Exponential quantum communication reductions from generalizations of the Boolean Hidden Matching problem

01/15/2020
by   João F. Doriguello, et al.
0

In this work we revisit the Boolean Hidden Matching communication problem, which was the first communication problem in the one-way model to demonstrate an exponential classical-quantum communication separation. In this problem, Alice's bits are matched into pairs according to a partition that Bob holds. These pairs are compressed using a Parity function and it is promised that the final bit-string is equal either to another bit-string Bob holds, or its complement. The problem is to decide which case is the correct one. Here we generalize the Boolean Hidden Matching problem by replacing the parity function with an arbitrary function f. Efficient communication protocols are presented depending on the sign-degree of f. If its sign-degree is less than or equal to 1, we show an efficient classical protocol. If its sign-degree is less than or equal to 2, we show an efficient quantum protocol. We then completely characterize the classical hardness of all symmetric functions f of sign-degree greater than or equal to 2, except for one family of specific cases. We also prove, via Fourier analysis, a classical lower bound for any function f whose pure high degree is greater than or equal to 2. Similarly, we prove, also via Fourier analysis, a quantum lower bound for any function f whose pure high degree is greater than or equal to 3. These results give a large family of new exponential classical-quantum communication separations.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
03/07/2023

Approximate degree lower bounds for oracle identification problems

The approximate degree of a Boolean function is the minimum degree of re...
research
07/07/2020

Lower Bounds for XOR of Forrelations

The Forrelation problem, introduced by Aaronson [A10] and Aaronson and A...
research
07/26/2023

Fourier Growth of Communication Protocols for XOR Functions

The level-k ℓ_1-Fourier weight of a Boolean function refers to the sum o...
research
06/26/2020

Quantum Communication Complexity of Distribution Testing

The classical communication complexity of testing closeness of discrete ...
research
03/01/2023

Memory-Sample Lower Bounds for Learning with Classical-Quantum Hybrid Memory

In a work by Raz (J. ACM and FOCS 16), it was proved that any algorithm ...
research
05/15/2021

Communication Complexity of Private Simultaneous Quantum Messages Protocols

The private simultaneous messages model is a non-interactive version of ...
research
08/05/2022

Towards Antisymmetric Neural Ansatz Separation

We study separations between two fundamental models (or Ansätze) of anti...

Please sign up or login with your details

Forgot password? Click here to reset