Exponential quantum communication reductions from generalizations of the
Boolean Hidden Matching problem
- URL: http://arxiv.org/abs/2001.05553v2
- Date: Tue, 17 Aug 2021 08:05:21 GMT
- Title: Exponential quantum communication reductions from generalizations of the
Boolean Hidden Matching problem
- Authors: Jo\~ao F. Doriguello, Ashley Montanaro
- Abstract summary: First communication problem in the one-way model to demonstrate an exponential classical-quantum communication separation.
Efficient communication protocols are presented depending on the sign-degree of $f$.
- Score: 0.05076419064097732
- License: http://creativecommons.org/publicdomain/zero/1.0/
- Abstract: 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 Boolean 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.
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