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Swap test is a simple quantum circuit to measure the inner product of two quantum states [Wiki: swap test], it only contains three quantum gates. However, due to the error in the real quantum computer (gate error, measurement error, decoherence, etc), the swap test can not be perfectly implemented.

How to design a quantum circuit that applies the quantum-error-correction method to handle the errors? Is it possible to handle arbitrary errors?

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How to design a quantum circuit that applies the quantum-error-correction method to handle the errors?

Each qubit in the circuit has to be encoded in a quantum error correction code. You need to choose a quantum error correction code for which you know how to fault-tolerantly implement a universal logical gate set. Then you have to decompose the controlled swap into this logical gate set.

Is it possible to handle arbitrary errors?

Yes, fault-tolerant quantum computing is possible.

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  • $\begingroup$ There is a subtlety here. If $|\psi\rangle$ and $|\phi\rangle$ are provided as single qubit states, then one way is to start by encoding them into $n$-qubit encoded states. But that process of encoding itself will have errors that can destroy the states. A better way would be to teleport the one-qubit input states onto pre-prepared $n$-qubit encoded zero states. The core idea being to aggressively minimize the number of identity or non-identity gates being applies to your input states. $\endgroup$ Aug 27, 2023 at 20:29

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