[1] This result was proven (for various error models) by the groups of Dorit Aharanov and Michael Ben-Or;[2] Emanuel Knill, Raymond Laflamme, and Wojciech Zurek;[3] and Alexei Kitaev[4] independently.
[3] These results built on a paper of Peter Shor,[5] which proved a weaker version of the threshold theorem.
Surprisingly, the quantum threshold theorem shows that if the error to perform each gate is a small enough constant, one can perform arbitrarily long quantum computations to arbitrarily good precision, with only some small added overhead in the number of gates.
According to quantum information theorist Scott Aaronson:"The entire content of the Threshold Theorem is that you're correcting errors faster than they're created.
"[7]Current estimates put the threshold for the surface code on the order of 1%,[8] though estimates range widely and are difficult to calculate due to the exponential difficulty of simulating large quantum systems.
[citation needed][a] At a 0.1% probability of a depolarizing error, the surface code would require approximately 1,000-10,000 physical qubits per logical data qubit,[9] though more pathological error types could change this figure drastically.