# Code Distance and Threshold Theorem Two ideas govern whether [[Quantum Error Correction]] actually helps: the **code distance** (how much protection a given code provides) and the **threshold theorem** (the condition under which adding protection wins). ## Code distance The **distance** $d$ of a code is the smallest number of physical errors needed to corrupt the [[Physical vs Logical Qubits|logical]] information undetected. A code of distance $d$ can correct up to $\lfloor (d-1)/2 \rfloor$ errors. Bigger distance = more resilience, bought with more physical [[Qubit|qubits]]. In the [[Surface Code]], distance is roughly the width of the qubit patch. ## The threshold theorem There exists a critical physical error rate $p_\text{th}$ — the **fault-tolerance threshold** — with a sharp dichotomy: - If the physical error rate $p < p_\text{th}$: increasing the distance drives the logical error rate **down**, exponentially. - If $p > p_\text{th}$: adding qubits makes things **worse** — the correction machinery introduces more errors than it removes. $ p < p_\text{th} \;\Rightarrow\; P_\text{logical} \sim \Big(\frac{p}{p_\text{th}}\Big)^{\,d/2} \xrightarrow{\;d\to\infty\;} 0 $ > [!intuition] The leaky-bucket tipping point > If you bail water faster than it leaks in, a bigger bucket eventually stays dry. If you bail slower, a bigger bucket just holds more water before it overflows. The threshold is the break-even bailing rate. ## "Below threshold" as a milestone Demonstrating **below threshold** means showing experimentally that *larger code distance yields fewer logical errors* — proof that the platform's [[Gate Fidelity|gates]] are good enough for error correction to pay off. It is the gateway to scalable, fault-tolerant computing. ## Related - [[Quantum Error Correction]] - [[Surface Code]] - [[Gate Fidelity]] - [[Physical vs Logical Qubits]] - [[Syndrome Extraction]]