Fault-tolerant syndrome extraction and cat state preparation with fewer qubits

Prithviraj Prabhu and Ben W. Reichardt

University of Southern California, Los Angeles, CA 90089, USA

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Abstract

We reduce the extra qubits needed for two fault-tolerant quantum computing protocols: error correction, specifically syndrome bit measurement, and cat state preparation. For distance-three 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 needs at most $\lceil \log_2 w \rceil + 1$ ancilla qubits. If qubits reset quickly, four ancillas suffice. We also study the preparation of entangled cat states, and prove that the overhead for distance-three fault tolerance is logarithmic in the cat state size. These results apply both to near-term experiments with a few qubits, and to the general study of the asymptotic resource requirements of syndrome measurement and state preparation.
With $a$ flag qubits, previous methods use $O(a)$ flag patterns to identify faults. In order to use the same flag qubits more efficiently, we show how to use nearly all $2^a$ possible flag patterns, by constructing maximal-length paths through the $a$-dimensional hypercube.

We propose advancements in two important aspects of fault-tolerant quantum computing: error correction and state preparation. Our protocols aim to reduce the number of extra qubits needed for these processes. We demonstrate that stabilizer measurement, crucial for error detection and correction, can be performed with overhead that is logarithmic in the weight of the stabilizer being measured. This contrasts with previous protocols which required a linearly growing overhead to tolerate a single fault. When qubits can be reset quickly, only four ancilla qubits are needed. These findings are applicable to both near-term experiments and to the general study of fault-tolerant resource requirements. For fault-tolerant cat state preparation, we prove that tolerance to one fault can again be achieved with overhead that is logarithmic in the cat state size. These improvements abound due to the optimized use of flag qubits used to identify faults, by constructing maximal-length paths through a hypercube. This method allows for efficient usage of nearly all $2^n$ flag patterns (for a hypercube of dimension $n$), an exponential improvement. Overall, these advancements contribute to more efficient and practical fault-tolerant quantum computing protocols.

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Cited by

[1] Dhruv Bhatnagar, Matthew Steinberg, David Elkouss, Carmen G. Almudever, and Sebastian Feld, 2023 IEEE International Conference on Quantum Computing and Engineering (QCE) 63 (2023) ISBN:979-8-3503-4323-6.

[2] Christopher Chamberland and Earl T. Campbell, "Circuit-level protocol and analysis for twist-based lattice surgery", Physical Review Research 4 2, 023090 (2022).

[3] Benjamin Anker and Milad Marvian, "Flag Gadgets based on Classical Codes", arXiv:2212.10738, (2022).

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