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1. Depth-First Search in Directed Planar Graphs, Revisited

   https://doi.org/10.4230/LIPIcs.MFCS.2021.7  

   Allender, Eric; Chauhan, Archit; Datta, Samir (January 2021, Leibniz international proceedings in informatics)

   Bonchi, Filippo; Puglisi, Simon J. (Ed.)

   We present an algorithm for constructing a depth-first search tree in planar digraphs; the algorithm can be implemented in the complexity class AC^1(UL ∩ co-UL), which is contained in AC^2. Prior to this (for more than a quarter-century), the fastest uniform deterministic parallel algorithm for this problem was O(log^10 n) (corresponding to the complexity class AC^10 ⊆ NC^11). We also consider the problem of computing depth-first search trees in other classes of graphs, and obtain additional new upper bounds. 

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2. Exponential separation between shallow quantum circuits and unbounded fan-in shallow classical circuits

   https://doi.org/10.1145/3313276.3316404  

   Watts, Adam Bene; Kothari, Robin; Schaeffer, Luke; Tal, Avishay (June 2019, Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing)

   Recently, Bravyi, Gosset, and Konig (Science, 2018) exhibited a search problem called the 2D Hidden Linear Function (2D HLF) problem that can be solved exactly by a constant-depth quantum circuit using bounded fan-in gates (or QNC^0 circuits), but cannot be solved by any constant-depth classical circuit using bounded fan-in AND, OR, and NOT gates (or NC^0 circuits). In other words, they exhibited a search problem in QNC^0 that is not in NC^0. We strengthen their result by proving that the 2D HLF problem is not contained in AC^0, the class of classical, polynomial-size, constant-depth circuits over the gate set of unbounded fan-in AND and OR gates, and NOT gates. We also supplement this worst-case lower bound with an average-case result: There exists a simple distribution under which any AC^0 circuit (even of nearly exponential size) has exponentially small correlation with the 2D HLF problem. Our results are shown by constructing a new problem in QNC^0, which we call the Parity Halving Problem, which is easier to work with. We prove our AC^0 lower bounds for this problem, and then show that it reduces to the 2D HLF problem. 

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3. Cryptographic Hardness Under Projections for Time-Bounded Kolmogorov Complexity

   https://doi.org/10.4230/LIPIcs.ISAAC.2021.54  

   Allender, Eric; Gouwar, John; Hirahara, Shuichi; Robelle, Caleb (November 2021, Leibniz international proceedings in informatics)

   Ahn, Hee-Kap; Sadakane, Kunihiko (Ed.)

   A version of time-bounded Kolmogorov complexity, denoted KT, has received attention in the past several years, due to its close connection to circuit complexity and to the Minimum Circuit Size Problem MCSP. Essentially all results about the complexity of MCSP hold also for MKTP (the problem of computing the KT complexity of a string). Both MKTP and MCSP are hard for SZK (Statistical Zero Knowledge) under BPP-Turing reductions; neither is known to be NP-complete. Recently, some hardness results for MKTP were proved that are not (yet) known to hold for MCSP. In particular, MKTP is hard for DET (a subclass of P) under nonuniform ≤^{NC^0}_m reductions. In this paper, we improve this, to show that the complement of MKTP is hard for the (apparently larger) class NISZK_L under not only ≤^{NC^0}_m reductions but even under projections. Also, the complement of MKTP is hard for NISZK under ≤^{P/poly}_m reductions. Here, NISZK is the class of problems with non-interactive zero-knowledge proofs, and NISZK_L is the non-interactive version of the class SZK_L that was studied by Dvir et al. As an application, we provide several improved worst-case to average-case reductions to problems in NP, and we obtain a new lower bound on MKTP (which is currently not known to hold for MCSP). 

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4. The complexity of NISQ

   https://doi.org/10.1038/s41467-023-41217-6  

   Chen, Sitan; Cotler, Jordan; Huang, Hsin-Yuan; Li, Jerry (December 2023, Nature Communications)

   The recent proliferation of NISQ devices has made it imperative to understand their power. In this work, we define and study the complexity class , which encapsulates problems that can be efficiently solved by a classical computer with access to noisy quantum circuits. We establish super-polynomial separations in the complexity among classical computation, , and fault-tolerant quantum computation to solve some problems based on modifications of Simon’s problems. We then consider the power of for three well-studied problems. For unstructured search, we prove that cannot achieve a Grover-like quadratic speedup over classical computers. For the Bernstein-Vazirani problem, we show that only needs a number of queries logarithmic in what is required for classical computers. Finally, for a quantum state learning problem, we prove that is exponentially weaker than classical computers with access to noiseless constant-depth quantum circuits. 

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5. Pseudodeterminism: promises and lowerbounds

   https://doi.org/10.1145/3519935.3520043  

   Dixon, Peter; Pavan, A.; Woude, Jason Vander; Vinodchandran, N. V. (July 2022, Symposium on Theory of Computing (STOC))

   Stefano Leonardi and Anupam Gupta (Ed.)

   A probabilistic algorithm A is pseudodeterministic if, on every input, there exists a canonical value that is output with high probability. If the algorithm outputs one of k canonical values with high probability, then it is called a k-pseudodeterministic algorithm. In the study of pseudodeterminism, the Acceptance Probability Estimation Problem (APEP), which is to additively approximate the acceptance probability of a Boolean circuit, is emerging as a central computational problem. This problem admits a 2-pseudodeterministic algorithm. Recently, it was shown that a pseudodeterministic algorithm for this problem would imply that any multi-valued function that admits a k-pseudodeterministic algorithm for a constant k (including approximation algorithms) also admits a pseudodeterministic algorithm (Dixon, Pavan, Vinodchandran; ITCS 2021). The contribution of the present work is two-fold. First, as our main conceptual contribution, we establish that the existence of a pseudodeterministic algorithm for APEP is fundamentally related to the gap between probabilistic promise classes and the corresponding standard complexity classes. In particular, we show the following equivalence: APEP has a pseudodeterministic approximation algorithm if and only if every promise problem in PromiseBPP has a solution in BPP. A conceptual interpretation of this equivalence is that the algorithmic gap between 2-pseudodeterminism and pseudodeterminism is equivalent to the gap between PromiseBPP and BPP. Based on this connection, we show that designing pseudodeterministic algorithms for APEP leads to the solution of some open problems in complexity theory, including new Boolean circuit lower bounds. This equivalence also explains how multi-pseudodeterminism is connected to problems in SearchBPP. In particular, we show that if APEP has a pseudodeterministic algorithm, then every problem that admits a k(n)-pseudodeterministic algorithm (for any polynomial k) is in SearchBPP and admits a pseudodeterministic algorithm. Motivated by this connection, we also explore its connection to probabilistic search problems and establish that APEP is complete for certain notions of search problems in the context of pseudodeterminism. Our second contribution is establishing query complexity lower bounds for multi-pseudodeterministic computations. We prove that for every k ≥ 1, there exists a problem whose (k+1)-pseudodeterministic query complexity, in the uniform query model, is O(1) but has a k-pseudodeterministic query complexity of Ω(n), even in the more general nonadaptive query model. A key contribution of this part of the work is the utilization of Sperner’s lemma in establishing query complexity lower bounds. 

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