We show that hard sets S for NP must have exponential density, i.e. |S=n| ges 2nepsi for some isin > 0 and infinitely many n, unless coNP sube NP/poly and the polynomial-time hierarchy collapses. This result holds for Turing reductions that make n1-isin queries. In addition we study the instance complexity o/NP- hard problems and show that hard sets also have an exponential amount of instances that have instance complexity n for some sigma > 0. This result also holds for Turing reductions that make n1-isin queries.
{"title":"NP-Hard Sets Are Exponentially Dense Unless coNP C NP/poly","authors":"H. Buhrman, J. M. Hitchcock","doi":"10.1109/CCC.2008.21","DOIUrl":"https://doi.org/10.1109/CCC.2008.21","url":null,"abstract":"We show that hard sets S for NP must have exponential density, i.e. |S=n| ges 2nepsi for some isin > 0 and infinitely many n, unless coNP sube NP/poly and the polynomial-time hierarchy collapses. This result holds for Turing reductions that make n1-isin queries. In addition we study the instance complexity o/NP- hard problems and show that hard sets also have an exponential amount of instances that have instance complexity n for some sigma > 0. This result also holds for Turing reductions that make n1-isin queries.","PeriodicalId":338061,"journal":{"name":"2008 23rd Annual IEEE Conference on Computational Complexity","volume":"70 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2008-06-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115814141","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the complexity of deciding whether a given homogeneous multivariate polynomial has a non- trivial root over a finite field. Given a homogeneous algebraic circuit C that computes an n- variate polynomial p(x) of degree d over a finite field Fq, we wish to determine if there exists a nonzero xisinFq n with C(x)=0. For constant n there are known algorithms for doing this efficiently. However for linear n, the problem becomes NP hard. In this paper, using interesting algebraic techniques, we show that if d is prime and n>d/2, the problem can be solved over sufficiently large finite fields in randomized polynomial time. We complement this result by showing that relaxing any of these constraints makes the problem intractable again.
{"title":"Detecting Rational Points on Hypersurfaces over Finite Fields","authors":"Swastik Kopparty, S. Yekhanin","doi":"10.1109/CCC.2008.36","DOIUrl":"https://doi.org/10.1109/CCC.2008.36","url":null,"abstract":"We study the complexity of deciding whether a given homogeneous multivariate polynomial has a non- trivial root over a finite field. Given a homogeneous algebraic circuit C that computes an n- variate polynomial p(x) of degree d over a finite field Fq, we wish to determine if there exists a nonzero xisinFq n with C(x)=0. For constant n there are known algorithms for doing this efficiently. However for linear n, the problem becomes NP hard. In this paper, using interesting algebraic techniques, we show that if d is prime and n>d/2, the problem can be solved over sufficiently large finite fields in randomized polynomial time. We complement this result by showing that relaxing any of these constraints makes the problem intractable again.","PeriodicalId":338061,"journal":{"name":"2008 23rd Annual IEEE Conference on Computational Complexity","volume":"57 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2008-06-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114325308","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We revisit the computational power of constant width polynomial size planar nondeterministic branching programs. We show that they are capable of computing any function computed by a Pi2 o CC0 o AC0 circuit in polynomial size. In the quasipolynomial size setting we obtain a characterization of ACC0 by constant width planar non-deterministic branching programs.
我们重新讨论了等宽多项式尺寸平面不确定性分支规划的计算能力。我们证明了它们能够以多项式大小计算任何由Pi2 o CC0 o AC0电路计算的函数。在拟多项式尺寸设置下，我们用等宽平面不确定性分支规划得到了ACC0的一个表征。
{"title":"Constant Width Planar Branching Programs Characterize ACC^0 in Quasipolynomial Size","authors":"Kristoffer Arnsfelt Hansen","doi":"10.1109/CCC.2008.30","DOIUrl":"https://doi.org/10.1109/CCC.2008.30","url":null,"abstract":"We revisit the computational power of constant width polynomial size planar nondeterministic branching programs. We show that they are capable of computing any function computed by a Pi<sub>2</sub> o CC<sup>0</sup> o AC<sup>0</sup> circuit in polynomial size. In the quasipolynomial size setting we obtain a characterization of ACC<sup>0</sup> by constant width planar non-deterministic branching programs.","PeriodicalId":338061,"journal":{"name":"2008 23rd Annual IEEE Conference on Computational Complexity","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2008-06-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130345878","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2008-04-04DOI: 10.4086/toc.2009.v005a001
S. Aaronson, Salman Beigi, Andrew Drucker, B. Fefferman, P. Shor
The class QMA(k), introduced by Kobayashi et al., consists of all languages that can be verified using k unentangled quantum proofs. Many of the simplest questions about this class have remained embarrassingly open: for example, can we give any evidence that k quantum proofs are more powerful than one? Can we show any upper bound on QMA(k), besides the trivial NEXP? Does QMA(k)=QMA(2) for kges2? Can QMA(k) protocols be amplified to exponentially small error? In this paper, we make progress on all of the above questions. *We give a protocol by which a verifier can be convinced that a 3SAT formula of size n is satisfiable, with constant soundness, given O tilde(radicn) unentangled quantum witnesses with O(log n) qubits each. Our protocol relies on Dinur's version of the PCP Theorem and is inherently non-relativizing. *We show that assuming the famous Additivity Conjecture from quantum information theory, any QMA(2) protocol can be amplified to exponentially small error, and QMA(k)=QMA(2) for all kges=2. *We give evidence that QMA(2) sube PSPACE, by showing that this would follow from "strong amplification" of QMA(2) protocols. *We prove the nonexistence of "perfect disentanglers" for simulating multiple Merlins with one.
{"title":"The Power of Unentanglement","authors":"S. Aaronson, Salman Beigi, Andrew Drucker, B. Fefferman, P. Shor","doi":"10.4086/toc.2009.v005a001","DOIUrl":"https://doi.org/10.4086/toc.2009.v005a001","url":null,"abstract":"The class QMA(k), introduced by Kobayashi et al., consists of all languages that can be verified using k unentangled quantum proofs. Many of the simplest questions about this class have remained embarrassingly open: for example, can we give any evidence that k quantum proofs are more powerful than one? Can we show any upper bound on QMA(k), besides the trivial NEXP? Does QMA(k)=QMA(2) for kges2? Can QMA(k) protocols be amplified to exponentially small error? In this paper, we make progress on all of the above questions. *We give a protocol by which a verifier can be convinced that a 3SAT formula of size n is satisfiable, with constant soundness, given O tilde(radicn) unentangled quantum witnesses with O(log n) qubits each. Our protocol relies on Dinur's version of the PCP Theorem and is inherently non-relativizing. *We show that assuming the famous Additivity Conjecture from quantum information theory, any QMA(2) protocol can be amplified to exponentially small error, and QMA(k)=QMA(2) for all kges=2. *We give evidence that QMA(2) sube PSPACE, by showing that this would follow from \"strong amplification\" of QMA(2) protocols. *We prove the nonexistence of \"perfect disentanglers\" for simulating multiple Merlins with one.","PeriodicalId":338061,"journal":{"name":"2008 23rd Annual IEEE Conference on Computational Complexity","volume":"65 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2008-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131323263","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A. Doherty, Yeong-Cherng Liang, B. Toner, S. Wehner
We study the quantum moment problem: given a conditional probability distribution together with some polynomial constraints, does there exist a quantum state rho and a collection of measurement operators such that (i) the probability of obtaining a particular outcome when a particular measurement is performed on rho is specified by the conditional probability distribution, and (ii) the measurement operators satisfy the constraints. For example, the constraints might specify that some measurement operators must commute. We show that if an instance of the quantum moment problem is unsatisfiable, then there exists a certificate of a particular form proving this. Our proof is based on a recent result in algebraic geometry, the noncommutative Positivstellensatz of Helton and McCullough [Trans. Amer. Math. Soc., 356(9):3721, 2004]. A special case of the quantum moment problem is to compute the value of one-round multi-prover games with entangled provers. Under the conjecture that the provers need only share states in finite-dimensional Hilbert spaces, we prove that a hierarchy of semidefinite programs similar to the one given by Navascues, Pironioand Acin [Phys. Rev. Lett., 98:010401, 2007] converges to the entangled value of the game. Under this conjecture, it would follow that the languages recognized by a multi-prover interactive proof system where the provers share entanglement are recursive.
{"title":"The Quantum Moment Problem and Bounds on Entangled Multi-prover Games","authors":"A. Doherty, Yeong-Cherng Liang, B. Toner, S. Wehner","doi":"10.1109/CCC.2008.26","DOIUrl":"https://doi.org/10.1109/CCC.2008.26","url":null,"abstract":"We study the quantum moment problem: given a conditional probability distribution together with some polynomial constraints, does there exist a quantum state rho and a collection of measurement operators such that (i) the probability of obtaining a particular outcome when a particular measurement is performed on rho is specified by the conditional probability distribution, and (ii) the measurement operators satisfy the constraints. For example, the constraints might specify that some measurement operators must commute. We show that if an instance of the quantum moment problem is unsatisfiable, then there exists a certificate of a particular form proving this. Our proof is based on a recent result in algebraic geometry, the noncommutative Positivstellensatz of Helton and McCullough [Trans. Amer. Math. Soc., 356(9):3721, 2004]. A special case of the quantum moment problem is to compute the value of one-round multi-prover games with entangled provers. Under the conjecture that the provers need only share states in finite-dimensional Hilbert spaces, we prove that a hierarchy of semidefinite programs similar to the one given by Navascues, Pironioand Acin [Phys. Rev. Lett., 98:010401, 2007] converges to the entangled value of the game. Under this conjecture, it would follow that the languages recognized by a multi-prover interactive proof system where the provers share entanglement are recursive.","PeriodicalId":338061,"journal":{"name":"2008 23rd Annual IEEE Conference on Computational Complexity","volume":"16 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2008-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127427291","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Tsuyoshi Ito, Hirotada Kobayashi, Daniel Preda, Xiaoming Sun, A. Yao
A central question in quantum information theory and computational complexity is how powerful nonlocal strategies are in cooperative games with imperfect information, such as multi-prover interactive proof systems. This paper develops a new method for proving limits of nonlocal strategies that make use of prior entanglement among players (or, provers, in the terminology of multi-prover interactive proofs). Instead of proving the limits for usual isolated provers who initially share entanglement, this paper proves the limits for "commuting-operator provers", who share private space, but can apply only such operators that are commutative with any operator applied by other provers. Obviously, these commuting-operator provers are at least as powerful as usual isolated but prior-entangled provers, and thus, limits in the model with commuting-operator provers immediately give limits in the usual model with prior-entangled provers. Using this method, we obtain an n-party generalization of the Tsirelson bound for the Clauser-Horne-Shimony-Holt inequality, for every n. Our bounds are tight in the sense that, in every n-party case, the equality is achievable by a usual nonlocal strategy with prior entanglement. We also apply our method to a three-prover one-round binary interactive proof system for NEXP. Combined with the technique developed by Kempe, Kobayashi, Matsumoto, Toner and Vidick to analyze the soundness of the proof system, it is proved to be NP-hard to distinguish whether the entangled value of a three-prover one-round binary-answer game is equal to one or at most 1-1/p(n) for some polynomial p, where n is the number of questions. This is in contrast to the two-prover one-round binary-answer case, where the corresponding problem is efficiently decidable. Alternatively, NEXP has a three-prover one-round binary interactive proof system with perfect completeness and soundness 1 middot 2-poly.
{"title":"Generalized Tsirelson Inequalities, Commuting-Operator Provers, and Multi-prover Interactive Proof Systems","authors":"Tsuyoshi Ito, Hirotada Kobayashi, Daniel Preda, Xiaoming Sun, A. Yao","doi":"10.1109/CCC.2008.12","DOIUrl":"https://doi.org/10.1109/CCC.2008.12","url":null,"abstract":"A central question in quantum information theory and computational complexity is how powerful nonlocal strategies are in cooperative games with imperfect information, such as multi-prover interactive proof systems. This paper develops a new method for proving limits of nonlocal strategies that make use of prior entanglement among players (or, provers, in the terminology of multi-prover interactive proofs). Instead of proving the limits for usual isolated provers who initially share entanglement, this paper proves the limits for \"commuting-operator provers\", who share private space, but can apply only such operators that are commutative with any operator applied by other provers. Obviously, these commuting-operator provers are at least as powerful as usual isolated but prior-entangled provers, and thus, limits in the model with commuting-operator provers immediately give limits in the usual model with prior-entangled provers. Using this method, we obtain an n-party generalization of the Tsirelson bound for the Clauser-Horne-Shimony-Holt inequality, for every n. Our bounds are tight in the sense that, in every n-party case, the equality is achievable by a usual nonlocal strategy with prior entanglement. We also apply our method to a three-prover one-round binary interactive proof system for NEXP. Combined with the technique developed by Kempe, Kobayashi, Matsumoto, Toner and Vidick to analyze the soundness of the proof system, it is proved to be NP-hard to distinguish whether the entangled value of a three-prover one-round binary-answer game is equal to one or at most 1-1/p(n) for some polynomial p, where n is the number of questions. This is in contrast to the two-prover one-round binary-answer case, where the corresponding problem is efficiently decidable. Alternatively, NEXP has a three-prover one-round binary interactive proof system with perfect completeness and soundness 1 middot 2-poly.","PeriodicalId":338061,"journal":{"name":"2008 23rd Annual IEEE Conference on Computational Complexity","volume":"139 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2007-12-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128992446","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We give the first exponential separation between quantum and classical multi-party communication complexity in the (non-interactive) one-way and simultaneous message passing settings. For every k, we demonstrate a relational communication problem between k parties that can be solved exactly by a quantum simultaneous message passing protocol of cost O (log n) and requires protocols of cost nc/k2, where c > 0 is a constant, in the classical non-interactive one-way message passing model with shared randomness and bounded error. Thus our separation of corresponding communication classes is superpolynomial as long as k =0 (radic log n/ log log n ) and exponential for k = O(1).
{"title":"Exponential Separation of Quantum and Classical Non-interactive Multi-party Communication Complexity","authors":"Dmitry Gavinsky, P. Pudlák","doi":"10.1109/CCC.2008.27","DOIUrl":"https://doi.org/10.1109/CCC.2008.27","url":null,"abstract":"We give the first exponential separation between quantum and classical multi-party communication complexity in the (non-interactive) one-way and simultaneous message passing settings. For every k, we demonstrate a relational communication problem between k parties that can be solved exactly by a quantum simultaneous message passing protocol of cost O (log n) and requires protocols of cost nc/k2, where c > 0 is a constant, in the classical non-interactive one-way message passing model with shared randomness and bounded error. Thus our separation of corresponding communication classes is superpolynomial as long as k =0 (radic log n/ log log n ) and exponential for k = O(1).","PeriodicalId":338061,"journal":{"name":"2008 23rd Annual IEEE Conference on Computational Complexity","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2007-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125939566","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A k-query Locally Decodable Code (LDC) encodes an n-bit message x as an N-bit codeword C(x), such that one can probabilistically recover any bit Xi of the message by querying only k bits of the codeword C(x), even after some constant fraction of codeword bits has been corrupted. The major goal of LDC related research is to establish the optimal trade-off between length and query complexity of such codes. Recently upper bounds for the length of LDCs were vastly improved via constructions that rely on existence of certain special (nice) subsets of finite fields. In this work we extend the constructions of LDCs from nice subsets. We argue that further progress on upper bounds for LDCs via these methods is tied to progress on an old number theory question regarding the size of the largest prime factors of Mersenne numbers. Specifically, we show that every Mersenne number m = 2t-1 that has a prime factor p > mUpsi yields a family of k(Upsi)-query locally decodable codes of length exp (nepsi) . Conversely, if for some fixed k and all epsi > 0 one can use the nice subsets technique to obtain a family of k-query LDCs of length exp (nepsi); then infinitely many Mersenne numbers have prime factors larger than known currently.
{"title":"Locally Decodable Codes From Nice Subsets of Finite Fields and Prime Factors of Mersenne Numbers","authors":"K. Kedlaya, S. Yekhanin","doi":"10.1137/070696519","DOIUrl":"https://doi.org/10.1137/070696519","url":null,"abstract":"A k-query Locally Decodable Code (LDC) encodes an n-bit message x as an N-bit codeword C(x), such that one can probabilistically recover any bit Xi of the message by querying only k bits of the codeword C(x), even after some constant fraction of codeword bits has been corrupted. The major goal of LDC related research is to establish the optimal trade-off between length and query complexity of such codes. Recently upper bounds for the length of LDCs were vastly improved via constructions that rely on existence of certain special (nice) subsets of finite fields. In this work we extend the constructions of LDCs from nice subsets. We argue that further progress on upper bounds for LDCs via these methods is tied to progress on an old number theory question regarding the size of the largest prime factors of Mersenne numbers. Specifically, we show that every Mersenne number m = 2t-1 that has a prime factor p > mUpsi yields a family of k(Upsi)-query locally decodable codes of length exp (nepsi) . Conversely, if for some fixed k and all epsi > 0 one can use the nice subsets technique to obtain a family of k-query LDCs of length exp (nepsi); then infinitely many Mersenne numbers have prime factors larger than known currently.","PeriodicalId":338061,"journal":{"name":"2008 23rd Annual IEEE Conference on Computational Complexity","volume":"39 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2007-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123141610","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We present a new variant of the quantum adversary method, a method for proving lower bounds on the quantum query complexity of a function. Adversary methods work as follows: one defines a progress function based on the state of the algorithm, and shows that for a successful algorithm there is a large gap between the initial and final value of the progress, and furthermore that the progress function cannot change by much with a single query. All known variants upper-bound the difference of the progress function, whereas our new variant upper-bounds the ratio and that is why we coin it the multiplicative adversary. Our new method is rooted in the quantum lower-bound method by Ambainis (2005, 2006), based on the analysis of eigenspaces of the density matrix. Ambainis's method is technically very complicated, it lacks intuition, and it only works for symmetric functions. Our method fits well into the adversary framework, has a simple formulation in terms of common block-diagonalization of two operators, and works for all functions. Furthermore, we prove an unconditional strong direct product theorem for the multiplicative quantum adversary bound.
{"title":"The Multiplicative Quantum Adversary","authors":"R. Spalek","doi":"10.1109/CCC.2008.9","DOIUrl":"https://doi.org/10.1109/CCC.2008.9","url":null,"abstract":"We present a new variant of the quantum adversary method, a method for proving lower bounds on the quantum query complexity of a function. Adversary methods work as follows: one defines a progress function based on the state of the algorithm, and shows that for a successful algorithm there is a large gap between the initial and final value of the progress, and furthermore that the progress function cannot change by much with a single query. All known variants upper-bound the difference of the progress function, whereas our new variant upper-bounds the ratio and that is why we coin it the multiplicative adversary. Our new method is rooted in the quantum lower-bound method by Ambainis (2005, 2006), based on the analysis of eigenspaces of the density matrix. Ambainis's method is technically very complicated, it lacks intuition, and it only works for symmetric functions. Our method fits well into the adversary framework, has a simple formulation in terms of common block-diagonalization of two operators, and works for all functions. Furthermore, we prove an unconditional strong direct product theorem for the multiplicative quantum adversary bound.","PeriodicalId":338061,"journal":{"name":"2008 23rd Annual IEEE Conference on Computational Complexity","volume":"2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2007-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114080354","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：