QCC 2022, Home Work 1
1. Two qubits are in the state
√1 |00⟩+ i|01⟩− 1|10⟩+ i 3|11⟩.
We measure these 2 qubits. Find the classical outputs, quantum out- puts, and the corresponding probabilities.
2. Four qubits are in the state
√1 |0000⟩ + i |0001⟩ − 1 |1010⟩ + i 3 |1011⟩
We measure qubits 1 and 2 only. Find the classical outputs, quantum outputs, and the corresponding probabilities.
If we get classical outputs j1 = 1, j2 = 0, what is the quantum state of qubits 3 and 4 after the measurement?
3. Show how S gate changes the state α|0⟩ + β|1⟩.
4. Find the unitary matrix that corresponds to the Toffoli gate.
5. In quantum teleportation protocol we assume that qubits 1 and 2 are in the state √1 (|00⟩ + |11⟩)).
2 Letusnowassumethatweprepared2qubitsinthestate√1 (|01⟩−|10⟩)
and sent qubit 1 to Alice and qubit 2 to Bob.
Draw a block diagram for quantum teleportation in these case, find states |ψ0⟩, |ψ1⟩, |ψ2⟩, and classical and quantum measurement outputs and the corresponding probabilities. Show what Bob has to do upon receiving classical measurement results.
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