Hyperdense Coding Modulo 6 with Filter-Machines
Abstract
We show how one can encode
n
bits with
n
o(1)
``wave-bits'' using still hypothetical filter-machines (here
o(1)
denotes a positive quantity which goes to 0 as
n
goes to infity). Our present result - in a completely different computational model - significantly improves on the quantum superdense-coding breakthrough of Bennet and Wiesner (1992) which encoded
n
bits by
⌈n/2⌉
quantum-bits. We also show that our earlier algorithm (Tech. Rep. TR03-001, ECCC, See this ftp URL) which used
n
o(1)
muliplication for computing a representation of the dot-product of two
n
-bit sequences modulo 6, and, similarly, an algorithm for computing a representation of the multiplication of two
n×n
matrices with
n
2+o(1)
multiplications can be turned to algorithms computing the exact dot-product or the exact matrix-product with the same number of multiplications with filter-machines. With classical computation, computing the dot-product needs
Ω(n)
multiplications and the best known algorithm for matrix multiplication (D. Coppersmith and S. Winograd, Matrix multiplication via arithmetic progressions, J. Symbolic Comput., 9(3):251--280, 1990) uses
n
2.376
multiplications.