Quadratically Tight Relations for Randomized Query Complexity

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Abstract

In this work we investigate the problem of quadratically tightly approximating the randomized query complexity of Boolean functions $$\mathsf {R}(f)$$. The certificate complexity $$\mathsf {C}(f)$$ is such a complexity measure for the zero-error randomized query complexity $$\mathsf {R}_0(f)$$: $$\mathsf {C}(f) \le \mathsf {R}_0(f) \le \mathsf {C}(f)^2$$. In the first part of the paper we introduce a new complexity measure, expectational certificate complexity $$\mathsf {EC}(f)$$, which is also a quadratically tight bound on $$\mathsf {R}_0(f)$$: $$\mathsf {EC}(f) \le \mathsf {R}_0(f) = O(\mathsf {EC}(f)^2)$$. For $$\mathsf {R}(f)$$, we prove that $$\mathsf {EC}^{2/3} \le \mathsf {R}(f)$$. We then prove that $$\mathsf {EC}(f) \le \mathsf {C}(f) \le \mathsf {EC}(f)^2$$ and show that there is a quadratic separation between the two, thus $$\mathsf {EC}(f)$$ gives a tighter upper bound for $$\mathsf {R}_0(f)$$. The measure is also related to the fractional certificate complexity $$\mathsf {FC}(f)$$ as follows: $$\mathsf {FC}(f) \le \mathsf {EC}(f) = O(\mathsf {FC}(f)^{3/2})$$. This also connects to an open question by Aaronson whether $$\mathsf {FC}(f)$$ is a quadratically tight bound for $$\mathsf {R}_0(f)$$, as $$\mathsf {EC}(f)$$ is in fact a relaxation of $$\mathsf {FC}(f)$$.

In the second part of the work, we investigate whether the corruption bound $$\mathsf {corr}_\epsilon (f)$$ quadratically approximates $$\mathsf {R}(f)$$. By Yao’s theorem, it is enough to prove that the square of the corruption bound upper bounds the distributed query complexity $$\mathsf {D}^\mu _\epsilon (f)$$ for all input distributions $$\mu$$. Here, we show that this statement holds for input distributions in which the various bits of the input are distributed independently. This is a natural and interesting subclass of distributions, and is also in the spirit of the input distributions studied in communication complexity in which the inputs to the two communicating parties are statistically independent. Our result also improves upon a result of Harsha et al. [2015], who proved a similar weaker statement. We also note that a similar statement in the communication complexity is open.

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Notes

1. 1.

Jain and Klauck in their paper defined $$\mathsf {prt}_\epsilon (f)$$ to be the value of the linear program, instead of the logarithm of the value of the program.

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Acknowledgements

This work is supported in part by the Singapore National Research Foundation under NRF RF Award No. NRF-NRFF2013-13, the Ministry of Education, Singapore under the Research Centres of Excellence programme by the Tier-3 grant. Grant “Random numbers from quantum processes” No. MOE2012-T3-1-009.

M.S. is partially funded by the ANR Blanc program under contract ANR-12-BS02-005 (RDAM project).

J.V. is supported by the ERC Advanced Grant MQC. Part of this work was done while J.V. was an intern at the Centre for Quantum Technologies at the National University of Singapore.

We thank Anurag Anshu for helpful discussions.

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Correspondence to Jevgēnijs Vihrovs .

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Jain, R. et al. (2018). Quadratically Tight Relations for Randomized Query Complexity. In: Fomin, F., Podolskii, V. (eds) Computer Science – Theory and Applications. CSR 2018. Lecture Notes in Computer Science(), vol 10846. Springer, Cham. https://doi.org/10.1007/978-3-319-90530-3_18

• DOI: https://doi.org/10.1007/978-3-319-90530-3_18

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• Publisher Name: Springer, Cham

• Print ISBN: 978-3-319-90529-7

• Online ISBN: 978-3-319-90530-3

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