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Fault-Tolerant Syndrome Extraction and Cat State Preparation with Fewer Qubits Prabhu, Prithviraj Reichardt, Ben W. Quantum error correction fault tolerance quantum state preparation combinatorics We reduce the extra qubits needed for two fault-tolerant quantum computing protocols: error correction, specifically syndrome bit measurement, and cat state preparation. For fault-tolerant syndrome extraction, we show an exponential reduction in qubit overhead over the previous best protocol. For a weight-w stabilizer, we demonstrate that stabilizer measurement tolerating one fault (distance-three) needs at most ⌈ log₂ w ⌉ + 1 ancillas. If qubits reset quickly, four ancillas suffice. We also study the preparation of cat states, simple yet versatile entangled states. We prove that the overhead needed for distance-three fault tolerance is only logarithmic in the cat state size. These results could be useful both for near-term experiments with a few qubits, and for the general study of the asymptotic resource requirements of syndrome measurement and state preparation. For 'a' measured flag bits, there are 2^a possible flag patterns that can identify faults. Hence our results come from solving a combinatorial problem: the construction of maximal-length paths in the a-dimensional hypercube, corresponding to maximal-weight stabilizers or maximal-weight cat states. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Prithviraj Prabhu and Ben W. Reichardt 2021 Is Part Of LIPIcs, Volume 197, 16th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2021) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.TQC.2021.5 urn:nbn:de:0030-drops-140001 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2021.5 eng https://creativecommons.org/licenses/by/4.0/legalcode