### Volume 2008

#### Simultaneous Communication Protocols with Quantum and Classical Messages

Dmitry Gavinsky
NEC Laboratories America, Inc.
4 Independence Way, Suite 200. Princeton
U.S.A.
Oded Regev
School of Computer Science
Tel-Aviv University
Tel-Aviv 69978
Israel,
and
Ronald de Wolf
CWI
Kruislaan 413, 1098SJ Amsterdam
The Netherlands
December 29, 2008
##### Abstract
We study the simultaneous message passing (SMP) model of communication complexity, for the case where one party is quantum and the other is classical. We show that in an SMP protocol that computes some function with the first party sending $q$ qubits and the second sending $c$ classical bits, the quantum message can be replaced by a randomized message of $O(qc)$ classical bits, as well as by a deterministic message of $O(q c \log q)$ classical bits. Our proofs rely heavily on earlier results due to Scott Aaronson. In particular, our results imply that quantum-classical protocols need to send $\Omega(\sqrt{n/ \log n})$ bits/qubits to compute EQUALITY on $n$-bit strings, and hence are not significantly better than classical-classical protocols (and are much worse than quantum-quantum protocols such as quantum fingerprinting). This essentially answers a recent question of Wim van Dam. Our results also imply, more generally, that there are no superpolynomial separations between quantum-classical and classical-classical SMP protocols for functional problems. This contrasts with the situation for relational problems, where exponential gaps between quantum-classical and classical-classical SMP protocols are known. We show that this surprising situation cannot arise in purely classical models: there, an exponential separation for a relational problem can be converted into an exponential separation for a functional problem.

DOI: 10.4086/cjtcs.2008.007
Volume 2008, Article 6 Volume 2009, Article 1
Volume 2008 Published articles