Simultaneous Communication Protocols with Quantum and Classical Messages

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(qclogq) classical bits. Our proofs rely heavily on earlier results due to Scott Aaronson [1, 2]. In particular, our results imply that quantum-classical protocols need to send ( p n/logn) 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 [7]. Our results also imply, more generally, that there are no superpolynomial separations between quantumclassical 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.

[1]  Ronald de Wolf,et al.  Quantum communication and complexity , 2002, Theor. Comput. Sci..

[2]  R. Cleve,et al.  Quantum fingerprinting. , 2001, Physical review letters.

[3]  Ilan Newman,et al.  Private vs. Common Random Bits in Communication Complexity , 1991, Inf. Process. Lett..

[4]  László Babai,et al.  Randomized simultaneous messages: solution of a problem of Yao in communication complexity , 1997, Proceedings of Computational Complexity. Twelfth Annual IEEE Conference.

[5]  Andrew Chi-Chih Yao,et al.  Probabilistic computations: Toward a unified measure of complexity , 1977, 18th Annual Symposium on Foundations of Computer Science (sfcs 1977).

[6]  Andris Ambainis,et al.  Communication complexity in a 3-computer model , 1996, Algorithmica.

[7]  Ilan Newman,et al.  Public vs. private coin flips in one round communication games (extended abstract) , 1996, STOC '96.

[8]  Thierry Paul,et al.  Quantum computation and quantum information , 2007, Mathematical Structures in Computer Science.

[9]  Scott Aaronson,et al.  The learnability of quantum states , 2006, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[10]  Ronald de Wolf,et al.  Bounded-error quantum state identification and exponential separations in communication complexity , 2005, STOC '06.

[11]  Ziv Bar-Yossef,et al.  Exponential separation of quantum and classical one-way communication complexity , 2004, STOC '04.

[12]  Ran Raz,et al.  Exponential Separation for One-Way Quantum Communication Complexity, with Applications to Cryptography , 2008, SIAM J. Comput..

[13]  Scott Aaronson,et al.  Limitations of quantum advice and one-way communication , 2004, Proceedings. 19th IEEE Annual Conference on Computational Complexity, 2004..