A Fault-Tolerant Honeycomb Memory
Craig Gidney, Michael Newman, Austin G. Fowler, Michael Broughton
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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>0.2</mml:mn><mml:mi mathvariant="normal">&#x0025;</mml:mi><mml:mo>&#x2212;</mml:mo><mml:mn>0.3</mml:mn><mml:mi mathvariant="normal">&#x0025;</mml:mi></mml:math> for the honeycomb code compared to a threshold of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>0.5</mml:mn><mml:mi mathvariant="normal">&#x0025;</mml:mi><mml:mo>&#x2212;</mml:mo><mml:mn>0.7</mml:mn><mml:mi mathvariant="normal">&#x0025;</mml:mi></mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>1.5</mml:mn><mml:mi mathvariant="normal">&#x0025;</mml:mi><mml:mo>&#x003C;</mml:mo><mml:mi>p</mml:mi><mml:mo>&#x003C;</mml:mo><mml:mn>2.0</mml:mn><mml:mi mathvariant="normal">&#x0025;</mml:mi></mml:math>, where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>p</mml:mi></mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mn>10</mml:mn><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>&#x2212;</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msup></mml:math>, we project that the honeycomb code can reach the teraquop regime with only <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>600</mml:mn></mml:math> physical qubits.