Using fault-tolerant constructions, computations performed with unreliable components can simulate their noiseless counterparts though the introduction of a modest amount of redundancy. Given the modest overhead required to achieve fault-tolerance, and the fact that increasing the reliability of basic components often comes at a cost, are there situations where fault-tolerance may be more economical? We present a general framework to account for this overhead cost in order to effectively compare fault-tolerant to non-fault-tolerant approaches for computation, in the limit of small logical error rates. Using this detailed accounting, we determine explicit boundaries at which fault-tolerant designs become more efficient than designs that achieve comparable reliability through direct consumption of resources. We find that the fault-tolerant construction is always preferred in the limit of high reliability in cases where the resources required to construct a basic unit grows faster than $\log(1 / \epsilon)$ asymptotically for small $\epsilon$.

翻译：借助容错构造，使用不可靠组件执行的计算可通过引入适度冗余来模拟其无噪声对应。考虑到实现容错所需的适度开销，以及提高基本组件可靠性通常伴随成本这一事实，是否存在某些情形下容错方案更为经济？我们提出一个通用框架来核算此类开销成本，以在极小逻辑错误率极限下有效比较容错与非容错计算方案。通过这种精细核算，我们确定了容错设计相较于通过直接资源消耗实现同等可靠性的设计变得更为高效的具体边界。研究发现，当构造基本单元所需资源在渐近意义下随小量ε的增长速率超过$\log(1 / \epsilon)$时，在极高可靠性极限下容错构造始终是更优选择。