How to calculate the threshold of Shor's code (9-qubit)

Hi guys,

If I have a Shor code (9-qubit), just for an example, and the noise is e=(1−p)I+p/3X+p/3Y+p/3Z. How can I get the threshold? Can I use Pennylane to verify it? I tried it but the threshold is not correct I believe.

Hi @learnerSWE,

I’m not an expert in quantum error correction, I work in quantum algorithms and quantum software, but let me give this a shot.

I believe that thresholds apply to fault-tolerance, which is different than error correction. In fault-tolerance, we are interested not only in correcting errors in states, but also errors in the operations that we apply. This is much more difficult because it’s hard to correct errors when the operations we apply to correct errors also give rise to errors!

As far as I know (and I may be wrong), the threshold of a fault-tolerant scheme is not a simple property of the underlying error-correction code. It depends on the code, the noise model, and the overall strategy for fault-tolerance. I therefore think that obtaining a fault-tolerant threshold for the 9-qubit code is not a straightforward task.

I would advise you to take a look at the literature to understand how thresholds are computed. For example, I found this nice reference after a quick search. You may find even better ones. Just keep in mind that most work focuses on other, better codes.

Finally, my understanding is that thresholds are mathematical properties that can’t be straightforwardly verified by experiments or simulation. Probably the best you can do is to perform a noisy simulation of the fault-tolerant protocol, then show that errors are corrected properly below threshold but fail above threshold. Since there is typically a large overhead, namely many more physical qubits than logical qubits, such a simulation would be very expensive. Proceed with caution!

You can use the default.mixed simulator in PennyLane to simulate noisy circuits, and the noise model you are mentioning is equivalent to the Depolarizing Channel.

Hope that helps!

Thank you. Pennylane is a little difficult to compute this (I am not an expert, sorry about that). Thank you for your paper! Really useful!

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