We propose a new model of quantum computation comprised of low-weight measurement sequences that simultaneously encode logical information, enable error correction, and apply logical gates. These measurement sequences constitute a new class of quantum error-correcting codes generalizing Floquet codes, which we call dynamic automorphism codes. We construct an explicit example, the dynamic automorphism color code, which is assembled from short measurement sequences that can realize all 72 automorphisms of the 2D color code. On a stack of N triangular patches, the dynamic automorphism color code encodes N logical qubits and can implement the full logical Clifford group by a sequence of two- and, more rarely, three-qubit Pauli measurements. We also make the first step towards universal quantum computation with dynamic automorphism codes by introducing a 3D dynamic automorphism color code and showing that a non-Clifford logical gate can be realized by adaptive two-qubit measurements.
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