 
Summary: On the power of algebraic branching programs of width two
Eric Allender
Department of Computer Science
Rutgers University
Piscataway, NJ 08855, USA
allender@cs.rutgers.edu
Fengming Wang
Department of Computer Science
Rutgers University
Piscataway, NJ 08855, USA
fengming@cs.rutgers.edu
May 21, 2011
Abstract
We show that there are families of polynomials having small depthtwo arithmetic circuits that cannot
be expressed by algebraic branching programs of width two. This clarifies the complexity of the problem
of computing the product of a sequence of twobytwo matrices, which arises in several settings.
1 Introduction
The nth Iterated Matrix Multiplication polynomial of degree d, denoted IMMd,n is the multilinear polyno
mial with d2n variables that is the result of multiplying n dbyd matrices of indeterminates. This family
plays a central role in the study of algebraic complexity. BenOr and Cleve showed that IMM3,n is complete
