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
n
such that there exists an identity
(
0.1
)
(
x
1
2
+
x
2
2
+
⋯
+
x
k
2
)
⋅
(
y
1
2
+
y
2
2
+
⋯
+
y
k
2
)
=
f
1
2
+
f
2
2
+
⋯
+
f
n
2
,
\begin{equation*} (0.1)\quad \quad (x_1^2+x_2^2+\cdots + x_k^2)\cdot (y_1^2+y_2^2+\cdots + y_k^2)= f_{1}^{2}+f_{2}^{2}+\dots +f_{n}^{2} , \quad \quad \end{equation*}
where each
f
i
=
f
i
(
X
,
Y
)
f_{i} = f_i(X,Y)
is a bilinear form in
X
=
{
x
1
,
…
,
x
k
}
X=\{x_{1},\dots ,x_{k}\}
and
Y
=
{
y
1
,
…
,
y
k
}
Y=\{y_{1},\dots , y_{k}\}
. Over the complex numbers, we show that a sufficiently strong superlinear lower bound on
n
n
in (0.1), namely,
n
≥
k
1
+
ϵ
n\geq k^{1+\epsilon }
with
ϵ
>
0
\epsilon >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 biqua- dratic polynomial
h
(
X
,
Y
)
h(X,Y)
, namely
(
0.2
)
h
(
X
,
Y
)
=
f
1
2
+
f
2
2
+
⋯
+
f
n
2
.
\begin{equation*} (0.2) \quad \quad \qquad \quad \quad \quad \quad h(X,Y) = f_{1}^{2}+f_{2}^{2}+\dots +f_{n}^{2} . \quad \quad \qquad \quad \quad \quad \quad \end{equation*}
Again, proving
n
≥
k
1
+
ϵ
n\geq k^{1+\epsilon }
in (0.2) for any explicit
h
h
over the complex numbers gives an exponential lower bound for the non-commutative permanent. Our proofs rely on several new structure theorems for non-commutative circuits, as well as a non-commutative analog of Valiant’s completeness of the permanent.
We prove such a superlinear bound in one special case. Over the real numbers, we construct an explicit biquadratic polynomial
h
h
such that
n
n
in (0.2) must be at least
Ω
(
k
2
)
\Omega (k^{2})
. Unfortunately, this result does 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 ordered 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 a non-commutative syntactically multilinear circuit computing an explicit polynomial. This polynomial is, however, not ordered and an unrestricted circuit lower bound does not follow.