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Is it known/proven what the smallest quantum error correction code is that can correct arbitrary two-qubit Pauli errors? I can think of the nested/concatenated 5-qubit code or a 25-qubit version of the Shor (repetition) code, but I am not sure if there are codes requiring fewer qubits.

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If you look in this paper, section 7, they give an [[11,1,5]] code, and show that it is the smallest you can have.

In general, for these sorts of questions, a great starting point is Gottesman's thesis. That's where I found this result stated.

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Another good place to find codes with your desired parameters is this website: http://www.codetables.de/

The standard format to describe a quantum code is [[n,k,d]], where n is the number of physical qubits whose joint entangled state stores the logical information, k is the number of logical qubits encoded, and d is the distance to which they are protected.

The table is n vs k, where the distance d is the # in the box, so the smallest distance-5 code is the (11,1) entry

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