3 research outputs found
Tight Bounds for the Randomized and Quantum Communication Complexities of Equality with Small Error
We investigate the randomized and quantum communication complexities of the
well-studied Equality function with small error probability , getting
the optimal constant factors in the leading terms in a number of different
models.
In the randomized model,
1) we give a general technique to convert public-coin protocols to
private-coin protocols by incurring a small multiplicative error, at a small
additive cost. This is an improvement over Newman's theorem [Inf. Proc.
Let.'91] in the dependence on the error parameter.
2) Using this we obtain a -cost private-coin
communication protocol that computes the -bit Equality function, to error
. This improves upon the upper bound
implied by Newman's theorem, and matches the best known lower bound, which
follows from Alon [Comb. Prob. Comput.'09], up to an additive
.
In the quantum model,
1) we exhibit a one-way protocol of cost , that uses only
pure states and computes the -bit Equality function to error .
This bound was implicitly already shown by Nayak [PhD thesis'99].
2) We show that any -error one-way protocol for -bit Equality
that uses only pure states communicates at least
qubits.
3) We exhibit a one-way protocol of cost , that
uses mixed states and computes the -bit Equality function to error
. This is also tight up to an additive ,
which follows from Alon's result.
Our upper bounds also yield upper bounds on the approximate rank and related
measures of the Identity matrix. This also implies improved upper bounds on
these measures for the distributed SINK function, which was recently used to
refute the randomized and quantum versions of the log-rank conjecture.Comment: 16 page