We describe a general technique that yields the first Statistical Query lower bounds} fora range of fundamental high-dimensional learning problems involving Gaussian distributions. Our main results are for the problems of (1) learning Gaussian mixture models (GMMs), and (2) robust (agnostic) learning of a single unknown Gaussian distribution. For each of these problems, we show a super-polynomial gap} between the (information-theoretic)sample complexity and the computational complexity of any} Statistical Query algorithm for the problem. Statistical Query (SQ) algorithms are a class of algorithms that are only allowed to query expectations of functions of the distribution rather than directly access samples. This class of algorithms is quite broad: a wide range of known algorithmic techniques in machine learning are known to be implementable using SQs.Moreover, for the unsupervised learning problems studied in this paper, all known algorithms with non-trivial performance guarantees are SQ or are easily implementable using SQs. Our SQ lower bound for Problem (1)is qualitatively matched by known learning algorithms for GMMs. At a conceptual level, this result implies that – as far as SQ algorithms are concerned – the computational complexity of learning GMMs is inherently exponential in the dimension of the latent space} – even though there is no such information-theoretic barrier. Our lower bound for Problem (2) implies that the accuracy of the robust learning algorithm in cite{DiakonikolasKKLMS16} is essentially best possible among all polynomial-time SQ algorithms. On the positive side, we also give a new (SQ) learning algorithm for Problem (2) achievingthe information-theoretically optimal accuracy, up to a constant factor, whose running time essentially matches our lower bound. Our algorithm relies on a filtering technique generalizing cite{DiakonikolasKKLMS16} that removes outliers based on higher-order tensors.Our SQ lower bounds are attained via a unified moment-matching technique that is useful in other contexts and may be of broader interest. Our technique yields nearly-tight lower bounds for a number of related unsupervised estimation problems. Specifically, for the problems of (3) robust covariance estimation in spectral norm, and (4) robust sparse mean estimation, we establish a quadratic statistical–computational tradeoff} for SQ algorithms, matching known upper bounds. Finally, our technique can be used to obtain tight sample complexitylower bounds for high-dimensional testing} problems. Specifically, for the classical problem of robustly testing} an unknown mean (known covariance) Gaussian, our technique implies an information-theoretic sample lower bound that scales linearly} in the dimension. Our sample lower bound matches the sample complexity of the corresponding robust learning} problem and separates the sample complexity of robust testing from standard (non-robust) testing. This separation is surpri
{"title":"Statistical Query Lower Bounds for Robust Estimation of High-Dimensional Gaussians and Gaussian Mixtures","authors":"Ilias Diakonikolas, D. Kane, Alistair Stewart","doi":"10.1109/FOCS.2017.16","DOIUrl":"https://doi.org/10.1109/FOCS.2017.16","url":null,"abstract":"We describe a general technique that yields the first Statistical Query lower bounds} fora range of fundamental high-dimensional learning problems involving Gaussian distributions. Our main results are for the problems of (1) learning Gaussian mixture models (GMMs), and (2) robust (agnostic) learning of a single unknown Gaussian distribution. For each of these problems, we show a super-polynomial gap} between the (information-theoretic)sample complexity and the computational complexity of any} Statistical Query algorithm for the problem. Statistical Query (SQ) algorithms are a class of algorithms that are only allowed to query expectations of functions of the distribution rather than directly access samples. This class of algorithms is quite broad: a wide range of known algorithmic techniques in machine learning are known to be implementable using SQs.Moreover, for the unsupervised learning problems studied in this paper, all known algorithms with non-trivial performance guarantees are SQ or are easily implementable using SQs. Our SQ lower bound for Problem (1)is qualitatively matched by known learning algorithms for GMMs. At a conceptual level, this result implies that – as far as SQ algorithms are concerned – the computational complexity of learning GMMs is inherently exponential in the dimension of the latent space} – even though there is no such information-theoretic barrier. Our lower bound for Problem (2) implies that the accuracy of the robust learning algorithm in cite{DiakonikolasKKLMS16} is essentially best possible among all polynomial-time SQ algorithms. On the positive side, we also give a new (SQ) learning algorithm for Problem (2) achievingthe information-theoretically optimal accuracy, up to a constant factor, whose running time essentially matches our lower bound. Our algorithm relies on a filtering technique generalizing cite{DiakonikolasKKLMS16} that removes outliers based on higher-order tensors.Our SQ lower bounds are attained via a unified moment-matching technique that is useful in other contexts and may be of broader interest. Our technique yields nearly-tight lower bounds for a number of related unsupervised estimation problems. Specifically, for the problems of (3) robust covariance estimation in spectral norm, and (4) robust sparse mean estimation, we establish a quadratic statistical–computational tradeoff} for SQ algorithms, matching known upper bounds. Finally, our technique can be used to obtain tight sample complexitylower bounds for high-dimensional testing} problems. Specifically, for the classical problem of robustly testing} an unknown mean (known covariance) Gaussian, our technique implies an information-theoretic sample lower bound that scales linearly} in the dimension. Our sample lower bound matches the sample complexity of the corresponding robust learning} problem and separates the sample complexity of robust testing from standard (non-robust) testing. This separation is surpri","PeriodicalId":311592,"journal":{"name":"2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"27 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125554997","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}
Miroslav Dudík, Nika Haghtalab, Haipeng Luo, R. Schapire, Vasilis Syrgkanis, Jennifer Wortman Vaughan
We consider the design of computationally efficient online learning algorithms in an adversarial setting in which the learner has access to an offline optimization oracle. We present an algorithm called Generalized Followthe- Perturbed-Leader and provide conditions under which it is oracle-efficient while achieving vanishing regret. Our results make significant progress on an open problem raised by Hazan and Koren [1], who showed that oracle-efficient algorithms do not exist in full generality and asked whether one can identify conditions under which oracle-efficient online learning may be possible. Our auction-design framework considers an auctioneer learning an optimal auction for a sequence of adversarially selected valuations with the goal of achieving revenue that is almost as good as the optimal auction in hindsight, among a class of auctions. We give oracle-efficient learning results for: (1) VCG auctions with bidder-specific reserves in singleparameter settings, (2) envy-free item-pricing auctions in multiitem settings, and (3) the level auctions of Morgenstern and Roughgarden [2] for single-item settings. The last result leads to an approximation of the overall optimal Myerson auction when bidders’ valuations are drawn according to a fast-mixing Markov process, extending prior work that only gave such guarantees for the i.i.d. setting.We also derive various extensions, including: (1) oracleefficient algorithms for the contextual learning setting in which the learner has access to side information (such as bidder demographics), (2) learning with approximate oracles such as those based on Maximal-in-Range algorithms, and (3) no-regret bidding algorithms in simultaneous auctions, which resolve an open problem of Daskalakis and Syrgkanis [3].
{"title":"Oracle-Efficient Online Learning and Auction Design","authors":"Miroslav Dudík, Nika Haghtalab, Haipeng Luo, R. Schapire, Vasilis Syrgkanis, Jennifer Wortman Vaughan","doi":"10.1145/3402203","DOIUrl":"https://doi.org/10.1145/3402203","url":null,"abstract":"We consider the design of computationally efficient online learning algorithms in an adversarial setting in which the learner has access to an offline optimization oracle. We present an algorithm called Generalized Followthe- Perturbed-Leader and provide conditions under which it is oracle-efficient while achieving vanishing regret. Our results make significant progress on an open problem raised by Hazan and Koren [1], who showed that oracle-efficient algorithms do not exist in full generality and asked whether one can identify conditions under which oracle-efficient online learning may be possible. Our auction-design framework considers an auctioneer learning an optimal auction for a sequence of adversarially selected valuations with the goal of achieving revenue that is almost as good as the optimal auction in hindsight, among a class of auctions. We give oracle-efficient learning results for: (1) VCG auctions with bidder-specific reserves in singleparameter settings, (2) envy-free item-pricing auctions in multiitem settings, and (3) the level auctions of Morgenstern and Roughgarden [2] for single-item settings. The last result leads to an approximation of the overall optimal Myerson auction when bidders’ valuations are drawn according to a fast-mixing Markov process, extending prior work that only gave such guarantees for the i.i.d. setting.We also derive various extensions, including: (1) oracleefficient algorithms for the contextual learning setting in which the learner has access to side information (such as bidder demographics), (2) learning with approximate oracles such as those based on Maximal-in-Range algorithms, and (3) no-regret bidding algorithms in simultaneous auctions, which resolve an open problem of Daskalakis and Syrgkanis [3].","PeriodicalId":311592,"journal":{"name":"2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"33 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117285263","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 regular expression membership testing: Given a regular expression of size m and a string of size n, decide whether the string is in the language described by the regular expression. Its classic O(nm) algorithm is one of the big success stories of the 70s, which allowed pattern matching to develop into the standard tool that it is today.Many special cases of pattern matching have been studied that can be solved faster than in quadratic time. However, a systematic study of tractable cases was made possible only recently, with the first conditional lower bounds reported by Backurs and Indyk [FOCS16]. Restricted to any type of homogeneous regular expressions of depth 2 or 3, they either presented a near-linear time algorithm or a quadratic conditional lower bound, with one exception known as the Word Break problem.In this paper we complete their work as follows:• We present two almost-linear time algorithms that generalize all known almost-linear time algorithms for special cases of regular expression membership testing.• We classify all types, except for the Word Break problem, into almost-linear time or quadratic time assuming the Strong Exponential Time Hypothesis. This extends the classification from depth 2 and 3 to any constant depth.• For the Word Break problem we give an improved O(nm1/3 + m) algorithm. Surprisingly, we also prove a matching conditional lower bound for combinatorial algorithms. This establishes Word Break as the only intermediate problem.In total, we prove matching upper and lower bounds for any type of bounded-depth homogeneous regular expressions, which yields a full dichotomy for regular expression member-ship testing.
{"title":"A Dichotomy for Regular Expression Membership Testing","authors":"K. Bringmann, A. Jørgensen, Kasper Green Larsen","doi":"10.1109/FOCS.2017.36","DOIUrl":"https://doi.org/10.1109/FOCS.2017.36","url":null,"abstract":"We study regular expression membership testing: Given a regular expression of size m and a string of size n, decide whether the string is in the language described by the regular expression. Its classic O(nm) algorithm is one of the big success stories of the 70s, which allowed pattern matching to develop into the standard tool that it is today.Many special cases of pattern matching have been studied that can be solved faster than in quadratic time. However, a systematic study of tractable cases was made possible only recently, with the first conditional lower bounds reported by Backurs and Indyk [FOCS16]. Restricted to any type of homogeneous regular expressions of depth 2 or 3, they either presented a near-linear time algorithm or a quadratic conditional lower bound, with one exception known as the Word Break problem.In this paper we complete their work as follows:• We present two almost-linear time algorithms that generalize all known almost-linear time algorithms for special cases of regular expression membership testing.• We classify all types, except for the Word Break problem, into almost-linear time or quadratic time assuming the Strong Exponential Time Hypothesis. This extends the classification from depth 2 and 3 to any constant depth.• For the Word Break problem we give an improved O(nm1/3 + m) algorithm. Surprisingly, we also prove a matching conditional lower bound for combinatorial algorithms. This establishes Word Break as the only intermediate problem.In total, we prove matching upper and lower bounds for any type of bounded-depth homogeneous regular expressions, which yields a full dichotomy for regular expression member-ship testing.","PeriodicalId":311592,"journal":{"name":"2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121789550","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}
Let X be a set of n points of norm at most 1 in the Euclidean space R^k, and suppose ≥0. An ≥-distance sketch for X is a data structure that, given any two points of X enables one to recover the square of the (Euclidean) distance between them up to an additive} error of ≥. Let f(n,k,≥) denote the minimum possible number of bits of such a sketch. Here we determine f(n,k,≥) up to a constant factor for all n ≥ k ≥ 1 and all ≥ ≥ frac{1}{n^{0.49}}. Our proof is algorithmic, and provides an efficient algorithm for computing a sketch of size O(f(n,k,≥)/n) for each point, so that the square of the distance between any two points can be computed from their sketches up to an additive error of ≥ in time linear in the length of the sketches. We also discuss the case of smaller ≥2/√ n and obtain some new results about dimension reduction in this range. In particular, we show that for any such ≥ and any k ≤ t=frac{log (2+≥^2 n)}{≥^2} there are configurations of n points in R^k that cannot be embedded in R^{ℓ} for ℓ
{"title":"Optimal Compression of Approximate Inner Products and Dimension Reduction","authors":"N. Alon, B. Klartag","doi":"10.1109/FOCS.2017.65","DOIUrl":"https://doi.org/10.1109/FOCS.2017.65","url":null,"abstract":"Let X be a set of n points of norm at most 1 in the Euclidean space R^k, and suppose ≥0. An ≥-distance sketch for X is a data structure that, given any two points of X enables one to recover the square of the (Euclidean) distance between them up to an additive} error of ≥. Let f(n,k,≥) denote the minimum possible number of bits of such a sketch. Here we determine f(n,k,≥) up to a constant factor for all n ≥ k ≥ 1 and all ≥ ≥ frac{1}{n^{0.49}}. Our proof is algorithmic, and provides an efficient algorithm for computing a sketch of size O(f(n,k,≥)/n) for each point, so that the square of the distance between any two points can be computed from their sketches up to an additive error of ≥ in time linear in the length of the sketches. We also discuss the case of smaller ≥2/√ n and obtain some new results about dimension reduction in this range. In particular, we show that for any such ≥ and any k ≤ t=frac{log (2+≥^2 n)}{≥^2} there are configurations of n points in R^k that cannot be embedded in R^{ℓ} for ℓ","PeriodicalId":311592,"journal":{"name":"2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"64 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115713868","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}
Adam Bouland, Lijie Chen, D. Holden, J. Thaler, Prashant Nalini Vasudevan
We examine the power of statistical zero knowledge proofs (captured by the complexity class SZK) and their variants. First, we give the strongest known relativized evidence that SZK contains hard problems, by exhibiting an oracle relative to which SZK (indeed, even NISZK) is not contained in the class UPP, containing those problems solvable by randomized algorithms with unbounded error. This answers an open question of Watrous from 2002. Second, we lift this oracle separation to the setting of communication complexity, thereby answering a question of Göös et al. (ICALP 2016). Third, we give relativized evidence that perfect zero knowledge proofs (captured by the class PZK) are weaker than general zero knowledge proofs. Specifically, we exhibit oracles which separate SZK from PZK, NISZK from NIPZK and PZK from coPZK. The first of these results answers a question raised in 1991 by Aiello and Håstad (Information and Computation), and the second answers a question of Lovett and Zhang (2016). We also describe additional applications of these results outside of structural complexity.The technical core of our results is a stronger hardness amplification theorem for approximate degree, which roughly says that composing the gapped-majority function with any function of high approximate degree yields a function with high threshold degree.
{"title":"On the Power of Statistical Zero Knowledge","authors":"Adam Bouland, Lijie Chen, D. Holden, J. Thaler, Prashant Nalini Vasudevan","doi":"10.1109/FOCS.2017.71","DOIUrl":"https://doi.org/10.1109/FOCS.2017.71","url":null,"abstract":"We examine the power of statistical zero knowledge proofs (captured by the complexity class SZK) and their variants. First, we give the strongest known relativized evidence that SZK contains hard problems, by exhibiting an oracle relative to which SZK (indeed, even NISZK) is not contained in the class UPP, containing those problems solvable by randomized algorithms with unbounded error. This answers an open question of Watrous from 2002. Second, we lift this oracle separation to the setting of communication complexity, thereby answering a question of Göös et al. (ICALP 2016). Third, we give relativized evidence that perfect zero knowledge proofs (captured by the class PZK) are weaker than general zero knowledge proofs. Specifically, we exhibit oracles which separate SZK from PZK, NISZK from NIPZK and PZK from coPZK. The first of these results answers a question raised in 1991 by Aiello and Håstad (Information and Computation), and the second answers a question of Lovett and Zhang (2016). We also describe additional applications of these results outside of structural complexity.The technical core of our results is a stronger hardness amplification theorem for approximate degree, which roughly says that composing the gapped-majority function with any function of high approximate degree yields a function with high threshold degree.","PeriodicalId":311592,"journal":{"name":"2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128262362","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}
{"title":"Optimality of the Johnson-Lindenstrauss Lemma","authors":"Kasper Green Larsen, Jelani Nelson","doi":"10.1109/FOCS.2017.64","DOIUrl":"https://doi.org/10.1109/FOCS.2017.64","url":null,"abstract":"For any d, n ≥ 2 and 1=(min{n, d})0.4999","PeriodicalId":311592,"journal":{"name":"2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"73 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122776853","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 streaming principal component analysis (PCA), that is to find, in O(dk) space, the top k eigenvectors of a d× d hidden matrix bold Sigma with online vectors drawn from covariance matrix bold Sigma.We provide global convergence for Ojas algorithm which is popularly used in practice but lacks theoretical understanding for k≈1. We also provide a modified variant mathsf{Oja}^{++} that runs even faster than Ojas. Our results match the information theoretic lower bound in terms of dependency on error, on eigengap, on rank k, and on dimension d, up to poly-log factors. In addition, our convergence rate can be made gap-free, that is proportional to the approximation error and independent of the eigengap.In contrast, for general rank k, before our work (1) it was open to design any algorithm with efficient global convergence rate; and (2) it was open to design any algorithm with (even local) gap-free convergence rate in O(dk) space.
{"title":"First Efficient Convergence for Streaming k-PCA: A Global, Gap-Free, and Near-Optimal Rate","authors":"Zeyuan Allen-Zhu, Yuanzhi Li","doi":"10.1109/FOCS.2017.51","DOIUrl":"https://doi.org/10.1109/FOCS.2017.51","url":null,"abstract":"We study streaming principal component analysis (PCA), that is to find, in O(dk) space, the top k eigenvectors of a d× d hidden matrix bold Sigma with online vectors drawn from covariance matrix bold Sigma.We provide global convergence for Ojas algorithm which is popularly used in practice but lacks theoretical understanding for k≈1. We also provide a modified variant mathsf{Oja}^{++} that runs even faster than Ojas. Our results match the information theoretic lower bound in terms of dependency on error, on eigengap, on rank k, and on dimension d, up to poly-log factors. In addition, our convergence rate can be made gap-free, that is proportional to the approximation error and independent of the eigengap.In contrast, for general rank k, before our work (1) it was open to design any algorithm with efficient global convergence rate; and (2) it was open to design any algorithm with (even local) gap-free convergence rate in O(dk) space.","PeriodicalId":311592,"journal":{"name":"2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128500631","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}
Ground states of local Hamiltonians can be generally highly entangled: any quantum circuit that generates them (even approximately) must be sufficiently deep to allow coupling (entanglement) between any pair of qubits. Until now this property was not known to be robust - the marginals of such states to a subset of the qubits containing all but a small constant fraction of them may be only locally entangled, and hence approximable by shallow quantum circuits. In this work we construct a family of 16-local Hamiltonians for which any 1-10^-8 fraction of qubits of any ground state must be highly entangled.This provides evidence that quantum entanglement is not very fragile, and perhaps our intuition about its instability is an artifact of considering local Hamiltonians which are not only local but spatially local. Formally, it provides positive evidence for two wide-open conjectures in condensed-matter physics and quantum complexity theory which are the qLDPC conjecture, positing the existence of good quantum LDPC codes, and the NLTS conjecture due to Freedman and Hastings positing the existence of local Hamiltonians in which any low-energy state is highly-entangled.Our Hamiltonian is based on applying the hypergraph product by Tillich-Zemor to the repetition code with checks from an expander graph. A key tool in our proof is a new lower bound on the vertex expansion of the output of low-depth quantum circuits, which may be of independent interest.
{"title":"Local Hamiltonians Whose Ground States Are Hard to Approximate","authors":"Lior Eldar, A. Harrow","doi":"10.1109/FOCS.2017.46","DOIUrl":"https://doi.org/10.1109/FOCS.2017.46","url":null,"abstract":"Ground states of local Hamiltonians can be generally highly entangled: any quantum circuit that generates them (even approximately) must be sufficiently deep to allow coupling (entanglement) between any pair of qubits. Until now this property was not known to be robust - the marginals of such states to a subset of the qubits containing all but a small constant fraction of them may be only locally entangled, and hence approximable by shallow quantum circuits. In this work we construct a family of 16-local Hamiltonians for which any 1-10^-8 fraction of qubits of any ground state must be highly entangled.This provides evidence that quantum entanglement is not very fragile, and perhaps our intuition about its instability is an artifact of considering local Hamiltonians which are not only local but spatially local. Formally, it provides positive evidence for two wide-open conjectures in condensed-matter physics and quantum complexity theory which are the qLDPC conjecture, positing the existence of good quantum LDPC codes, and the NLTS conjecture due to Freedman and Hastings positing the existence of local Hamiltonians in which any low-energy state is highly-entangled.Our Hamiltonian is based on applying the hypergraph product by Tillich-Zemor to the repetition code with checks from an expander graph. A key tool in our proof is a new lower bound on the vertex expansion of the output of low-depth quantum circuits, which may be of independent interest.","PeriodicalId":311592,"journal":{"name":"2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115610658","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}
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