Error correction without syndrome measurements

Question

Active error correction is usually described as a three-step process -- measuring syndromes, decoding the error, and applying the correcting operator. There seem to be alternatives to this approach, such as self-correcting dissipative memories. I will now describe an error correction procedure that is basically unitary. My questions will be if this procedure has a name, can be generalized or could be useful.

Consider the following circuit

One can check that for any statevector which lies in the repetition code subspace $|\psi\rangle=\alpha|000\rangle+\beta|111\rangle$ up to a single-qubit $X$ error (i.e. including $X_1|\psi\rangle, X_2|\psi\rangle$ and $X_3|\psi\rangle$) it acts as $|\psi\rangle=\alpha|000\rangle+\beta|111\rangle\to (\alpha|0\rangle+\beta|1\rangle)|state_{23}\rangle$. I.e. the first qubit acquires the decoded logical state and isn't entangled with the last two. For details see a table at the end. Therefore, this unitary circuit both corrects a single-qubit error $X$ and decodes a logical state. I will call it a decoder-corrector. Applying this decoder-corrector + resetting the ancilla qubits + making the repetition encoding again has the same effect as the usual error correction process, but without any syndrome measurements and decoding.

Although I do not have a nice circuit, I know that [[5, 1, 3]] code admits correction of arbitrary 1q errors in the same way (unitary decoder-corrector+ancilla reset+re-encoding).

1. Does this procedure have a name? Does it exist for any error correcting code?
2. Given that measuring syndromes+decoding is a costly process, could there be a setting where a similar procedure outperforms the standard active error correction?

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Decoder-corrector circuit acting on the code space of the repetition code+single X errors.

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Answer 1

What you are doing is applying the deferred measurement principle to move classical corrections back into the quantum circuit.

There are some practical problems with doing this.

1. It means your corrective circuit experiences as much noise as the computation it's protecting. For example, what if there is an X error on the target during the toffoli? That error is not detected. You want to get the correction away from the quantum computer, so that it's reliable.

2. You still need to dissipate the entropy that accumulated into the ancillas qubits. In practice you need to continuously error correct, so you need to keep those ancillas qubits clean. So you need to dissipate their state into the environment, which isn't exactly like a measurement but it's basically a measurement.

Answer 2

If I understand your question correctly then yes this is possible for any stabilizer code. You can look at this paper Efficient Quantum Circuit for Encoding and Decoding of the [[8,3,5]] Stabilizer Code (Fig. 2) for an example. There was an open access version at some point but now it's behind a paywall. The code is actually a distance 3 code ($[[8,3,3]]$) so that's a mistake in the paper. The process isn't too different from the approach where you measure the syndrome into a classical register and then apply the correction based on classical bits; the difference is that you apply the correction using a series of multi-qubit control gates based on the possible syndromes. I have software that generates the entire circuit automatically and I tested it for small codes; it's not python and it will take a while to translate it...Note that this needs as many ancilla qubits as stabilizers; I am not aware of a way to do this without ancilla. Where did you see the $[[5,1,3]]$ code decoded without ancilla?