# 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. Jan 28, 2020 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 $$e^{\frac{t_{gate}}{t_{decoherence}}}$$
• Could you please tell what is $t_{\mathrm{gate}}$? Dec 28, 2019 at 19:15
Where decoherence rates tends to matter more is the depth of the algorithm you can run. If you have a 5$$\mu$$s decoherence time and each of your qubit operations takes 100ns, then you can only usefully perform 50 operations at best.