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CS 190/264: Quantum Computation Homework 1
Note: graduate students are not required to do problems 1, 2, and 4, but are required to probelm 9. Undergraduates are required to do problems 1-8.
1. Let α and β be two complex numbers.
(a) Prove that (αβ) ∗ = α ∗β ∗ .
(b) Prove that (α + β) ∗ = α ∗ + β ∗ .
2. Let α = 3e iπ2/3 and β = 2e −iπ/4 .
(a) What is |α|?
(b) What is α ∗ ?
(c) Express α in standard (a + bi) form.
(d) What is α · β? (e) What is α ∗ · β?
3. Define
|φi = √ 3 4 |000i+ 1 4 |001i+ 1 2 √ 2 |010i+ 1 2 √ 2 |011i+ 1 2 √ 2 |100i+ 1 2 √ 2 |101i+ 1 2 √ 2 |110i+ 1 2 √ 2 |111i.
(a) Suppose that all three qubits are measured. What is the probability that the outcome of the measurement is 001?
(b) If the outcome of the measurement is 001, then what is the state after measurement?
(c) Consider the state |φi again before measurement. Suppose now that only the last bit is measured. What’s the probability that the outcome of the measurement is 1?
(d) If the last bit is measured and the outcome is 1, then what is the state after measurement?
4. Consider the state of a single qubit
|φi = e iπ/6 √ 3 |0i + √ 2e −iπ/4 √ 3 |1i.
If the qubit is measured, what’s the probability that the outcome is 0?
5. Define the states |φi and |ψi as: |φi = 1/ √ 3 (1 + i)/ √ 6 0 i/√ 3 and |ψi = i/2 (1 − i)/4 (1 + i)/4 1/2 .
(a) Express hψ| as a row or column vector.
(b) Express √ 3 2 |ψi + 1 4 |φi as a row or column vector.
(c) Calculate hφ|ψi.
(d) What is the L2 norm of |ψi?
6. Suppose that |φi ∈ C N . Prove that hφ|φi is always a real non-negative number.
7. Consider the following four states of a 2-qubit system:
• |φ0i = 1 2 (|00i + |01i + |10i + |11i)
• |φ1i = 1 2 (|00i − |01i + |10i − |11i)
• |φ2i = 1 2 (|00i + |01i − |10i − |11i)
• |φ3i = 1 2 (|00i − |01i − |10i + |11i)
(a) Do the states |φ0i, |φ1i, |φ2i, and |φ3i form an orthonormal basis of the Hilbert space spanned by the 2-qubit system?
(b) Express |01i as a linear combination of the |φj i’s.
(c) Express the following linear operator in matrix form: |φ2ihφ0|,
(d) Without creating the matrix, determine how the operator |φ3ihφ2| acts on a state |φi = α0 α1 α2 α3 .
That is, express the result of the operator |φ3ihφ2| acting on |φi as a function of the α’s.
8. Recall the definitions for |+i and |−i which form an orhonormal basis of C 2 : |+i = 1/ √ 2 1/ √ 2 and |−i = 1/ √ 2 −1/ √ 2 . Express the following linear operator using outer-bracket notation and the |+i, |−i basis: A = 0 −i i 0 .
For graduate students: skip problems 1, 2, and 4, and do the following problem:
9. Let A be a linear operator acting on C N . |v1i, . . . , |vN i and |w1i, . . . , |wN i are two different ortho-normal bases of C N . We will define two N × N matrics A0 and A00. The elements of A0 and A00 are A0 ij = hvi |A|vj i and A00 ij = hwi |A|wj i. Characterize the relationship between A0 and A00. In other words, describe a way to transform matrix A0 into the matrix A00 .