(57) A quantum computer implemented probabilistic error correction method for use with
quantum circuits with a noise channel N(X) that has a Choi rank not greater than two,
such as N(X) = N0XN0† + N1XN1†. The method comprising steps of: encoding a first qubit into an encoded state |ψ〉, putting a second qubit into an arbitrary fixed state |d1〉, implementing two qubit unitary operation UE on the first qubit and the second qubit to obtain an encoded state UE(||ψ〉 ⊗ |d1〉), and after the encoded state is affected by the noise N(X), implementing two qubit
unitary operation UD, followed by measuring the second qubit in the {|d2〉,
} basis measurement to obtain a classical label i ∈ {0,1}, next preparing a third qubit in the arbitrary fixed state |d3〉, and implementing two qubit unitary operation VD on the first qubit and the third qubit, followed by measuring the third qubit in
the {|d4〉,
} basis measurement to obtain a classical label j ∈ {0,1}, and when (i,j) = (0,0) accepting as an output δexp a decoded state of the first qubit, and when (i,j) ≠ (0,0) rejecting the output δexp of a decoded state of the first qubit.
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