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How can one do an error correction simulation after a state injection? I understand that we can keep the subspace with the syndrome measurements, but how can a logical error be verified, if the X and Z are no more stabilizers of the system and should give random results? Can it be done numerically?
For example, how can one compute the threshold and logical error rate for state injection and distillation, especially under a complex circuit-based noise which makes analytical analysis hard?
There are a few approaches you can take, which have various degrees of difficulty. Most of these would also work as experiments on hardware.
Inject stabilizer states. Instead of trying to inject $|T\rangle = |0\rangle + \exp(45^\circ)|1\rangle$, inject the X and Y and Z eigenstates. These can then be measured directly by the error correcting code. If those all work, you have good reason to expect other single qubit states to work well.
Inject entanglement. Do two simultaneous injection experiments, with the injection points right next to each other, injecting $|00\rangle + |11\rangle$ across the two logical qubits. Then perform Bell basis measurement within the error correcting code at the end. If this works well, injecting any single qubit state should also work well.
Use magic state ejection. If you time-reverse an injection circuit, you get a circuit that measures in an arbitrary basis. Start with an injection, end with an ejection, and check that everything went well. Beware that if you're accidentally preparing the wrong state, time reversing that will unprepare that same state and hide the mistake.
Use tomography. The $|T\rangle$ state, when projected onto the X, Y, and Z axes has expectations of $\sqrt{0.5}$, $\sqrt{0.5}$, and 0 respectively. So do three variations of the experiment, one ending in an X measurement, one in Y, and on in Z, to view these three different expectations. This tells you where on the Bloch sphere the states are ending up.
(Simulator only; doesn't work in experiment) Replace your measurements with a magical confirmation that the logical state is correct or not. You have the simulator's state; it knows everything. Pull out that knowledge of the state.
Use distillation. Relevant magic states can be distilled to higher fidelity. The distillation process fails more often if the magic states are worse (as long as they're not so bad that they're below threshold). So you can use the distillation process' failure rate to understand how good the magic states are, if the magic states are pretty good to start with.
