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Assume one runs a purely classical algorithm on $n$ logical qubits on a physical device with some bit flip probability.

Can implementing error correction in this case be any easier than in the case of a general quantum circuit?

I guess, given that the hardware is "quantum", to answer this question we also need to make assumptions on probabilities of hardware errors other than bit flips.

Basically I'm trying to understand whether quantum error correction has anything to offer to classical computation, at a cost less then the "full" quantum error correction.

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    $\begingroup$ Quantum error correction is made for qubit ensembles not for classical bits. $\endgroup$
    – Mauricio
    Oct 10, 2021 at 15:34

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Can implementing error correction in this case be any easier than in the case of a general quantum circuit?

Yes, for example you could use a classical error correcting code such as a repetition code. Or, and I really want to emphasize how much more efficient this would be as a strategy for running the computation, you could throw the quantum computer into a wood chipper and then run the classical computation directly on a classical computer.

I'm trying to understand whether quantum error correction has anything to offer to classical computation, at a cost less then the "full" quantum error correction.

Error correction involves doing syndrome decoding. The amount of classical compute you need to do syndrome decoding to keep the quantum computer on track will be millions of times worse than if you'd just run the computation directly on a classical computer [1] [2].

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  • $\begingroup$ I guess, my question was whether one can do syndrome detection (without measuring all qubits) and correction, if no entanglement is required for the original algorithm. $\endgroup$
    – mavzolej
    Oct 10, 2021 at 17:52
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    $\begingroup$ @mavzolej Yes, you can cut corners when only one observable has to be protected instead of an anticommuting pair. But cutting those corners will make it clear that further corners can be cut, and those will expose even more corners, and ultimately you will find that you converge on the entire process running in the classical control system with the quantum computer sitting idle. Failing to do so would indicate you aren't cutting enough corners. $\endgroup$
    – Craig Gidney
    Oct 10, 2021 at 21:27
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Peter Shor's error correction bit flip circuit

Peter Shor has two error correcting methods. One is the bit flip method and the other is the phase shift method. The bit flip method is similar to what you could use in classical computing, and is what I would recommend you use when comparing the two. The phase shift method is unique to quantum computing. This is a great link describing it. This circuit diagram shows how the bit flip error, where the qubits' computational state flips from 0 to 1 or vice versa, is corrected. This 3-qubit circuit is using two ancillary qubits to correct one qubit.

https://quantumcomputinguk.org/tutorials/quantum-error-correction-bit-flip-code-in-qiskit

The code works by first using CNOT gates to transfer the computational state of the main qubit to the other ancillary qubits. Then if an error occurs the first qubits state will be flipped. To correct the bit flip CNOT gates are applied to the ancillary qubits again and then a toffoli gate is applied to the first qubit which will correct its state.

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