Midterm Exam, 2022
|v⟩ = 12i|011⟩ − 34|101⟩ − i12|111⟩
2. (2) (The vector |v⟩ = √1 (|000⟩ + |111⟩) is a quantum state of how many qubits?
3. (5) Among the following vectors
|ψ1⟩ = √1 (e2πi 41 |00⟩ + e2πi 34 |11⟩), 2
|ψ2⟩ = √1 (e2πi 41 |01⟩ + e2πi 34 |10⟩), 2
|ψ3⟩ = √1 (|10⟩ − |01⟩), 2
|ψ4⟩ = √1 (|01⟩ + |10⟩), 2
there are two that define the same quantum state. Find them.
4. (3) Write in Dirac’s notation the quantum state
Problem 1 Quantum state. 1. (3) Why the vector
is not a quantum state?
−1/2 v=0.
√ 1/2
Problem 2 (7) Using CNOT, H, NOT, and Z gates construct a circuit that moves the input state |0⟩|0⟩ into √1 (|01⟩ − |10⟩).
(Hint: look at lecture notes for the circuit for creating the EPR pair √1 (|00⟩ + |11⟩). 2
Problem 3 Assume we have the following quantum circuit
1. (10) Find the unitary operator U that corresponds to this circuit. 1
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2. (4) Use this U to find the output quantum state in linear algebra notation.
3. (5) Track the evolution of the input quantum state along the circuit using Dirac’s notation
to get the output state.
Problem 4 (10) Let us measure qubits 1 and 3 of the quantum state
1|0000⟩ + i√1 |0101⟩ + 2244
3|1010⟩ − i1|1011⟩. Find all classical and quantum outputs with the corresponding probabilities.
Problem 5 (11) Assume we have two qubits in the state √1 (|01⟩ + |10⟩) and send the 1st qubit 2
to Alice and the 2nd qubit to Bob. Alice and Bob want to use these qubits for superdense coding. Define Alice’s actions and construct the quantum circuit that should be used by Bob.
Problem 6 Let x = 3 and N = 5 and we want to construct a quantum circuit for finding the order of x modulo N.
1. (11) Find the unitary transformation U that we should use. 2. (4) Draw the entire circuit with t = 2.
Problem 7 The previous problem continued.
1. (7)Findallvectors|uj⟩, j=0,…,r−1(explicitly).
2. (5) Find the quantum state before the measurement blocks.
3. (5) Find all possible classical and quantum measurement outputs with the corresponding probabilities.
Problem 8 Assume we have two qubits in the state √1 (|01⟩ − |10⟩) and send the 1st qubit to 2
Alice and the 2nd qubit to Bob. At a moment t Bob measures his qubit and gets classical result 1. Next, at a moment t + δ Alice measures her qubit.
1. (4) What can you say about Alice’s classical measurement result?
2. (4) What can you say about the joint quantum state of these two qubits after the measure- ments?
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