Equation 27 · Error Correction Is the Whole Problem in Quantum Computing

What does this equation mean?

P_L(d) \;\approx\; A \left( \frac{p}{p_{\mathrm{th}}} \right)^{\left\lfloor (d+1)/2 \right\rfloor} ,

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This equation gives an approximation: it relates the quantities while allowing an approximation. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.

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P_L

Symbol P_L

$P_L$ is a part of this expression. Its role is fixed by the surrounding article and by the operations shown in the formula.

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d

Symbol d

d is a part of this expression. Its role is fixed by the surrounding article and by the operations shown in the formula.

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A

Symbol A

A is a part of this expression. Its role is fixed by the surrounding article and by the operations shown in the formula.

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p

Symbol p

well below $p_{\mathrm{th}}$ , not marginally below it.

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p_{\mathrm{th}}

Symbol p_th

$p_t$ h occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.

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fraction

Divide the expression above the line by the one below it.

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≈

Approximately equal to; the equality is not exact.

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addition

Add the term after the plus sign to the term or group before it.

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subscript

The lower label selects a particular version, component, or indexed member of the quantity. For example, x₀ and xₜ can be values at different positions.

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superscript

A raised number can be a power. When it is a label or bound, it selects a case or the upper limit of a sum; the formula’s structure distinguishes these uses.

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How to interpret it

With a fixed numerator, increasing a nonzero denominator reduces the fraction. Its accuracy depends on the assumptions and range of use described in the article.

What the article says around this equation

The practical form of the promise is a scaling law. Below threshold, the logical error rate per round of a distance- d code falls approximately as $P_L(d) \;\approx\; A \left( \frac{p}{p_{\mathrm{th}}} \right)^{\left\lfloor (d+1)/2 \right\rfloor}$ . so that each increase of the code distance by two multiplies the protection by a factor conventionally written

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Sources cited in the article section

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