Quantum 3-Qubit Error Correction Circuit

[Ted55]

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

Hello! I have this problem regarding the analysis of an error correction circuit. This problem set has been set for hand-in before the lectures regarding error-correction take place, so I am struggling. With that in mind I have read about the initialisation of the first qubit with two others, and the single-qubit error moves the quantum state into a subspace spanned by the computational basis states with distance 0 and 1 to the logical qubits.

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!

 

[mark726]

Hello,

Thank you for your post and for sharing your thoughts on the problem. It seems like you have a good understanding of the concepts involved in the error correction circuit. I would like to offer some clarification and guidance to help you with the problem.

For the first part, the input state is indeed given in the question. However, you are correct in thinking that you may need to find a matrix representation for the initial state and the circuit. This will make it easier to perform the necessary operations and understand the effects of the error correction circuit.

In terms of the error, the unitary matrix represented by the 4-term first bracket is correct. This matrix represents the single-qubit error that occurs. However, the second bracket does not represent the initialized logical qubits. It represents the ancillary qubits into which the logical qubits have been moved to. The |00> term is the initial state of these ancillary qubits.

For the second part, you are correct in thinking that you need to perform CNOT swaps on the relevant qubits based on the syndrome diagnosis. This will move the logical qubits back into their correct positions.

For the recovery operations, you will need to apply the associated Toffoli and Pauli-X gates to correct the errors. This can be done by finding a matrix representation for the circuit and performing the operations on the matrix.

I would recommend finding a matrix representation for the initial state and the circuit, as this will make it easier to perform the necessary operations and understand the effects of the error correction circuit. It may also be helpful to re-write the circuit in terms of the basis qubit |Φ> = α|0> + β|1> and apply the necessary operations on the matrix.

I hope this helps and good luck with your analysis of the error correction circuit! Let me know if you have any further questions or need any additional clarification.