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I'm studying about the surface code in quantum computing. I'm targeting NxN lattice surface code. While I'm reading this following paper; https://iopscience.iop.org/article/10.1088/1367-2630/aa5918/pdf ; I really wonder how many holes (that occur because of fabrication error) a specific code distance 'd' surface code can handle. In the research paper, they only handle 1 defect in the surface code.

If you have any references about this question, please let me know.

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I really wonder how many holes (occur because of fabrication error, defected qubit) can specific code distance 'd' surface code can handle and do quantum computation correctly

In this paper, the authors prove that noise can be suppressed for a finite density of fabrication defects https://arxiv.org/pdf/2111.06432.pdf.

Here is a quote from the abstract:

We present a threshold theorem showing that an arbitrarily large quantum computation can be completed with a vanishing probability of failure using a two-dimensional array of noisy qubits with a finite density of fabrication defects.

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