Fault-Tolerant Syndrome Extraction and Cat State Preparation with Fewer Qubits

Authors Prithviraj Prabhu , Ben W. Reichardt

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Author Details

Prithviraj Prabhu
  • Department of Electrical and Computer Engineering, University of Southern California, Los Angeles, CA, USA
Ben W. Reichardt
  • Department of Electrical and Computer Engineering, University of Southern California, Los Angeles, CA, USA


The authors would like to thank Rui Chao, Sourav Kundu and Zhang Jiang for insightful conversations.

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Prithviraj Prabhu and Ben W. Reichardt. Fault-Tolerant Syndrome Extraction and Cat State Preparation with Fewer Qubits. In 16th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 197, pp. 5:1-5:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


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.

Subject Classification

ACM Subject Classification
  • Hardware → Quantum error correction and fault tolerance
  • Theory of computation → Quantum computation theory
  • Quantum error correction
  • fault tolerance
  • quantum state preparation
  • combinatorics


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