ECCC-Report TR19-006https://eccc.weizmann.ac.il/report/2019/006Comments and Revisions published for TR19-006en-usMon, 04 Mar 2019 08:27:41 +0200
Revision 1
| Upper Bounds on Communication in terms of Approximate Rank |
Anna Gal,
Ridwan Syed
https://eccc.weizmann.ac.il/report/2019/006#revision1We 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.
Mon, 04 Mar 2019 08:27:41 +0200https://eccc.weizmann.ac.il/report/2019/006#revision1
Paper TR19-006
| Upper Bounds on Communication in terms of Approximate Rank |
Anna Gal,
Ridwan Syed
https://eccc.weizmann.ac.il/report/2019/006We 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 results yield lower bounds on approximate rank. We also obtain a strengthening of Newman's theorem with respect to approximate rank.
Thu, 17 Jan 2019 00:58:33 +0200https://eccc.weizmann.ac.il/report/2019/006