  Goldilocks quantum cellular automata (QCA) have been simulated on quantum
hardware and produce emergent small-world correlation networks. In Goldilocks
QCA, a single-qubit unitary is applied to each qubit in a one-dimensional chain
subject to a balance constraint: a qubit is updated if its neighbors are in
opposite basis states. Here, we prove that a subclass of Goldilocks QCA --
including the one implemented experimentally -- map onto free fermions and
therefore can be classically simulated efficiently. We support this claim with
two independent proofs, one involving a Jordan--Wigner transformation and one
mapping the integrable six-vertex model to QCA. We compute local conserved
quantities of these QCA and predict experimentally measurable expectation
values. These calculations can be applied to test large digital quantum
computers against known solutions. In contrast, typical Goldilocks QCA have
equilibration properties and quasienergy-level statistics that suggest
nonintegrability. Still, the latter QCA conserve one quantity useful for error
mitigation. Our work provides a parametric quantum circuit with tunable
integrability properties with which to test quantum hardware.
