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In pursuit of large-scale fault-tolerant quantum computation, quantum low-density parity-check (LPDC) codes have been established as promising candidates for low-overhead memory when compared to conventional approaches based on surface codes. Performing fault-tolerant logical computation on QLDPC memory, however, has been a long standing challenge in theory and in practice. In this work, we propose a new primitive, which we call an $\textit{extractor system}$, that can augment any QLDPC memory into a computational block well-suited for Pauli-based computation. In particular, any logical Pauli operator supported on the memory can be fault-tolerantly measured in one logical cycle, consisting of $O(d)$ physical syndrome measurement cycles, without rearranging qubit connectivity. We further propose a fixed-connectivity, LDPC architecture built by connecting many extractor-augmented computational (EAC) blocks with bridge systems. When combined with any user-defined source of high fidelity $|T\rangle$ states, our architecture can implement universal quantum circuits via parallel logical measurements, such that all single-block Clifford gates are compiled away. The size of an extractor on an $n$ qubit code is $\tilde{O}(n)$, where the precise overhead has immense room for practical optimizations.