Consider the single bit flip error detecting and correcting circuit as shown below. (Note that, the choice of convention does not affect the answer in this problem.) Given that the error gate ‘E’ is of the form I1⊗X2⊗I3{“version”:”1.1″,”math”:”\(I_1 \otimes X_2 \otimes I_3\)”}.What is the state of the ancillary qubit A2{“version”:”1.1″,”math”:”\(A_2\)”} at the ‘Stage’ marked in red?

In the following circuit, what is the probability for the measurement outcome of Qubit 1 to be in state |1⟩? Here, the states are expressed in the convention |Qubit2⊗Qubit1⟩{“version”:”1.1″,”math”:”\(\vert Qubit_2 \otimes Qubit_1 \rangle\)”}.

Consider two qubits both of which are initialized in the |1⟩{“version”:”1.1″,”math”:”\(\vert 1 \rangle\)”} state. Here, please choose the convention that the left qubit is the control while the right qubit is the target i.e., CNOT(|Control⟩⊗|Target⟩){“version”:”1.1″,”math”:”\(CNOT(\vert Control \rangle \otimes \vert Target \rangle)\)”} .Which of the following gate sequences acting on the initial state |1⟩⊗|1⟩{“version”:”1.1″,”math”:”\(\vert 1 \rangle \otimes \vert 1 \rangle\)”} create the Bell basis state |ψ+⟩=12[|0⟩|1⟩+|1⟩|0⟩]{“version”:”1.1″,”math”:”\(\vert \psi^+ \rangle = \dfrac{1}{\sqrt{2}} [\vert 0 \rangle \vert 1\rangle + \vert 1\rangle \vert 0 \rangle]\)”}?

Consider the single bit flip error detecting and correcting circuit as shown below. (Note that, the choice of convention does not affect the answer in this problem.) Given that the error gate ‘E’ is of the form I1⊗X2⊗I3{“version”:”1.1″,”math”:”\(I_1 \otimes X_2 \otimes I_3\)”}.What is the state of the ancillary qubit A2{“version”:”1.1″,”math”:”\(A_2\)”} at the ‘Stage’ marked in red?

Consider the single bit flip error detecting and correcting circuit as shown below. (Note that, the choice of convention does not affect the answer in this problem.) Given that the error gate ‘E’ is of the form I1⊗X2⊗I3{“version”:”1.1″,”math”:”\(I_1 \otimes X_2 \otimes I_3\)”}.What is the state of the ancillary qubit A1{“version”:”1.1″,”math”:”\(A_1\)”} at the ‘Stage’ marked in red?

Consider the single bit flip error detecting and correcting circuit as shown below. (Note that, the choice of convention does not affect the answer in this problem.) Given that the error gate ‘E’ is of the form I1⊗X2⊗I3{“version”:”1.1″,”math”:”\(I_1 \otimes X_2 \otimes I_3\)”}.What is the state of the ancillary qubit A1{“version”:”1.1″,”math”:”\(A_1\)”} at the ‘Stage’ marked in red?

Consider a qubit to be in the y-basis state |ψ⟩=12(|0⟩+i|1⟩){“version”:”1.1″,”math”:”\(\vert \psi \rangle = \dfrac{1}{\sqrt{2}} (\vert 0 \rangle + i \vert 1 \rangle)\)”}. The Hadamard gate operation on the y-basis state |ψ⟩{“version”:”1.1″,”math”:”\(\vert \psi \rangle\)”} is

Consider a quantum universe to be composed of a system qubit labeled `s’ and a three level quantum environment labeled by ‘e’, i.e. environment state space has three levels (|0⟩e{“version”:”1.1″,”math”:”\(\vert 0 \rangle_{e}\)”}, |1⟩e{“version”:”1.1″,”math”:”\(\vert 1 \rangle_{e}\)”}, |2⟩e{“version”:”1.1″,”math”:”\(\vert 2 \rangle_{e}\)”}). Let the quantum state of the universe be |ψ⟩=12[|0⟩s|0⟩e+|0⟩s|2⟩e−|1⟩s|0⟩e−|1⟩s|2⟩e]{“version”:”1.1″,”math”:”\(\vert \psi \rangle = \dfrac{1}{2} \left[ \vert 0 \rangle_{s} \vert 0 \rangle_{e} + \vert 0 \rangle_{s} \vert 2 \rangle_{e} – \vert 1 \rangle_{s} \vert 0 \rangle_{e} – \vert 1 \rangle_{s} \vert 2 \rangle_{e} \right]\)”}. What is the effective density matrix describing the system qubit ρsys{“version”:”1.1″,”math”:”\(\rho_{sys}\)”} (can be obtained by taking partial trace over the environment)?

Consider a qubit to be in the state |1⟩{“version”:”1.1″,”math”:”\(\vert 1 \rangle\)”}. What is the probability to find the spin projection along the y-axis (corresponding operator σy{“version”:”1.1″,”math”:”\(\sigma_y\)”}) to be -1{“version”:”1.1″,”math”:”-1″}?

Consider a qubit which is found to be in the state |0⟩≡|+⟩+|−⟩2{“version”:”1.1″,”math”:”\(\vert 0 \rangle \equiv \dfrac{\vert + \rangle + \vert – \rangle}{\sqrt{2}}\)”} with probability 5/8 and in the state |1⟩=|+⟩−|−⟩2{“version”:”1.1″,”math”:”\(\vert 1 \rangle = \dfrac{\vert + \rangle – \vert – \rangle}{\sqrt{2}}\)”} with probability 3/8.  What is the density matrix of the qubit in the sign basis space, i.e. in the {|+⟩,|−⟩}{“version”:”1.1″,”math”:”\(\{\vert + \rangle, \vert – \rangle\}\)”} basis?