IA Scholar Query: Nisan-Wigderson Pseudorandom Generators for Circuits with Polynomial Threshold Gates.
https://scholar.archive.org/
Internet Archive Scholar query results feedeninfo@archive.orgMon, 11 Jul 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440Improved Pseudorandom Generators for AC⁰ Circuits
https://scholar.archive.org/work/3zsnecmcr5earcpzz6jsykixb4
We give PRG for depth-d, size-m AC⁰ circuits with seed length O(log^{d-1}(m)log(m/ε)log log(m)). Our PRG improves on previous work [Luca Trevisan and Tongke Xue, 2013; Rocco A. Servedio and Li-Yang Tan, 2019; Zander Kelley, 2021] from various aspects. It has optimal dependence on 1/ε and is only one "log log(m)" away from the lower bound barrier. For the case of d = 2, the seed length tightly matches the best-known PRG for CNFs [Anindya De et al., 2010; Avishay Tal, 2017]. There are two technical ingredients behind our new result; both of them might be of independent interest. First, we use a partitioning-based approach to construct PRGs based on restriction lemmas for AC⁰. Previous works [Luca Trevisan and Tongke Xue, 2013; Rocco A. Servedio and Li-Yang Tan, 2019; Zander Kelley, 2021] usually built PRGs on the Ajtai-Wigderson framework [Miklós Ajtai and Avi Wigderson, 1989]. Compared with them, the partitioning approach avoids the extra "log(n)" factor that usually arises from the Ajtai-Wigderson framework, allowing us to get the almost-tight seed length. The partitioning approach is quite general, and we believe it can help design PRGs for classes beyond constant-depth circuits. Second, improving and extending [Luca Trevisan and Tongke Xue, 2013; Rocco A. Servedio and Li-Yang Tan, 2019; Zander Kelley, 2021], we prove a full derandomization of the powerful multi-switching lemma [Johan Håstad, 2014]. We show that one can use a short random seed to sample a restriction, such that a family of DNFs simultaneously simplifies under the restriction with high probability. This answers an open question in [Zander Kelley, 2021]. Previous derandomizations were either partial (that is, they pseudorandomly choose variables to restrict, and then fix those variables to truly-random bits) or had sub-optimal seed length. In our application, having a fully-derandomized switching lemma is crucial, and the randomness-efficiency of our derandomization allows us to get an almost-tight seed length.Xin Lyu, Shachar Lovettwork_3zsnecmcr5earcpzz6jsykixb4Mon, 11 Jul 2022 00:00:00 GMTThe Acrobatics of BQP
https://scholar.archive.org/work/k4o6xtbpinggbfo75ecnugayri
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) can be remarkably decoupled from that of classical complexity classes like 𝖭𝖯. Specifically: -There exists an oracle relative to which 𝖭𝖯^𝖡𝖰𝖯⊄𝖡𝖰𝖯^𝖯𝖧, resolving a 2005 problem of Fortnow. As a corollary, there exists an oracle relative to which 𝖯=𝖭𝖯 but 𝖡𝖰𝖯≠𝖰𝖢𝖬𝖠. -Conversely, there exists an oracle relative to which 𝖡𝖰𝖯^𝖭𝖯⊄𝖯𝖧^𝖡𝖰𝖯. -Relative to a random oracle, 𝖯𝖯=𝖯𝗈𝗌𝗍𝖡𝖰𝖯 is not contained in the "𝖰𝖬𝖠 hierarchy" 𝖰𝖬𝖠^𝖰𝖬𝖠^𝖰𝖬𝖠^⋯. -Relative to a random oracle, Σ_k+1^𝖯⊄𝖡𝖰𝖯^Σ_k^𝖯 for every k. -There exists an oracle relative to which 𝖡𝖰𝖯=𝖯^# 𝖯 and yet 𝖯𝖧 is infinite. -There exists an oracle relative to which 𝖯=𝖭𝖯≠𝖡𝖰𝖯=𝖯^# 𝖯. To achieve these results, we build on the 2018 achievement by Raz and Tal of an oracle relative to which 𝖡𝖰𝖯⊄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 𝖠𝖢^0 circuits, and a (provable) analogue of the Aaronson-Ambainis Conjecture for sparse oracles.Scott Aaronson, DeVon Ingram, William Kretschmerwork_k4o6xtbpinggbfo75ecnugayriMon, 28 Feb 2022 00:00:00 GMTQuantum learning algorithms imply circuit lower bounds
https://scholar.archive.org/work/kfgtaee2pbbfbozediiqye6y64
How to cite: Please refer to published version for the most recent bibliographic citation information.Srinivasan Arunachalam, Alex B. Grilo, Tom Gur, Igor C. Oliveira, Aarthi Sundaramwork_kfgtaee2pbbfbozediiqye6y64Improved Pseudorandom Generators for AC 0 Circuits
https://scholar.archive.org/work/a2wgan4fxnenzhdvtafpakl2iy
We give PRG for depth-d, size-m AC 0 circuits with seed length O(log d−1 (m) log(m/ε) log log(m)). Our PRG improves on previous work [TX13, ST19, Kel21] from various aspects. It has optimal dependence on 1 ε and is only one "log log(m)" away from the lower bound barrier. For the case of d = 2, the seed length tightly matches the best-known PRG for CNFs [DETT10, Tal17]. There are two technical ingredients behind our new result; both of them might be of independent interest. First, we develop a "partitioning-based" approach to construct PRGs based on restriction lemmas for AC 0 . Previous works [TX13, ST19, Kel21] usually built PRGs on the Ajtai-Wigderson framework [AW89]. Compared with them, our new approach avoids the extra "log(n)" factor that usually arises from the Ajtai-Wigderson framework, allowing us to get the almost-tight seed length. Our partitioning-based approach is quite general, and we believe it can help design PRGs for classes beyond constant-depth circuits. Second, improving and extending [TX13, ST19, Kel21], we prove a full derandomization of the powerful multi-switching lemma [Hås14] . We show that one can use a short random seed to sample a restriction, such that a family of DNFs simultaneously simplifies under the restriction with high probability. This answers an open question in [Kel21] . Previous derandomizations were either partial (that is, they pseudorandomly choose variables to restrict, and then fix those variables to truly-random bits) or had sub-optimal seed length. In our application, having a fullyderandomized switching lemma is crucial, and the randomness-efficiency of our derandomization allows us to get an almost-tight seed length.Xin Lyuwork_a2wgan4fxnenzhdvtafpakl2iyQuantum learning algorithms imply circuit lower bounds
https://scholar.archive.org/work/o36ohi3qdjhj3mkpffin67w2b4
We establish the first general connection between the design of quantum algorithms and circuit lower bounds. Specifically, let ℭ be a class of polynomial-size concepts, and suppose that ℭ can be PAC-learned with membership queries under the uniform distribution with error 1/2 - γ by a time T quantum algorithm. We prove that if γ^2 · T ≪ 2^n/n, then 𝖡𝖰𝖤⊈ℭ, where 𝖡𝖰𝖤 = 𝖡𝖰𝖳𝖨𝖬𝖤[2^O(n)] is an exponential-time analogue of 𝖡𝖰𝖯. This result is optimal in both γ and T, since it is not hard to learn any class ℭ of functions in (classical) time T = 2^n (with no error), or in quantum time T = 𝗉𝗈𝗅𝗒(n) with error at most 1/2 - Ω(2^-n/2) via Fourier sampling. In other words, even a marginal improvement on these generic learning algorithms would lead to major consequences in complexity theory. Our proof builds on several works in learning theory, pseudorandomness, and computational complexity, and crucially, on a connection between non-trivial classical learning algorithms and circuit lower bounds established by Oliveira and Santhanam (CCC 2017). Extending their approach to quantum learning algorithms turns out to create significant challenges. To achieve that, we show among other results how pseudorandom generators imply learning-to-lower-bound connections in a generic fashion, construct the first conditional pseudorandom generator secure against uniform quantum computations, and extend the local list-decoding algorithm of Impagliazzo, Jaiswal, Kabanets and Wigderson (SICOMP 2010) to quantum circuits via a delicate analysis. We believe that these contributions are of independent interest and might find other applications.Srinivasan Arunachalam, Alex B. Grilo, Tom Gur, Igor C. Oliveira, Aarthi Sundaramwork_o36ohi3qdjhj3mkpffin67w2b4Wed, 01 Dec 2021 00:00:00 GMT