1 Introduction Computational Complexity is the subfield of Theoretical Computer Science that aims to understand "how much" computation is necessary and sufficient to perform certain computational tasks. For example, given a computational problem it tries to establish tight upper and lower bounds on the length of the computation (or on other resources, like space). Unfortunately, for many, practically relevant, computational problems no tight bounds are known. An illustrative example is the well known P versus NP problem: for all NPcomplete problems the current upper and lower bounds lie exponentially ...
The quantum query complexity of approximating the median and related statistics
 STOC'99
, 1999
"... Let X = (z,, , z,,) be a sequence of n numbers. For 6> 0, we say that 5; is an eapproximate median if the number of elements strictly less than zi and the number of elements strictly greater than zi are each less than (1 + 6):. We consider the quantum query complexity of computing an capproxima ..."
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Cited by 65 (1 self)
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Let X = (z,, , z,,) be a sequence of n numbers. For 6> 0, we say that 5; is an eapproximate median if the number of elements strictly less than zi and the number of elements strictly greater than zi are each less than (1 + 6):. We consider the quantum query complexity of computing an capproximate median, given the sequence X as an oracle. We prove a lower bound of n(min{t,n}) queries for any quantum algorithm that computes an rapproximate median with any constant probability greater than l/2. We also show how an capproximate median may be computed with 0 ( $ log(t) log log ( $)) oracle queries, which rep resents an improvement over an earlier algorithm due to Grover [ll, 121. Thus, the lower bound we obtain is essentially optimal. The upper and the lower bound both hold in the comparison tree model as well. Our lower bound result is an application of the polynomial paradigm recently introduced to quantum complexity theory by Be & et ol. [l]. The main ingredient in the proof is a polynomial degree lower bound far real multilinear polynomials that “approximate” symmetric partial boolean functions. The degree bound extends a result of Patti [15] and also immediately yields lower bounds for the problems of approximating the kthsmallest element, approximating the mean of a sequence of numbers, and approximately counting the number of ones of a boolean function. All bounds obtained come within a polylogarithmic factor of the optimal (as we show by presenting algorithms where no such optimal or near optimal algorithms were known), thus demonstrating the power of the polynomial method.
Quantum Communication and Complexity
 Theoretical Computer Science
, 2000
"... In the setting of communication complexity, two distributed parties want to compute a function depending on both their inputs, using as little communication as possible. The required communication can sometimes be significantly lowered if we allow the parties the use of quantum communication. We sur ..."
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Cited by 32 (13 self)
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In the setting of communication complexity, two distributed parties want to compute a function depending on both their inputs, using as little communication as possible. The required communication can sometimes be significantly lowered if we allow the parties the use of quantum communication. We survey the main results of the young area of quantum communication complexity: its relation to teleportation and dense coding, the main examples of fast quantum communication protocols, lower bounds, and some applications. 1 Introduction The area of communication complexity deals with the following type of problem. There are two separated parties, called Alice and Bob. Alice receives some input x 2 X, Bob receives some y 2 Y , and together they want to compute some function f(x; y). As the value f(x; y) will generally depend on both x and y, neither Alice nor Bob will have sufficient information to do the computation by themselves, so they will have to communicate in order to achieve their go...
Improved quantum communication complexity bounds for disjointness and equality
 In Proc. Intl. Symp. on Theoretical Aspects of Computer Science (STACS
, 2002
"... Abstract. We prove new bounds on the quantum communication complexity of the disjointness and equality problems. For the case of exact and nondeterministic protocols we show that these complexities are all equal to n+1, the previous best lower bound being n/2. We show this by improving a general bo ..."
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Cited by 27 (5 self)
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Abstract. We prove new bounds on the quantum communication complexity of the disjointness and equality problems. For the case of exact and nondeterministic protocols we show that these complexities are all equal to n+1, the previous best lower bound being n/2. We show this by improving a general bound for nondeterministic protocols of de Wolf. We also give an O ( √ n·c log ∗ n)qubit boundederror protocol for disjointness, modifying and improving the earlier O ( √ n log n) protocol of Buhrman, Cleve, and Wigderson, and prove an Ω ( √ n) lower bound for a class of protocols that includes the BCWprotocol as well as our new protocol. 1
Quantum oracle interrogation: getting all information for almost half the price
 In Proc. of the 39th IEEE FOCS
, 1998
"... Consider a quantum computer in combination with a binary oracle of domain size N. It is shown how N/2 + √ N calls to the oracle are sufficient to guess the whole content of the oracle (being an N bit string) with probability greater than 95%. This contrasts the power of classical computers which wou ..."
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Cited by 26 (0 self)
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Consider a quantum computer in combination with a binary oracle of domain size N. It is shown how N/2 + √ N calls to the oracle are sufficient to guess the whole content of the oracle (being an N bit string) with probability greater than 95%. This contrasts the power of classical computers which would require N calls to achieve the same task. From this result it follows that any function with the N bits of the oracle as input can be calculated using N/2 + √ N queries if we allow a small probability of error. It is also shown that this error probability can be made arbitrary small by using N/2 + O ( √ N) oracle queries. In the second part of the article ‘approximate interrogation ’ is considered. This is when only a certain fraction of the N oracle bits are requested. Also for this scenario does the quantum algorithm outperform the classical protocols. An example is given where a quantum procedure with N/10 queries returns a string of which 80 % of the bits are correct. Any classical protocol would need 6N/10 queries to establish such a correctness ratio. ∗ c○1998 IEEE. Published in the Proceedings of FOCS’98, 811 November 1998 in Palo Alto, CA. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works, must be obtained from the ieee. Contact: Manager, Copyrights and
An introduction to quantum complexity theory
 Collected Papers on Quantum Computation and Quantum Information Theory
, 2000
"... ..."
Lower bounds of quantum blackbox complexity and degree of approximating polynomials by influence of Boolean variables
 Inform. Process. Lett
"... Boolean variables ..."