Separations in Communication Complexity Using Cheat Sheets and Information Complexity

While exponential separations are known between quantum and randomized communication complexity for partial functions (Raz, STOC 1999), the best known separation between these measures for a total function is quadratic, witnessed by the disjointness function. We give the first super-quadratic separation between quantum and randomized communication complexity for a total function, giving an example exhibiting a power 2. 5 gap. We further present a 1. 5 power separation between exact quantum and randomized communication complexity, improving on the previous ≅ 1. 15 separation by Ambainis (STOC 2013). Finally, we present a nearly optimal quadratic separation between randomized communication complexity and the logarithm of the partition number, improving upon the previous best power 1. 5 separation due to Goos, Jayram, Pitassi, and Watson. Our results are the communication analogues of separations in query complexity proved using the recent cheat sheet framework of Aaronson, Ben-David, and Kothari (STOC 2016). Our main technical results are randomized communication and information complexity lower bounds for a family of functions, called lookup functions, that generalize and port the cheat sheet framework to communication complexity.

Authors

• Anurag Anshu
• Aleksandrs Belovs
• Shalev Ben-David
• Mika Göös
• Rahul Jain 0001
• Robin Kothari
• Troy Lee
• Miklos Santha

Keywords

• Complexity theory
• Computational modeling
• Protocols
• Upper bound
• Computer science
• Sea measurements
• Complex Communication
• Cheat Sheet
• Lower Bound
• Family Functioning
• Part Of Function
• Quantum Information
• Optimal Separation
• Complex Quantum
• Complex Functions
• Error Probability
• Communication Performance
• Analogous Results
• General Theorem
• Input Bits
• Valid Proof
• Specific Fashion
• communication complexity
• randomized algorithms
• quantum algorithms