Quantum error correction is essential to perform reliable quantum computation at scale. Here we report experimental and theoretical progress towards scalable error-corrected computation. First, we report the realization of a programmable quantum processor based on encoded logical qubits, utilizing logical-level hardware-efficient control in reconfigurable neutral atom arrays. Using this logical processor, involving storage, entangling, and readout zones, we demonstrate improvement of a two-qubit logic gate by scaling surface code distance from d=3 to d=7, preparation of color codes with break-even fidelities, fault-tolerant creation of logical GHZ states, and operation of 40 color code qubits. In performing such algorithms, we observe that the experimental performance can be substantially improved by accounting for physical error propagation during transversal entangling gates and decoding the logical qubits jointly. We study this correlated decoding technique theoretically and find that it can significantly improve the thresholds of transversal entangling gates. We apply correlated decoding to deep logical circuits with noisy syndrome extraction and find that it enables reaching higher fidelities by reducing the number of rounds of noisy syndrome extraction per gate. This correlated decoding technique offers key advantages in early fault-tolerant computation, as well as the possibility for reduction in the spacetime cost of logical algorithms.
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