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.
@InProceedings{prabhu_et_al:LIPIcs.TQC.2021.5, author = {Prabhu, Prithviraj and Reichardt, Ben W.}, title = {{Fault-Tolerant Syndrome Extraction and Cat State Preparation with Fewer Qubits}}, booktitle = {16th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2021)}, pages = {5:1--5:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-198-6}, ISSN = {1868-8969}, year = {2021}, volume = {197}, editor = {Hsieh, Min-Hsiu}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2021.5}, URN = {urn:nbn:de:0030-drops-140001}, doi = {10.4230/LIPIcs.TQC.2021.5}, annote = {Keywords: Quantum error correction, fault tolerance, quantum state preparation, combinatorics} }
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