Non-commutative circuits and the sum-of-squares problem

We initiate a direction for proving lower bounds on the size of non-commutative arithmetic circuits. This direction is based on a connection between lower bounds on the size of non-commutative arithmetic circuits and a problem about commutative degree four polynomials, the classical sum-of-squares problem: find the smallest n such that there exists an identity (x2 1 + x2 2 + ? ? ? + x2 k) ? (y2 1 + y2 2 + ? ? ? + y2 k) = f2 1 + f2 2 + ? ? ? + f2 n, (0.1) where each fi = fi(X, Y ) is a bilinear form in X = {x1, . . . , xk} and Y = {y1, . . . , yk}. Over the complex numbers, we show that a sufficiently strong super-linear lower bound on n in (0.1), namely, n k1+ with > 0, implies an exponential lower bound on the size of arithmetic circuits computing the non-commutative permanent. More generally, we consider such sum-of-squares identities for any biquadratic polynomial h(X, Y ), namely h(X, Y ) = f2 1 + f2 2 + ? ? ? + f2 n. (0.2) Again, proving n k1+ in (0.2) for any explicit h ov...

• Type: Conference
• Year: 2010
• Where: STOC
• Authors: Pavel Hrubes, Avi Wigderson and Amir Yehudayoff
• Keywords: Algorithms, Non-commutative Arithmetic Circuits, Non-commutative Circuits, Non-commutative Polynomial, STOC 2010