Bounded-Depth Circuits Cannot Sample Good Codes

Lovett, Shachar; Viola, Emanuele

doi:10.1007/s00037-012-0039-3

Cited by 26 publications

(32 citation statements)

References 20 publications

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“…Last, the main body in [ASTS + 03] studies a bounded-error generation, and showed an exponential separation (O(log n) versus Ω( √ n)), while ours aims to generate the exact target correlation, and showed an "infinite" separation (1 versus log 2 n unconditionally, and 1 versus n assuming a conjecture). Studies of computational issues of probabilistic distributions instead of Boolean functions has recently be advocated by Viola [Vio10,LV10]. It is our hope that studies of the correlation complexity of distributions later help to sharpen our understandings of various complexity questions for Boolean functions.…”

Section: More Related Workmentioning

confidence: 99%

Quantum strategic game theory

Zhang

2012

Proceedings of the 3rd Innovations in Theoretical Computer Science Conference

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We propose a simple yet rich model to extend strategic games to the quantum setting, in which we define quantum Nash and correlated equilibria and study the relations between classical and quantum equilibria. Unlike all previous work that focused on qualitative questions on specific games of very small sizes, we quantitatively address the following fundamental question for general games of growing sizes:How much "advantage" can playing quantum strategies provide, if any?Two measures of the advantage are studied.1. A natural measure is the increase of payoff. We consider natural mappings between classical and quantum states, and study how well those mappings preserve the equilibrium properties. Among other results, we exhibit a correlated equilibrium p whose quantum superposition counterpart s p(s)|s is far from being a quantum correlated equilibrium; actually a player can increase her payoff from almost 0 to almost 1 in a [0, 1]-normalized game. We achieve this by a tensor product construction on carefully designed base cases.2. Another measure is the hardness of generating correlated equilibria, for which we propose to study correlation complexity, a new complexity measure for correlation generation. We show that there are n-bit correlated equilibria which can be generated by only one EPR pair followed by local operation (without communication), but need at least log 2 (n) classical shared random bits plus communication. The randomized lower bound can be improved to n, the best possible, assuming (even a much weaker version of) a recent conjecture in linear algebra. We believe that the correlation complexity, as a complexity-theoretical counterpart of the celebrated Bell's inequality, has independent interest in both physics and computational complexity theory and deserves more explorations.

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Section: More Related Workmentioning

confidence: 99%

Quantum strategic game theory

Zhang

2012

Proceedings of the 3rd Innovations in Theoretical Computer Science Conference

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“…It follows immediately from bounds on the noise-sensitivity of small AC 0 circuits [20,6] that small AC 0 circuits cannot compute good codes. This result was generalized in [32,21] (cf. [5]).…”

Section: Introductionmentioning

confidence: 75%

Tight bounds on computing error-correcting codes by bounded-depth circuits with arbitrary gates

Gál

Hansen

Koucký

et al. 2012

Proceedings of the Forty-Fourth Annual ACM Symposium on Theory of Computing

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We bound the minimum number w of wires needed to compute any (asymptotically good) error-correcting code C : {0, 1} Ω(n) → {0, 1} n with minimum distance Ω(n), using unbounded fan-in circuits of depth d with arbitrary gates. Our main results are:, and the * operation gives how many times one has to iterate the function λ i to reach a value at most 1 from the argument n.(4) If d = lg * n then w = O(n).For depth d = 2, our Ω(n(lg n/ lg lg n) 2 ) lower bound gives the largest known lower bound for computing any linear map. The upper bounds imply that a (necessarily * A preliminary version of this paper appeared in: A. Gál, K. A. Hansen, M. Koucký, P. Pudlák, E. Viola. Tight bounds on computing error-correcting codes by bounded-depth circuits with arbitrary gates. 1 dense) generator matrix for our code can be written as the product of two sparse matrices. Using known techniques we also obtain similar (but not tight) bounds for computing pairwise-independent hash functions.Our lower bounds are based on a superconcentrator-like condition that the graphs of circuits computing good codes must satisfy. This condition is provably intermediate between superconcentrators and their weakenings considered before.

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“…We were not able to do so, and we raise this as another challenge. (Some recent progress on this question appears in [LV10]. )…”

Section: Definition 12 a Function F : {0 1} → {0 1}mentioning

confidence: 99%

“…Recently, Lovett and the author [LV10] prove that small AC 0 circuits cannot generate the uniform distribution over any good error-correcting codes. This result does not solve Challenge 1.1 -it does not apply to distributions like (X, b(X)) -although it does answer a question asked in a preliminary version of this work.…”

Section: Definition 12 a Function F : {0 1} → {0 1}mentioning

confidence: 99%

The Complexity of Distributions

Viola¹

2012

SIAM J. Comput.

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Complexity theory typically studies the complexity of computing a function h(x) : {0, 1} m → {0, 1} n of a given input x. A few works have suggested to study the complexity of generating -or sampling -the distribution h(x) for uniform x, given random bits. We further advocate this study, with a new emphasis on lower bounds for restricted computational models. Our main results are:1. Any function f : {0, 1} → {0, 1} n such that (i) each output bit f i depends on o(log n) input bits, and (ii) ≤ log 2 n αn + n 0.99 , has output distribution f (U ) at statistical distance ≥ 1 − 1/n 0.49 from the uniform distribution over n-bit strings of hamming weight αn. We also prove lower bounds for generating (X, b(X)) for boolean b, and in the case in which each bit f i is a small-depth decision tree. These lower bounds seem to be the first of their kind; the proofs use anticoncentration results for the sum of random variables.2. Lower bounds for generating distributions imply succinct data structures lower bounds. As a corollary of (1), we obtain the first lower bound for the membership problem of representing a set S ⊆ [n] of size αn, in the case where 1/α is a power of 2: If queries "i ∈ S?" are answered by non-adaptively probing o(log n) bits, then the representation uses ≥ log 2 n αn + Ω(log n) bits. 3. Upper bounds complementing the bounds in (1) for various settings of parameters. Uniform randomized AC0 circuits of poly(n) size and depth d = O(1) with error can be simulated by uniform randomized AC 0 circuits of poly(n) size and depth d + 1 with error + o(1) using ≤ (log n) O(log log n) random bits. Previous derandomizations [Ajtai and Wigderson '85; Nisan '91] increase the depth by a constant factor, or else have poor seed length.

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Bounded-Depth Circuits Cannot Sample Good Codes

Cited by 26 publications

References 20 publications

Quantum strategic game theory

Quantum strategic game theory

Tight bounds on computing error-correcting codes by bounded-depth circuits with arbitrary gates

The Complexity of Distributions

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