Quantum accuracy threshold for concatenated distance-3 codes
Panos Aliferis, Daniel Gottesman, John Preskill
TL;DR
The paper proves a rigorous quantum accuracy threshold for concatenated distance-3 codes, establishing a lower bound ε0 ≥ 2.73 × 10^{-5} under adversarial independent stochastic noise and extending the framework to higher-distance codes and local non-Markovian noise. It builds a recursive fault-tolerance analysis based on extended rectangles (exRecs) and the exRec-Cor principle, using a threshold dance to prove that good level-k rectangles remain correct at higher levels. A detailed construction and analysis of fault-tolerant gadgets for the Steane [[7,1,3]] code yield a concrete threshold bound and demonstrate the practicality of the approach, including cat-state and encoded-Bell-pair syndrome techniques. The work also presents an alternative proof scheme and generalizes the theory to non-Markovian noise, highlighting the robustness and broad applicability of concatenated-distance-3 threshold results and setting a rigorous foundation for Knill-style fault-tolerance concepts.
Abstract
We prove a new version of the quantum threshold theorem that applies to concatenation of a quantum code that corrects only one error, and we use this theorem to derive a rigorous lower bound on the quantum accuracy threshold epsilon_0. Our proof also applies to concatenation of higher-distance codes, and to noise models that allow faults to be correlated in space and in time. The proof uses new criteria for assessing the accuracy of fault-tolerant circuits, which are particularly conducive to the inductive analysis of recursive simulations. Our lower bound on the threshold, epsilon_0 > 2.73 \times 10^{-5} for an adversarial independent stochastic noise model, is derived from a computer-assisted combinatorial analysis; it is the best lower bound that has been rigorously proven so far.