# How is the decoherence rate connected to the error rate?

I'm reading about the threshold theorem, which states that "a quantum computer with a physical error rate below a certain threshold can, through the application of quantum error correction schemes, suppress the logical error rate to arbitrarily low levels."

Now I know that decoherence leads to errors, and I know how to calculate decoherence rates, but I don't understand how I can take a decoherence rate (let's say $$5 \mathrm{\mu s}$$) and turn this into an error rate. I'm also not sure how to compare a given error rate to it's threshold and the threshold theorem.

Any ideas?

• It depends largely on what kind of decoherence is happening. There's no simple relationship, and there's a trove of literature investigating many different cases. You can start you search using Mike & Ike. – psitae Jan 28 at 13:36

You will need to know how long it takes for each gate of the circuit to be performed. Then the decoherence error rate is simply $$e^{t_{gate}/t_{decoherence}}$$
• Could you please tell what is $t_{\mathrm{gate}}$? – Martin Vesely Dec 28 '19 at 19:15