The Quantum Communication Complexity of the Pointer Chasing Problem: The Bit Version

Abstract

We consider the two-party quantum communication complexity of the bit version of the pointer chasing problem when the ‘wrong’ player starts, originally studied by Klauck, Nayak, Ta-Shma and Zuckerman [7]. We show that in any quantum protocol for this problem, the two players must exchangeΩ(n/k⁴) qubits. This improves the previous best lower bound of $\Omega \left( {\frac{n} {{2^{2^{O(k)} } }}} \right)$ in [7], and comes significantly closer to the best upper bounds known: O(n + k log n) (classical deterministic [12]) and $O(k\log n + \frac{n} {k}(\log ^{(\left\lceil {k/2} \right\rceil )} n + \log k))$ (classical randomised [7]). Our result demonstrates a separation between the communication complexity of k and k - 1 round bounded error quantum protocols, for all k < O((m/log² m)^1/5), where m is the size of the inputs to Alice and Bob. Earlier works could prove such a separation for much smaller k only. Our proof uses a round elimination argument for a class of quantum sampling protocols with correlated input generation, making better use of information-theoretic tools than previous works.

Supported by the EU 5th framework project QAIP IST-1999-11234, and by CNRS/STIC 01N80/0502 and 01N80/0607 grants.