Equation 18 · Error Correction Is the Whole Problem in Quantum Computing
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the weight of the smallest undetectable logical error. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.
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The overhead is not a fixed multiplier. It is a function of how far below threshold the hardware sits and how long the algorithm must run, because the required logical error rate is set by the total number of logical operations. An algorithm with 10^{10} logical operations needs a logical error rate well below 10^{-10} per operation, and the distance required to reach that number depends on the physical error rate in a way examined in the next section. This is why “a thousand physical qubits per logical qubit” is a slogan rather than a specification: the true ratio depends jointly on p , on d , on the code, on the decoder, and on the depth of the computation being run.
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