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Upper Bounds on Communication in Terms of Approximate Rank

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Abstract

We show that any Boolean function with approximate rank r can be computed by bounded-error quantum protocols without prior entanglement of complexity \(O( \sqrt{r} \log r)\). In addition, we show that any Boolean function with approximate rank r and discrepancy \(\delta \) can be computed by deterministic protocols of complexity O(r), and private coin bounded-error randomized protocols of complexity \(O((\frac{1}{\delta })^2 + \log r)\). Our deterministic upper bound in terms of approximate rank is tight up to constant factors, and the dependence on discrepancy in our randomized upper bound is tight up to taking square-roots. Our results can be used to obtain lower bounds on approximate rank. We also obtain a strengthening of Newman’s theorem with respect to approximate rank.

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Notes

  1. Our protocol does not use fingerprinting in the usual sense since we will really be encoding the factorization vectors corresponding to Alice and Bob’s inputs rather than their actual inputs.

  2. It suffices for \(S_0\) to negate the amplitudes on states where all channel qubits are 0.

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Acknowledgements

We thank Shachar Lovett for allowing us to include his argument improving our bound in Lemma 6, and Adi Shraibman for pointing us to [3]. We would also like to thank the anonymous referees of a previous version of this paper for helpful comments. The second author thanks Patrick Rall and Daniel Liang for helpful discussions.

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  1. University of Texas, Austin, TX, USA

    Anna Gál & Ridwan Syed

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  1. Anna Gál
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  2. Ridwan Syed
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Correspondence to Anna Gál.

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Gál, A., Syed, R. Upper Bounds on Communication in Terms of Approximate Rank. Theory Comput Syst (2023). https://doi.org/10.1007/s00224-023-10158-4

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