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In the original paper by Bravyi and Kitaev, the distillation procedure takes 5 physical qubits, each prepared in a faulty T state. The qubits are in a product state, and error can be detected by stabilizer measurements. If no errors occur, the output of this circuit is a single physical qubit with distill T state, i.e. a new T state with lower error probability than each initial prepared T states.

In contrast, most other distillation protocols operate on the logical level: one prepare, for example, a single physical qubit in a faulty T state, inject it to a surface code, and then the distillation is done on the logical level (for example: encode 15 surface code in the [[15, 1, 3]] code, and distill one error-corrected surface code).

My question is why should one preferred the former or the latter? In principle, isn't it better to distill a physical qubit first? Or maybe because the injection procedure is not fault tolerant (e.g. the CNOT required for stabilizer measurement is faulty and cannot be corrected at first step), so it will be insufficient for a practical use in a logical circuit?

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You would distill at the physical level whenever the fidelity of the output state is lower than the fidelity of your physical operations and storage.

For magic states, my guess is you would never do distillation physically. The initial state is produced by your physical operations, so it's already limited by the fidelity of your physical operations. If you try to distill physically, your output state will be limited by the same thing so you won't get any improvement. Might as well encode into logical qubits right away. As consolation prize, you can prepare the the state while encoding to make the process better than physical preparation followed by encoding. This is how Li injection and hook injection can give you starting fidelities better than the single qubit gates and two qubit gates used to perform the injection.

A context where you could do physical distillation is entanglement purification. Entanglement purification factories work basically the same way as magic state distillation factories, just with two computers comparing checks instead of one computer by itself. The initial entangled pairs come from a noisy channel between the two computers, not just from the computers' operations, so it could easily be much noisier than your physical operations. Like maybe it would be 90% fidelity instead of 99.9% fidelity. So for purification it can be beneficial to do physical distillation, to push the fidelity closer to 99%, before encoding into logical qubits so you can reach truly useful fidelities like 99.999999999%.


As an example, look at this figure of an entanglement purification sequence from my paper "Tetrationally Compact Entanglement Purification ". The error rates are proportional to the entries in the vectors other than the 1. The top row could probably be done physically. The second row probably needs logical qubits. The third row definitely needs logical qubits. The fourth row is intended to be funny; it would need logical qubits as big as the observable universe or maybe even bigger than that.

enter image description here

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  • $\begingroup$ Thank you very much! $\endgroup$ Jan 2 at 16:16

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