Yumeng He, Matan Kotler-Berkowitz, Harry Liuson, Zeyu Nie
In this paper, we examine the claims made by the paper "A polynomial-time algorithm for 3-SAT" by Lizhi Du. The paper claims to provide a polynomial-time algorithm for solving the NP-complete problem 3-SAT. In examining the paper's argument, we find a flaw in one of the main sections of its algorithm. We argue that this flaw causes the paper's algorithm to incorrectly decide that an infinite family of satisfiable 3-CNF boolean formulas are not satisfiable. Therefore, the paper does not establish that P = NP.
在本文中，我们对杜立志的论文 "A polynomial-timealgorithm for 3-SAT "所提出的主张进行了研究。该论文声称提供了一种多项式时间算法来求解 NP-完全问题 3-SAT。在研究该论文的论证过程中，我们发现其算法的一个主要部分存在缺陷。我们认为，这个缺陷导致论文的算法错误地判定可满足的 3-CNF 布尔型公式的无限族不可满足。
{"title":"A Critique of Du's \"A Polynomial-Time Algorithm for 3-SAT","authors":"Yumeng He, Matan Kotler-Berkowitz, Harry Liuson, Zeyu Nie","doi":"arxiv-2404.04395","DOIUrl":"https://doi.org/arxiv-2404.04395","url":null,"abstract":"In this paper, we examine the claims made by the paper \"A polynomial-time\u0000algorithm for 3-SAT\" by Lizhi Du. The paper claims to provide a polynomial-time\u0000algorithm for solving the NP-complete problem 3-SAT. In examining the paper's\u0000argument, we find a flaw in one of the main sections of its algorithm. We argue\u0000that this flaw causes the paper's algorithm to incorrectly decide that an\u0000infinite family of satisfiable 3-CNF boolean formulas are not satisfiable.\u0000Therefore, the paper does not establish that P = NP.","PeriodicalId":501024,"journal":{"name":"arXiv - CS - Computational Complexity","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140583148","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}
Considerable effort in experimental quantum computing is devoted to noisy intermediate scale quantum computers (NISQ computers). Understanding the effect of noise is important for various aspects of this endeavor including notable claims for achieving quantum supremacy and attempts to demonstrate quantum error correcting codes. In this paper we use Fourier methods combined with statistical analysis to study the effect of noise. In particular, we use Fourier analysis to refine the linear cross-entropy fidelity estimator. We use both analytical methods and simulations to study the effect of readout and gate errors, and we use our analysis to study the samples of Google's 2019 quantum supremacy experiment.
{"title":"Random Circuit Sampling: Fourier Expansion and Statistics","authors":"Gil Kalai, Yosef Rinott, Tomer Shoham","doi":"arxiv-2404.00935","DOIUrl":"https://doi.org/arxiv-2404.00935","url":null,"abstract":"Considerable effort in experimental quantum computing is devoted to noisy\u0000intermediate scale quantum computers (NISQ computers). Understanding the effect\u0000of noise is important for various aspects of this endeavor including notable\u0000claims for achieving quantum supremacy and attempts to demonstrate quantum\u0000error correcting codes. In this paper we use Fourier methods combined with\u0000statistical analysis to study the effect of noise. In particular, we use\u0000Fourier analysis to refine the linear cross-entropy fidelity estimator. We use\u0000both analytical methods and simulations to study the effect of readout and gate\u0000errors, and we use our analysis to study the samples of Google's 2019 quantum\u0000supremacy experiment.","PeriodicalId":501024,"journal":{"name":"arXiv - CS - Computational Complexity","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140583144","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}
The recent MIP*=RE theorem of Ji, Natarajan, Vidick, Wright, and Yuen shows that the complexity class MIP* of multiprover proof systems with entangled provers contains all recursively enumerable languages. Prior work of Grilo, Slofstra, and Yuen [FOCS '19] further shows (via a technique called simulatable codes) that every language in MIP* has a perfect zero knowledge (PZK) MIP* protocol. The MIP*=RE theorem uses two-prover one-round proof systems, and hence such systems are complete for MIP*. However, the construction in Grilo, Slofstra, and Yuen uses six provers, and there is no obvious way to get perfect zero knowledge with two provers via simulatable codes. This leads to a natural question: are there two-prover PZK-MIP* protocols for all of MIP*? In this paper, we show that every language in MIP* has a two-prover one-round PZK-MIP* protocol, answering the question in the affirmative. For the proof, we use a new method based on a key consequence of the MIP*=RE theorem, which is that every MIP* protocol can be turned into a family of boolean constraint system (BCS) nonlocal games. This makes it possible to work with MIP* protocols as boolean constraint systems, and in particular allows us to use a variant of a construction due to Dwork, Feige, Kilian, Naor, and Safra [Crypto '92] which gives a classical MIP protocol for 3SAT with perfect zero knowledge. To show quantum soundness of this classical construction, we develop a toolkit for analyzing quantum soundness of reductions between BCS games, which we expect to be useful more broadly. This toolkit also applies to commuting operator strategies, and our argument shows that every language with a commuting operator BCS protocol has a two prover PZK commuting operator protocol.
{"title":"Two prover perfect zero knowledge for MIP*","authors":"Kieran Mastel, William Slofstra","doi":"arxiv-2404.00926","DOIUrl":"https://doi.org/arxiv-2404.00926","url":null,"abstract":"The recent MIP*=RE theorem of Ji, Natarajan, Vidick, Wright, and Yuen shows\u0000that the complexity class MIP* of multiprover proof systems with entangled\u0000provers contains all recursively enumerable languages. Prior work of Grilo,\u0000Slofstra, and Yuen [FOCS '19] further shows (via a technique called simulatable\u0000codes) that every language in MIP* has a perfect zero knowledge (PZK) MIP*\u0000protocol. The MIP*=RE theorem uses two-prover one-round proof systems, and\u0000hence such systems are complete for MIP*. However, the construction in Grilo,\u0000Slofstra, and Yuen uses six provers, and there is no obvious way to get perfect\u0000zero knowledge with two provers via simulatable codes. This leads to a natural\u0000question: are there two-prover PZK-MIP* protocols for all of MIP*? In this paper, we show that every language in MIP* has a two-prover one-round\u0000PZK-MIP* protocol, answering the question in the affirmative. For the proof, we\u0000use a new method based on a key consequence of the MIP*=RE theorem, which is\u0000that every MIP* protocol can be turned into a family of boolean constraint\u0000system (BCS) nonlocal games. This makes it possible to work with MIP* protocols\u0000as boolean constraint systems, and in particular allows us to use a variant of\u0000a construction due to Dwork, Feige, Kilian, Naor, and Safra [Crypto '92] which\u0000gives a classical MIP protocol for 3SAT with perfect zero knowledge. To show\u0000quantum soundness of this classical construction, we develop a toolkit for\u0000analyzing quantum soundness of reductions between BCS games, which we expect to\u0000be useful more broadly. This toolkit also applies to commuting operator\u0000strategies, and our argument shows that every language with a commuting\u0000operator BCS protocol has a two prover PZK commuting operator protocol.","PeriodicalId":501024,"journal":{"name":"arXiv - CS - Computational Complexity","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140583130","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 develop a new theory of strong subalgebras and linear congruences that are defined globally. Using this theory we provide a new proof of the correctness of Zhuk's algorithm for all tractable CSPs on a finite domain, and therefore a new simplified proof of the CSP Dichotomy Conjecture. Additionally, using the new theory we prove that composing a weak near-unanimity operation of an odd arity $n$ we can derive an $n$-ary operation that is symmetric on all two-element sets. Thus, CSP over a constraint language $Gamma$ on a finite domain is tractable if and only if there exist infinitely many polymorphisms of $Gamma$ that are symmetric on all two-element sets.
{"title":"A simplified proof of the CSP Dichotomy Conjecture and XY-symmetric operations","authors":"Dmitriy Zhuk","doi":"arxiv-2404.01080","DOIUrl":"https://doi.org/arxiv-2404.01080","url":null,"abstract":"We develop a new theory of strong subalgebras and linear congruences that are\u0000defined globally. Using this theory we provide a new proof of the correctness\u0000of Zhuk's algorithm for all tractable CSPs on a finite domain, and therefore a\u0000new simplified proof of the CSP Dichotomy Conjecture. Additionally, using the\u0000new theory we prove that composing a weak near-unanimity operation of an odd\u0000arity $n$ we can derive an $n$-ary operation that is symmetric on all\u0000two-element sets. Thus, CSP over a constraint language $Gamma$ on a finite\u0000domain is tractable if and only if there exist infinitely many polymorphisms of\u0000$Gamma$ that are symmetric on all two-element sets.","PeriodicalId":501024,"journal":{"name":"arXiv - CS - Computational Complexity","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140583149","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 a graph partition problem where we are given a directed acyclic graph (DAG) whose vertices and arcs can be respectively regarded as tasks and dependencies among tasks. The objective of the problem is to minimize the total energy consumed for completing these tasks by assigning the tasks to k heterogeneous machines. We first show that the problem is NP-hard. Then, we present polynomial-time algorithms for two special cases where there are only two machines and where the input DAG is a directed path. Finally, we study a natural variant where there are only two machines with one of them being capable of executing a limited number of tasks. We show that this special case remains computationally hard.
我们研究的是一个图分割问题，即给定一个有向无环图（DAG），其顶点和弧可分别视为任务和任务间的依赖关系。问题的目标是通过将任务分配给 k 台异构机器，使完成这些任务所消耗的总能量最小。我们首先证明了该问题的 NP 难度。然后，我们提出了只有两台机器和输入 DAG 为有向路径的两种特殊情况下的多项式时间算法。最后，我们研究了一种自然变体，即只有两台机器，其中一台只能执行数量有限的任务。我们证明，这种特殊情况仍然难以计算。
{"title":"On the Complexity of Minimizing Energy Consumption of Partitioning DAG Tasks","authors":"Wei Liu, Jian-Jia Chen, Yongjie Yang","doi":"arxiv-2404.01022","DOIUrl":"https://doi.org/arxiv-2404.01022","url":null,"abstract":"We study a graph partition problem where we are given a directed acyclic\u0000graph (DAG) whose vertices and arcs can be respectively regarded as tasks and\u0000dependencies among tasks. The objective of the problem is to minimize the total\u0000energy consumed for completing these tasks by assigning the tasks to k\u0000heterogeneous machines. We first show that the problem is NP-hard. Then, we\u0000present polynomial-time algorithms for two special cases where there are only\u0000two machines and where the input DAG is a directed path. Finally, we study a\u0000natural variant where there are only two machines with one of them being\u0000capable of executing a limited number of tasks. We show that this special case\u0000remains computationally hard.","PeriodicalId":501024,"journal":{"name":"arXiv - CS - Computational Complexity","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140583134","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}
Yuting Fang, Lianna Hambardzumyan, Nathaniel Harms, Pooya Hatami
We prove that the class of communication problems with public-coin randomized constant-cost protocols, called $BPP^0$, does not contain a complete problem. In other words, there is no randomized constant-cost problem $Q in BPP^0$, such that all other problems $P in BPP^0$ can be computed by a constant-cost deterministic protocol with access to an oracle for $Q$. We also show that the $k$-Hamming Distance problems form an infinite hierarchy within $BPP^0$. Previously, it was known only that Equality is not complete for $BPP^0$. We introduce a new technique, using Ramsey theory, that can prove lower bounds against arbitrary oracles in $BPP^0$, and more generally, we show that $k$-Hamming Distance matrices cannot be expressed as a Boolean combination of any constant number of matrices which forbid large Greater-Than subproblems.
{"title":"No Complete Problem for Constant-Cost Randomized Communication","authors":"Yuting Fang, Lianna Hambardzumyan, Nathaniel Harms, Pooya Hatami","doi":"arxiv-2404.00812","DOIUrl":"https://doi.org/arxiv-2404.00812","url":null,"abstract":"We prove that the class of communication problems with public-coin randomized\u0000constant-cost protocols, called $BPP^0$, does not contain a complete problem.\u0000In other words, there is no randomized constant-cost problem $Q in BPP^0$,\u0000such that all other problems $P in BPP^0$ can be computed by a constant-cost\u0000deterministic protocol with access to an oracle for $Q$. We also show that the\u0000$k$-Hamming Distance problems form an infinite hierarchy within $BPP^0$.\u0000Previously, it was known only that Equality is not complete for $BPP^0$. We\u0000introduce a new technique, using Ramsey theory, that can prove lower bounds\u0000against arbitrary oracles in $BPP^0$, and more generally, we show that\u0000$k$-Hamming Distance matrices cannot be expressed as a Boolean combination of\u0000any constant number of matrices which forbid large Greater-Than subproblems.","PeriodicalId":501024,"journal":{"name":"arXiv - CS - Computational Complexity","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140583143","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 prove that Reed-Solomon (RS) codes with random evaluation points are list recoverable up to capacity with optimal output list size, for any input list size. Namely, given an input list size $ell$, a designated rate $R$, and any $varepsilon > 0$, we show that a random RS code is list recoverable from $1-R-varepsilon$ fraction of errors with output list size $L = O(ell/varepsilon)$, for field size $q=exp(ell,1/varepsilon) cdot n^2$. In particular, this shows that random RS codes are list recoverable beyond the ``list recovery Johnson bound''. Such a result was not even known for arbitrary random linear codes. Our technique follows and extends the recent line of work on list decoding of random RS codes, specifically the works of Brakensiek, Gopi, and Makam (STOC 2023), and of Guo and Zhang (FOCS 2023).
{"title":"Random Reed-Solomon Codes are List Recoverable with Optimal List Size","authors":"Dean Doron, S. Venkitesh","doi":"arxiv-2404.00206","DOIUrl":"https://doi.org/arxiv-2404.00206","url":null,"abstract":"We prove that Reed-Solomon (RS) codes with random evaluation points are list\u0000recoverable up to capacity with optimal output list size, for any input list\u0000size. Namely, given an input list size $ell$, a designated rate $R$, and any\u0000$varepsilon > 0$, we show that a random RS code is list recoverable from\u0000$1-R-varepsilon$ fraction of errors with output list size $L =\u0000O(ell/varepsilon)$, for field size $q=exp(ell,1/varepsilon) cdot n^2$. In\u0000particular, this shows that random RS codes are list recoverable beyond the\u0000``list recovery Johnson bound''. Such a result was not even known for arbitrary\u0000random linear codes. Our technique follows and extends the recent line of work\u0000on list decoding of random RS codes, specifically the works of Brakensiek,\u0000Gopi, and Makam (STOC 2023), and of Guo and Zhang (FOCS 2023).","PeriodicalId":501024,"journal":{"name":"arXiv - CS - Computational Complexity","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140583035","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}
Marcel Jackiewicz, Adam Kasperski, Paweł Zieliński
In this paper the recoverable robust shortest path problem is investigated. Discrete budgeted interval uncertainty representation is used to model uncertain second-stage arc costs. The known complexity results for this problem are strengthened. It is shown that it is Sigma_3^p-hard for the arc exclusion and the arc symmetric difference neighborhoods. Furthermore, it is also proven that the inner adversarial problem for these neighborhoods is Pi_2^p-hard.
{"title":"Computational Complexity of the Recoverable Robust Shortest Path Problem with Discrete Recourse","authors":"Marcel Jackiewicz, Adam Kasperski, Paweł Zieliński","doi":"arxiv-2403.20000","DOIUrl":"https://doi.org/arxiv-2403.20000","url":null,"abstract":"In this paper the recoverable robust shortest path problem is investigated.\u0000Discrete budgeted interval uncertainty representation is used to model\u0000uncertain second-stage arc costs. The known complexity results for this problem\u0000are strengthened. It is shown that it is Sigma_3^p-hard for the arc exclusion\u0000and the arc symmetric difference neighborhoods. Furthermore, it is also proven\u0000that the inner adversarial problem for these neighborhoods is Pi_2^p-hard.","PeriodicalId":501024,"journal":{"name":"arXiv - CS - Computational Complexity","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140583137","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 consider the task of locally correcting, and locally list-correcting, multivariate linear functions over the domain ${0,1}^n$ over arbitrary fields and more generally Abelian groups. Such functions form error-correcting codes of relative distance $1/2$ and we give local-correction algorithms correcting up to nearly $1/4$-fraction errors making $widetilde{mathcal{O}}(log n)$ queries. This query complexity is optimal up to $mathrm{poly}(loglog n)$ factors. We also give local list-correcting algorithms correcting $(1/2 - varepsilon)$-fraction errors with $widetilde{mathcal{O}}_{varepsilon}(log n)$ queries. These results may be viewed as natural generalizations of the classical work of Goldreich and Levin whose work addresses the special case where the underlying group is $mathbb{Z}_2$. By extending to the case where the underlying group is, say, the reals, we give the first non-trivial locally correctable codes (LCCs) over the reals (with query complexity being sublinear in the dimension (also known as message length)). The central challenge in constructing the local corrector is constructing ``nearly balanced vectors'' over ${-1,1}^n$ that span $1^n$ -- we show how to construct $mathcal{O}(log n)$ vectors that do so, with entries in each vector summing to $pm1$. The challenge to the local-list-correction algorithms, given the local corrector, is principally combinatorial, i.e., in proving that the number of linear functions within any Hamming ball of radius $(1/2-varepsilon)$ is $mathcal{O}_{varepsilon}(1)$. Getting this general result covering every Abelian group requires integrating a variety of known methods with some new combinatorial ingredients analyzing the structural properties of codewords that lie within small Hamming balls.
{"title":"Local Correction of Linear Functions over the Boolean Cube","authors":"Prashanth Amireddy, Amik Raj Behera, Manaswi Paraashar, Srikanth Srinivasan, Madhu Sudan","doi":"arxiv-2403.20305","DOIUrl":"https://doi.org/arxiv-2403.20305","url":null,"abstract":"We consider the task of locally correcting, and locally list-correcting,\u0000multivariate linear functions over the domain ${0,1}^n$ over arbitrary fields\u0000and more generally Abelian groups. Such functions form error-correcting codes\u0000of relative distance $1/2$ and we give local-correction algorithms correcting\u0000up to nearly $1/4$-fraction errors making $widetilde{mathcal{O}}(log n)$\u0000queries. This query complexity is optimal up to $mathrm{poly}(loglog n)$\u0000factors. We also give local list-correcting algorithms correcting $(1/2 -\u0000varepsilon)$-fraction errors with $widetilde{mathcal{O}}_{varepsilon}(log\u0000n)$ queries. These results may be viewed as natural generalizations of the classical work\u0000of Goldreich and Levin whose work addresses the special case where the\u0000underlying group is $mathbb{Z}_2$. By extending to the case where the\u0000underlying group is, say, the reals, we give the first non-trivial locally\u0000correctable codes (LCCs) over the reals (with query complexity being sublinear\u0000in the dimension (also known as message length)). The central challenge in constructing the local corrector is constructing\u0000``nearly balanced vectors'' over ${-1,1}^n$ that span $1^n$ -- we show how to\u0000construct $mathcal{O}(log n)$ vectors that do so, with entries in each vector\u0000summing to $pm1$. The challenge to the local-list-correction algorithms, given\u0000the local corrector, is principally combinatorial, i.e., in proving that the\u0000number of linear functions within any Hamming ball of radius\u0000$(1/2-varepsilon)$ is $mathcal{O}_{varepsilon}(1)$. Getting this general\u0000result covering every Abelian group requires integrating a variety of known\u0000methods with some new combinatorial ingredients analyzing the structural\u0000properties of codewords that lie within small Hamming balls.","PeriodicalId":501024,"journal":{"name":"arXiv - CS - Computational Complexity","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140583043","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}
Sabee Grewal, Vishnu Iyer, William Kretschmer, Daniel Liang
We show that any pseudoentangled state ensemble with a gap of $t$ bits of entropy requires $Omega(t)$ non-Clifford gates to prepare. This bound is tight up to polylogarithmic factors if linear-time quantum-secure pseudorandom functions exist. Our result follows from a polynomial-time algorithm to estimate the entanglement entropy of a quantum state across any cut of qubits. When run on an $n$-qubit state that is stabilized by at least $2^{n-t}$ Pauli operators, our algorithm produces an estimate that is within an additive factor of $frac{t}{2}$ bits of the true entanglement entropy.
{"title":"Pseudoentanglement Ain't Cheap","authors":"Sabee Grewal, Vishnu Iyer, William Kretschmer, Daniel Liang","doi":"arxiv-2404.00126","DOIUrl":"https://doi.org/arxiv-2404.00126","url":null,"abstract":"We show that any pseudoentangled state ensemble with a gap of $t$ bits of\u0000entropy requires $Omega(t)$ non-Clifford gates to prepare. This bound is tight\u0000up to polylogarithmic factors if linear-time quantum-secure pseudorandom\u0000functions exist. Our result follows from a polynomial-time algorithm to\u0000estimate the entanglement entropy of a quantum state across any cut of qubits.\u0000When run on an $n$-qubit state that is stabilized by at least $2^{n-t}$ Pauli\u0000operators, our algorithm produces an estimate that is within an additive factor\u0000of $frac{t}{2}$ bits of the true entanglement entropy.","PeriodicalId":501024,"journal":{"name":"arXiv - CS - Computational Complexity","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140583042","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：