Researchers have computed the exact acceptance and logical error rates for d=3 and d=5 quantum magic state cultivation circuits. These states, particularly T states, are crucial for universal quantum computation. Unlike previous studies that used approximations, this work directly analyzes T gates, considering every fault order at various circuit-level noise strengths (p). The analytical results, presented as a series expansion up to order (p/(1-p))¹⁰, accurately recover numerical values from prior work by Clifft and SOFT within their sampling uncertainty.

The methodology employed relies on Pauli propagation and binary tensor contraction, enabling an exact evaluation of error rates. This approach provides a deeper understanding of how faults propagate and affect the fidelity of generated quantum states. The precision of these calculations is fundamental for the design and optimization of future fault-tolerant quantum computers, where the ability to generate high-fidelity magic states is an indispensable requirement.

A significant finding of the study is that the d=3 and d=5 circuits actually have fault distances of d_fault=2 and d_fault=3, respectively. This degradation of the effective fault distance explains similar degradation effects observed in a companion work from 2026. Understanding this discrepancy between the nominal code distance and the actual fault distance is vital for accurately assessing the robustness of quantum error correction schemes and for improving the efficiency of magic state generation, a key bottleneck in quantum computing.