Universal quantum computation via scalable measurement-free error correction

We show that universal quantum computation can be concretely made fault-tolerant without mid-circuit measurements. To this end, we introduce a measurement-free deformation protocol of the Bacon-Shor code to realize a logical $\mathit{CCZ}$ gate. Combined with a fold-transversal logical Hadamard gate, this enables a universal set of fault-tolerant operations using only transversal gates and qubit permutations. For the purpose of benchmarking under circuit-level noise, we develop an efficient method to simulate non-Clifford circuits with a small number of Hadamard gates. Separately, we demonstrate that certain CSS codes can be concatenated without measurements or having to rely on a universal logical gate set. This is made possible by means of a resource-efficient gadget---termed the `disposable Toffoli gadget'---that realizes the error-correcting feedback. Then, under concatenation of the Bacon-Shor code, we observe a fault-tolerance threshold at a circuit-level depolarizing noise rate of approximately $0.12\,\%$. Together, the deformation and concatenation protocols outline a blueprint for a fully fault-tolerant architecture without any feed-forward operation, particularly suited to state-of-the-art neutral-atom platforms.