(5/5)

1. Bra-Ket Gymnastics. (15 Points)

(a) Let ψ and ϕ be normalized states. Prove that ψ ϕ 2 1. (5 Points)

(b) Suppose we are given an orthonormal basis set i : i = 1, . . . , n . Write the identity matrix in terms of these basis vectors. (5 Points)

(c) Let A = ϕ ψ and B = ϕ′ ψ′ . Write down Tr(AB) in terms of inner products. (Here Tr(X) is the trace of the matrix X.) (5 Points)

2. Measurement and distinguishability in different bases (40 Points)

(a) Show that no measurement can distinguish ϕ and eiθ ϕ for any θ R. Here “distinguishing” means that the statistical distribution of the measurement outcome is different. (10 Points)

(b) Suppose we are given |+⟩ = √1 (|0⟩ + |1⟩) and |−⟩ = √1 (|0⟩ − |1⟩). Can we distinguish these two

states using a measurement in the 0 , 1 basis? Explain you reasoning. (10 Points)

(c) Suppose we are given two states, ψ and ϕ , which are orthogonal to each other. Explain why there exists a measurement basis that can perfectly distinguish the two states. That is, there is a measure- ment basis in which the statistical distribution of the measurement outcome of ψ and ϕ are perfectly distinguishable from each other, with zero error. (10 Points)

(d) Suppose we are given two states, ψ and ϕ , which have a nonzero overlap, i.e., ψ ϕ 2 > 0. Explain why there is no measurement that can perfectly distinguish the two states. (10 Points)

3. Partial Measurements (20 Points)

(a) Consider a quantum state over two qubits

1|ψ⟩ = √2 (|00⟩ + |11⟩).

Suppose we meausure the first qubit in the {|+⟩, |−⟩} basis, where |±⟩ =

√1 (|0⟩ ± |1⟩). Write down the

post-measurement state (over the second qubit) and their respective probabilities. (5 Points)

(b) Using partial measurement, we can teleport a qubit. This means that we can send an unknown quantum state from one location to another. Here I will explain the procedure, while omitting the last step. Figure out this last step.

1. Start with the state |ψ⟩|+⟩, where |ψ⟩ is an arbitrary single-qubit state and |+⟩ = √1 (|0⟩ + |1⟩).

2. Apply the CZ gate to this state. The action of the CZ gate on the basis states is defined as

CZ|00⟩ = |00⟩, CZ|01⟩ = |01⟩, CZ|10⟩ = |10⟩, and CZ|11⟩ = −|11⟩.

3. Measure the first qubit.

4. Depending on the measurement outcome, apply some unitary operator to the second qubit.

5. The end result: The state of the second qubit is |ψ⟩.

The question is this: What operator should be applied in the fourth step and under what condition? (15 Points)

4. Pauli Matrices (15 Points)

(a) Prove that all single-qubit Pauli operators square to one. (5 Points)

(b) Write the eigenvalues of the single-qubit Pauli operators. (Hint: You can use the answer in (a).) (5 Points)

(c) Write all possible eigenvalues of multi-qubit Pauli operators. (5 Points)

5. Clifford Hierarchy (10 Points)

(a) The full Clifford group can be generated by H, S, and CNOT . Explain why that is true. (5 Points)

(b) Another useful gate in quantum computing is CCZ gate, defined as CCZ x y z = x y z only if x = y = z = 1 and CCZ x y z = x y z otherwise. (Here x, y, z 0, 1 .) Where does this gate belong in the Clifford hierarchy? (5 Points)

6. (Optional Challenge Problem) Find a way to implement Toffoli using Clifford + T . Conversely, find a way to implement Clifford + T using Toffoli + H. (20 Points)

(5/5)

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