74 research outputs found

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    Bounded Relativization

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    Relativization is one of the most fundamental concepts in complexity theory, which explains the difficulty of resolving major open problems. In this paper, we propose a weaker notion of relativization called bounded relativization. For a complexity class ?, we say that a statement is ?-relativizing if the statement holds relative to every oracle ? ? ?. It is easy to see that every result that relativizes also ?-relativizes for every complexity class ?. On the other hand, we observe that many non-relativizing results, such as IP = PSPACE, are in fact PSPACE-relativizing. First, we use the idea of bounded relativization to obtain new lower bound results, including the following nearly maximum circuit lower bound: for every constant ? > 0, BPE^{MCSP}/2^{?n} ? SIZE[2?/n]. We prove this by PSPACE-relativizing the recent pseudodeterministic pseudorandom generator by Lu, Oliveira, and Santhanam (STOC 2021). Next, we study the limitations of PSPACE-relativizing proof techniques, and show that a seemingly minor improvement over the known results using PSPACE-relativizing techniques would imply a breakthrough separation NP ? L. For example: - Impagliazzo and Wigderson (JCSS 2001) proved that if EXP ? BPP, then BPP admits infinitely-often subexponential-time heuristic derandomization. We show that their result is PSPACE-relativizing, and that improving it to worst-case derandomization using PSPACE-relativizing techniques implies NP ? L. - Oliveira and Santhanam (STOC 2017) recently proved that every dense subset in P admits an infinitely-often subexponential-time pseudodeterministic construction, which we observe is PSPACE-relativizing. Improving this to almost-everywhere (pseudodeterministic) or (infinitely-often) deterministic constructions by PSPACE-relativizing techniques implies NP ? L. - Santhanam (SICOMP 2009) proved that pr-MA does not have fixed polynomial-size circuits. This lower bound can be shown PSPACE-relativizing, and we show that improving it to an almost-everywhere lower bound using PSPACE-relativizing techniques implies NP ? L. In fact, we show that if we can use PSPACE-relativizing techniques to obtain the above-mentioned improvements, then PSPACE ? EXPH. We obtain our barrier results by constructing suitable oracles computable in EXPH relative to which these improvements are impossible

    Hybrid quantum-classical and quantum-inspired classical algorithms for solving banded circulant linear systems

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    Solving linear systems is of great importance in numerous fields. In particular, circulant systems are especially valuable for efficiently finding numerical solutions to physics-related differential equations. Current quantum algorithms like HHL or variational methods are either resource-intensive or may fail to find a solution. We present an efficient algorithm based on convex optimization of combinations of quantum states to solve for banded circulant linear systems whose non-zero terms are within distance KK of the main diagonal. By decomposing banded circulant matrices into cyclic permutations, our approach produces approximate solutions to such systems with a combination of quantum states linear to KK, significantly improving over previous convergence guarantees, which require quantum states exponential to KK. We propose a hybrid quantum-classical algorithm using the Hadamard test and the quantum Fourier transform as subroutines and show its PromiseBQP-hardness. Additionally, we introduce a quantum-inspired algorithm with similar performance given sample and query access. We validate our methods with classical simulations and actual IBM quantum computer implementation, showcasing their applicability for solving physical problems such as heat transfer.Comment: 21 pages, 12 figure

    Quantum fine-grained complexity

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    Covid Conspiracy Theories in Global Perspective

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    Covid Conspiracy Theories in Global Perspective examines how conspiracy theories and related forms of misinformation and disinformation about the Covid-19 pandemic have circulated widely around the world. Covid conspiracy theories have attracted considerable attention from researchers, journalists, and politicians, not least because conspiracy beliefs have the potential to negatively affect adherence to public health measures. While most of this focus has been on the United States and Western Europe, this collection provides a unique global perspective on the emergence and development of conspiracy theories through a series of case studies. The chapters have been commissioned by recognized experts on area studies and conspiracy theories. The chapters present case studies on how Covid conspiracism has played out (some focused on a single country, others on regions), using a range of methods from a variety of disciplinary perspectives, including history, politics, sociology, anthropology, and psychology. Collectively, the authors reveal that, although there are many narratives that have spread virally, they have been adapted for different uses and take on different meanings in local contexts. This volume makes an important contribution to the rapidly expanding field of academic conspiracy theory studies, as well as being of interest to those working in the media, regulatory agencies, and civil society organizations, who seek to better understand the problem of how and why conspiracy theories spread

    On Oracles and Algorithmic Methods for Proving Lower Bounds

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    This paper studies the interaction of oracles with algorithmic approaches to proving circuit complexity lower bounds, establishing new results on two different kinds of questions. 1) We revisit some prominent open questions in circuit lower bounds, and provide a clean way of viewing them as circuit upper bound questions. Let Missing-String be the (total) search problem of producing a string that does not appear in a given list L containing M bit-strings of length N, where M < 2?. We show in a generic way how algorithms and uniform circuits (from restricted classes) for Missing-String imply complexity lower bounds (and in some cases, the converse holds as well). We give a local algorithm for Missing-String, which can compute any desired output bit making very few probes into the input, when the number of strings M is small enough. We apply this to prove a new nearly-optimal (up to oracles) time hierarchy theorem with advice. We show that the problem of constructing restricted uniform circuits for Missing-String is essentially equivalent to constructing functions without small non-uniform circuits, in a relativizing way. For example, we prove that small uniform depth-3 circuits for Missing-String would imply exponential circuit lower bounds for ?? EXP, and depth-3 lower bounds for Missing-String would imply non-trivial circuits (relative to an oracle) for ?? EXP problems. Both conclusions are longstanding open problems in circuit complexity. 2) It has been known since Impagliazzo, Kabanets, and Wigderson [JCSS 2002] that generic derandomizations improving subexponentially over exhaustive search would imply lower bounds such as NEXP ? ? ?/poly. Williams [SICOMP 2013] showed that Circuit-SAT algorithms running barely faster than exhaustive search would imply similar lower bounds. The known proofs of such results do not relativize (they use techniques from interactive proofs/PCPs). However, it has remained open whether there is an oracle under which the generic implications from circuit-analysis algorithms to circuit lower bounds fail. Building on an oracle of Fortnow, we construct an oracle relative to which the circuit approximation probability problem (CAPP) is in ?, yet EXP^{NP} has polynomial-size circuits. We construct an oracle relative to which SAT can be solved in "half-exponential" time, yet exponential time (EXP) has polynomial-size circuits. Improving EXP to NEXP would give an oracle relative to which ?? ? has "half-exponential" size circuits, which is open. (Recall it is known that ?? ? is not in "sub-half-exponential" size, and the proof relativizes.) Moreover, the running time of the SAT algorithm cannot be improved: relative to all oracles, if SAT is in "sub-half-exponential" time then EXP does not have polynomial-size circuits

    What is in# P and what is not?

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    For several classical nonnegative integer functions, we investigate if they are members of the counting complexity class #P or not. We prove #P membership in surprising cases, and in other cases we prove non-membership, relying on standard complexity assumptions or on oracle separations. We initiate the study of the polynomial closure properties of #P on affine varieties, i.e., if all problem instances satisfy algebraic constraints. This is directly linked to classical combinatorial proofs of algebraic identities and inequalities. We investigate #TFNP and obtain oracle separations that prove the strict inclusion of #P in all standard syntactic subclasses of #TFNP-1

    A Relativization Perspective on Meta-Complexity

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    Meta-complexity studies the complexity of computational problems about complexity theory, such as the Minimum Circuit Size Problem (MCSP) and its variants. We show that a relativization barrier applies to many important open questions in meta-complexity. We give relativized worlds where: 1) MCSP can be solved in deterministic polynomial time, but the search version of MCSP cannot be solved in deterministic polynomial time, even approximately. In contrast, Carmosino, Impagliazzo, Kabanets, Kolokolova [CCC'16] gave a randomized approximate search-to-decision reduction for MCSP with a relativizing proof. 2) The complexities of MCSP[2^{n/2}] and MCSP[2^{n/4}] are different, in both worst-case and average-case settings. Thus the complexity of MCSP is not "robust" to the choice of the size function. 3) Levin’s time-bounded Kolmogorov complexity Kt(x) can be approximated to a factor (2+ε) in polynomial time, for any ε > 0. 4) Natural proofs do not exist, and neither do auxiliary-input one-way functions. In contrast, Santhanam [ITCS'20] gave a relativizing proof that the non-existence of natural proofs implies the existence of one-way functions under a conjecture about optimal hitting sets. 5) DistNP does not reduce to GapMINKT by a family of "robust" reductions. This presents a technical barrier for solving a question of Hirahara [FOCS'20]

    The Acrobatics of BQP

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    One can fix the randomness used by a randomized algorithm, but there is no analogous notion of fixing the quantumness used by a quantum algorithm. Underscoring this fundamental difference, we show that, in the black-box setting, the behavior of quantum polynomial-time (BQP\mathsf{BQP}) can be remarkably decoupled from that of classical complexity classes like NP\mathsf{NP}. Specifically: -There exists an oracle relative to which NPBQP⊄BQPPH\mathsf{NP^{BQP}}\not\subset\mathsf{BQP^{PH}}, resolving a 2005 problem of Fortnow. As a corollary, there exists an oracle relative to which P=NP\mathsf{P}=\mathsf{NP} but BQP≠QCMA\mathsf{BQP}\neq\mathsf{QCMA}. -Conversely, there exists an oracle relative to which BQPNP⊄PHBQP\mathsf{BQP^{NP}}\not\subset\mathsf{PH^{BQP}}. -Relative to a random oracle, PP=PostBQP\mathsf{PP}=\mathsf{PostBQP} is not contained in the "QMA\mathsf{QMA} hierarchy" QMAQMAQMA⋯\mathsf{QMA}^{\mathsf{QMA}^{\mathsf{QMA}^{\cdots}}}. -Relative to a random oracle, Σk+1P⊄BQPΣkP\mathsf{\Sigma}_{k+1}^\mathsf{P}\not\subset\mathsf{BQP}^{\mathsf{\Sigma}_{k}^\mathsf{P}} for every kk. -There exists an oracle relative to which BQP=P#P\mathsf{BQP}=\mathsf{P^{\# P}} and yet PH\mathsf{PH} is infinite. -There exists an oracle relative to which P=NP≠BQP=P#P\mathsf{P}=\mathsf{NP}\neq\mathsf{BQP}=\mathsf{P^{\# P}}. To achieve these results, we build on the 2018 achievement by Raz and Tal of an oracle relative to which BQP⊄PH\mathsf{BQP}\not \subset \mathsf{PH}, and associated results about the Forrelation problem. We also introduce new tools that might be of independent interest. These include a "quantum-aware" version of the random restriction method, a concentration theorem for the block sensitivity of AC0\mathsf{AC^0} circuits, and a (provable) analogue of the Aaronson-Ambainis Conjecture for sparse oracles.Comment: 63 pages. V2: various writing improvement
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