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47
ErrorBounded Probabilistic Computations Between MA and AM
 IN PROCEEDINGS 28TH MATHEMATICAL FOUNDATIONS OF COMPUTER SCIENCE
, 2002
"... We introduce the probabilistic class SBP which is defined in a BPPlike manner. This class emerges from BPP by keeping the promise of a probability gap but decreasing the probability limit from 1/2 to exponentially small values. We show ..."
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We introduce the probabilistic class SBP which is defined in a BPPlike manner. This class emerges from BPP by keeping the promise of a probability gap but decreasing the probability limit from 1/2 to exponentially small values. We show
Quantum versus Classical Learnability
 IN: SIXTEENTH CONFERENCE ON COMPUTATIONAL COMPLEXITY (CCC
, 2000
"... This paper studies fundamental questions in computational learning theory from a quantum computation perspective. We consider quantum versions of two wellstudied classical learning models: Angluin's model of exact learning from membership queries and Valiant's Probably Approximately Corre ..."
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This paper studies fundamental questions in computational learning theory from a quantum computation perspective. We consider quantum versions of two wellstudied classical learning models: Angluin's model of exact learning from membership queries and Valiant's Probably Approximately Correct (PAC) model of learning from random examples. We give positive and negative results for quantum versus classical learnability. For each of the two learning models described above, we show that any concept class is informationtheoretically learnable from polynomially many quantum examples if and only if it is informationtheoretically learnable from polynomially many classical examples. In contrast to this informationtheoretic equivalence betwen quantum and classical learnability, though, we observe that a separation does exist between ecient quantum and classical learnability. For both the model of exact learning from membership queries and the PAC model, we show that under a widely held computational hardness assumption for classical computation (the intractability of factoring), there is a concept class which is polynomialtime learnable in the quantum version but not in the classical version of the model.
Equivalences and separations between quantum and classical learnability
 SIAM J. Comput
, 2004
"... Abstract. We consider quantum versions of two wellstudied models of learning Boolean functions: Angluin’s model of exact learning from membership queries and Valiant’s Probably Approximately Correct (PAC) model of learning from random examples. For each of these two learning models we establish a p ..."
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Abstract. We consider quantum versions of two wellstudied models of learning Boolean functions: Angluin’s model of exact learning from membership queries and Valiant’s Probably Approximately Correct (PAC) model of learning from random examples. For each of these two learning models we establish a polynomial relationship between the number of quantum versus classical queries required for learning. These results contrast known results which show that testing blackbox functions for various properties, as opposed to learning, can require exponentially more classical queries than quantum queries. We also show that under a widely held computational hardness assumption (the intractability of factoring Blum integers) there is a class of Boolean functions which is polynomialtime learnable in the quantum version but not the classical version of each learning model. For the model of exact learning from membership queries, we establish a stronger separation by showing that if any oneway function exists, then there is a class of functions which is polynomialtime learnable in the quantum setting but not in the classical setting. Thus, while quantum and classical learning are equally powerful from an information theory perspective, the models are different when viewed from a computational complexity perspective.
Onesided Versus Twosided Randomness
 In Proceedings of the 16th Symposium on Theoretical Aspects of Computer Science. Lecture Notes in Computer Science
, 1998
"... We demonstrate how to use Lautemann's proof that BPP is in \Sigma p 2 to exhibit that BPP is in RP PromiseRP . Immediate consequences show that if PromiseRP is easy or if there exist quick hitting set generators then P = BPP. Our proof vastly simplifies the proofs of the later result due to ..."
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We demonstrate how to use Lautemann's proof that BPP is in \Sigma p 2 to exhibit that BPP is in RP PromiseRP . Immediate consequences show that if PromiseRP is easy or if there exist quick hitting set generators then P = BPP. Our proof vastly simplifies the proofs of the later result due to Andreev, Clementi and Rolim and Andreev, Clementi, Rolim and Trevisan. Clementi, Rolimand Trevisan question whether the promise is necessary for the above results, i.e. whether BPP ` RP RP for instance. We give a relativized world where P = RP 6= BPP and thus the promise is indeed needed. 1 Introduction Andreev, Clementi and Rolim [ACR98] show how given access to a quick hitting set generator, one can approximate the size of easily describable sets. As an immediate consequence one gets that if quick hitting set generators exist then P = BPP. Andreev, Clementi, Rolim and Trevisan [ACRT97] simplify the proof and apply the result to simulating BPP with weak random sources. Much earlier, Lautema...
The Isomorphism Conjecture Holds and Oneway Functions Exist Relative to an Oracle
 Journal of Computer and System Sciences
, 1994
"... In this paper we demonstrate an oracle relative to which there are oneway functions but every paddable 1lidegree collapses to an isomorphism type, thus yielding a relativized failure of the JosephYoung Conjecture (JYC) [JY85]. We then use this result to construct an oracle relative to which t ..."
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In this paper we demonstrate an oracle relative to which there are oneway functions but every paddable 1lidegree collapses to an isomorphism type, thus yielding a relativized failure of the JosephYoung Conjecture (JYC) [JY85]. We then use this result to construct an oracle relative to which the Isomorphism Conjecture (IC) is true but oneway functions exist, which answers an open question of Fenner, Fortnow, and Kurtz [FFK92]. Thus, there are now relativizations realizing every one of the four possible states of affairs between the IC and the existence of oneway functions. 1 Introduction Berman and Hartmanis [BH76, BH77] showed that if two languages A and B are equivalent to one another under polynomialtime manytoone reductions and if they are both paddable then they are polynomialtime isomorphic. After surveying all of the thenknown NPcomplete languages and discovering that each was indeed paddable, they posed: The Isomorphism Conjecture (IC) Every NPcomplete lan...
A General Method to Construct Oracles Realizing Given Relationships between Complexity Classes
, 1994
"... We present a method to prove oracle theorems of the following type. ..."
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Cited by 9 (1 self)
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We present a method to prove oracle theorems of the following type.
The npcompleteness column: Finding needles in haystacks
 ACM Transactions on Algorithms
, 2007
"... Abstract. This is the 26th edition of a column that covers new developments in the theory of NPcompleteness. The presentation is modeled on that which M. R. Garey and I used in our book “Computers and Intractability: A Guide to the Theory of NPCompleteness, ” W. H. Freeman & Co., New York, 197 ..."
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Cited by 8 (0 self)
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Abstract. This is the 26th edition of a column that covers new developments in the theory of NPcompleteness. The presentation is modeled on that which M. R. Garey and I used in our book “Computers and Intractability: A Guide to the Theory of NPCompleteness, ” W. H. Freeman & Co., New York, 1979, hereinafter referred to as “[G&J]. ” Previous columns, the first 23 of which appeared in J. Algorithms, will be referred to by a combination of their sequence number and year of appearance, e.g., “Column 1 [1981]. ” Full bibliographic details on the previous columns, as well as downloadable unofficial versions of them, can be found at
A Tight Relationship between Generic Oracles and Type2 Complexity Theory
, 1997
"... We show that any two complexity classes satisfying some general conditions are distinct relative to a generic oracle iff the corresponding type2 classes are distinct. ..."
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Cited by 7 (1 self)
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We show that any two complexity classes satisfying some general conditions are distinct relative to a generic oracle iff the corresponding type2 classes are distinct.
Lower bounds for quantum search and derandomization
, 1998
"... We prove lower bounds on the error probability of a quantum algorithm for searching through an unordered list of N items, as a function of the number T of queries it makes. In particular, if T ∈ O ( √ N) then the error is lower bounded by a constant. If we want error ≤ 1/2 N then we need T ∈ Ω(N) q ..."
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Cited by 7 (3 self)
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We prove lower bounds on the error probability of a quantum algorithm for searching through an unordered list of N items, as a function of the number T of queries it makes. In particular, if T ∈ O ( √ N) then the error is lower bounded by a constant. If we want error ≤ 1/2 N then we need T ∈ Ω(N) queries. We apply this to show that a quantum computer cannot do much better than a classical computer when amplifying the success probability of an RPmachine. A classical computer can achieve error ≤ 1/2 k using k applications of the RPmachine, a quantum computer still needs at least ck applications for this (when treating the machine as a blackbox), where c> 0 is a constant independent of k. Furthermore, we prove a lower bound of Ω ( √ log N / loglog N) queries for quantum boundederror search of an ordered list of N items. 1
Analysis of quantum functions
 in Proceedings of the 19th International Conference on Foundations of Software Technology and Theoretical Computer Science, Lecture Notes in Computer Science, Vol.1738
, 1999
"... Abstract. Quantum functions are functions that are defined in terms of quantum mechanical computation. Besides quantum computable functions, we study quantum probability functions, which compute the acceptance probability of quantum computation. We also investigate quantum gap functions, which compu ..."
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Abstract. Quantum functions are functions that are defined in terms of quantum mechanical computation. Besides quantum computable functions, we study quantum probability functions, which compute the acceptance probability of quantum computation. We also investigate quantum gap functions, which compute the gap between acceptance and rejection probabilities of quantum computation. 1