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 <i>non-commutative</i> arithmetic circuits and a problem about <i>commutative</i> degree four polynomials, the classical sum-of-squares problem: find the smallest n such that there exists an identity (x₁²+x₂²+•• + x_k²)• (y₁^2+y₂²+•• + y_k²)= f₁²+f₂²+ ... +f_n², where each f_i = f_i(X,Y) is bilinear in X={x₁,... ,x_k} and Y={y₁,..., y_k}. Over the complex numbers, we show that a sufficiently strong <i>super-linear</i> lower bound on n in, namely, n ≥ k^1+ε with ε >0, implies an <i>exponential</i> lower bound on the size of arithmetic circuits computing the non-commutative permanent. More generally, we consider such sum-of-squares identities for any M polynomial h(X,Y), namely: h(X,Y) = f₁²+f₂²+...+f_n². Again, proving n ≥ k^1+ε in for <i>any</i> explicit h over the complex numbers gives an <i>exponential</i> lower bound for the non-commutative permanent. Our proofs relies on several new structure theorems for non-commutative circuits, as well as a non-commutative analog of Valiant's completeness of the permanent. We proceed to prove such super-linear bounds in some restricted cases. We prove that n ≥ Ω(k^6/5) in (1), if f₁,..., f_n are required to have <i>integer</i> coefficients. Over the <i>real</i> numbers, we construct an explicit M polynomial h such that n in (2) must be at least Ω(k²). Unfortunately, these results do not imply circuit lower bounds. We also present other structural results about non-commutative arithmetic circuits. We show that any non-commutative circuit computing an <i>ordered</i> non-commutative polynomial can be efficiently transformed to a syntactically multilinear circuit computing that polynomial. The permanent, for example, is ordered. Hence, lower bounds on the size of syntactically multilinear circuits computing the permanent imply unrestricted non-commutative lower bounds. We also prove an exponential lower bound on the size of non-commutative syntactically multilinear circuit computing an explicit polynomial. This polynomial is, however, not ordered and an unrestricted circuit lower bound does not follow.