How is the decoherence rate connected to the error rate?

Question

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?

Answer 1

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 $$e^{\frac{t_{gate}}{t_{decoherence}}}$$

Answer 2

I don't understand how I can take a decoherence rate (let's say 5μs) and turn this into an error rate

Unfortunately, I think you're trying to compare apples to oranges a little bit here. For most physical realizations, individual gate error rates typically arise from non-idealities of the analog quantum system you're trying to use as a discrete qubit (e.g. slight microwave or laser power fluctuations between pulses, having nonzero linewidths for your transitions, external noise, etc). Most papers that characterize the performance of quantum gates report the metric "fidelity," which measures how well that physical system realizes an ideal qubit.

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.

Now as for the threshold theorem, you can apply it on both fronts. With a sufficiently low error rate, I can initialize many entangled copies of the same state, run them through a set of gates, do some syndrome measurements, and correct any gate errors that may have arisen. Similarly, every so often (faster than the decoherence rate) I can have a number of entangled copies, measure their syndromes of to diagnose any decoherence effects, and fix my state, effectively resetting the decoherence time.