  The zero-error capacity of a channel (or Shannon capacity of a graph)
quantifies how much information can be transmitted with no risk of error. In
contrast to the Shannon capacity of a channel, the zero-error capacity has not
even been shown to be computable: we have no convergent upper bounds. In this
work, we present a new quantity, the zero-error {\em unitary} capacity, and
show that it can be succinctly represented as the tensor product value of a
quantum game. By studying the structure of finite automata, we show that the
unitary capacity is within a controllable factor of the zero-error capacity.
This allows new upper bounds through the sum-of-squares hierarchy, which
converges to the commuting operator value of the game. Under the conjecture
that the commuting operator and tensor product value of this game are equal,
this would yield an algorithm for computing the zero-error capacity.
