- #1

Ted55

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- Homework Statement
- Consider the error-correction circuit

shown below. Determine the

input state |ψ> = (aI + bX1 + cX2 +

dX3)(α|000> + β|111>)|00> describing the

logical qubit with single-qubit spin-flip

errors, the state after the syndrome

diagnosis and the state after the recovery

operation.

- Relevant Equations
- Kronecker Product, ⊗

Matrix Multiplications

My two thoughts for the first part, the input state, is it not already given in the question? The error is the unitary matrix represented by the 4 term first bracket, the second bracket represents the initialised logic qubits from the first, and the |00> term is the two ancillary qubits into which the logical qubits have been moved to. Am I misinterpreting what it is asking for, should I look for a matrix representation, or perhaps re-write it in terms of the basis qubit |Φ> = α|0> + β|1> -> U|Φ> where U is the unitary matrix representing the single-qubit error? And do the same for the two bits that initialise the qubit and so on?

For the second part,the state after syndrome diagnosis, should I simply perform the associated CNOT swaps on the relevant qubit? I.e) If the second bit in the logical qubit is in the state |0> it would flip the first ancilla qubit, and so on?

For the recovery operations, do I simply take the state I've found in the second part and apply the associated Toffoli gates and Pauli-X gates, analogous to the procedure in the second part?

Would it perhaps be easier to find a matrix representation for the initial state, and doing the same for the circuit - equivalent matrices for both the syndrome diagnosis and recovery operations and then simply performing them?

Many thanks for any advice anyone can give me!