A Fault-Tolerant Honeycomb Memory

Craig Gidney, Michael Newman, Austin Fowler, and Michael Broughton

Google Quantum AI, Santa Barbara, California 93117, USA

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Abstract

Recently, Hastings & Haah introduced a quantum memory defined on the honeycomb lattice. Remarkably, this honeycomb code assembles weight-six parity checks using only two-local measurements. The sparse connectivity and two-local measurements are desirable features for certain hardware, while the weight-six parity checks enable robust performance in the circuit model.
In this work, we quantify the robustness of logical qubits preserved by the honeycomb code using a correlated minimum-weight perfect-matching decoder. Using Monte Carlo sampling, we estimate the honeycomb code's threshold in different error models, and project how efficiently it can reach the "teraquop regime" where trillions of quantum logical operations can be executed reliably. We perform the same estimates for the rotated surface code, and find a threshold of $0.2\%-0.3\%$ for the honeycomb code compared to a threshold of $0.5\%-0.7\%$ for the surface code in a controlled-not circuit model. In a circuit model with native two-body measurements, the honeycomb code achieves a threshold of $1.5\% \lt p \lt 2.0\%$, where $p$ is the collective error rate of the two-body measurement gate - including both measurement and correlated data depolarization error processes. With such gates at a physical error rate of $10^{−3}$, we project that the honeycomb code can reach the teraquop regime with only $600$ physical qubits.

We checked how well a new quantum error correcting code works. The new code is called the honeycomb code and was found by researchers at Microsoft. We did simulations to estimate the quality and quantity of noisy qubits needed to reach error rates as low as 1 in a trillion. We compared the honeycomb code to the surface code (the current state of the art).

We used new open source tools: Stim and PyMatching. These tools allowed us to get prototype results in weeks, instead of months, even though the honeycomb code is a very recent and unusual quantum code.

We found that the honeycomb code is slightly worse than the surface code. However, the honeycomb code works on hardware with fewer connections between qubits than the surface code. Also, its performance is very good when the hardware can directly measure two-qubit parities – better than previous codes built entirely out of two-qubit parity measurements. This makes the honeycomb code interesting for quantum computer architectures with low connectivity or native parity measurements.

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Cited by

[1] Basudha Srivastava, Anton Frisk Kockum, and Mats Granath, "The XYZ2 hexagonal stabilizer code", Quantum 6, 698 (2022).

[2] Oscar Higgott, "PyMatching: A Python Package for Decoding Quantum Codes with Minimum-Weight Perfect Matching", ACM Transactions on Quantum Computing 3 3, 1 (2022).

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[5] Jeongwan Haah and Matthew B. Hastings, "Boundaries for the Honeycomb Code", arXiv:2110.09545, Quantum 6, 693 (2022).

[6] Benjamin A. Cordier, Nicolas P. D. Sawaya, Gian G. Guerreschi, and Shannon K. McWeeney, "Biology and medicine in the landscape of quantum advantages", arXiv:2112.00760.

[7] David Aasen, Zhenghan Wang, and Matthew B. Hastings, "Adiabatic paths of Hamiltonians, symmetries of topological order, and automorphism codes", arXiv:2203.11137.

[8] Julia Wildeboer, Thomas Iadecola, and Dominic J. Williamson, "Symmetry-Protected Infinite-Temperature Quantum Memory from Subsystem Codes", PRX Quantum 3 2, 020330 (2022).

[9] Adam Paetznick, Christina Knapp, Nicolas Delfosse, Bela Bauer, Jeongwan Haah, Matthew B. Hastings, and Marcus P. da Silva, "Performance of planar Floquet codes with Majorana-based qubits", arXiv:2202.11829.

[10] Christophe Vuillot, "Planar Floquet Codes", arXiv:2110.05348.

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[13] Jason Bennett, "Fractons: gauging spin models and tensor gauge theory", arXiv:2206.14028.

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