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CS 190/264: Quantum Computation Instructor: Sandy Irani Homework 2
Note: graduate students are not required to do problems 3-6, but are required to probelm 9, 10, and 11. Undergraduates are required to do problems 1-8.
1. Suppose a Hermetian operator A that operatorates on vectors in C 4 has eigenvectors |v1i, |v2i, |v3i, |v4i. Moreover the set |v1i, |v2i, |v3i, |v4i form an ortho-normal basis of C 4 . The eigenvalue of |vii is λi . Suppose that λ1 > λ2 > λ3 = λ4. Give a different set of four egenvectors of A that also form an ortho-normal basis of C 4 .
2. Consider a linear operator A with eigenstates |v1i, . . . , |vN i, and corresponding eigenvalues λ1, . . . , λN .
(a) Express A in outer-bracket notation in the {|vii} basis.
(b) Let A2 be the linear operator obtained by applying A twice. The matrix representation of A2 is just the matrix representation of A squared. Express A2 in outer-bracket notation in the {|vii} basis. (Hint: take your outer-bracket notation for A in the previous question and see what happens when you apply it twice. Then simplify the expression as much as possible.)
3. Suppose the state |φi = 1 2 |0i − √ 3 2 |1i is measured in the |+i, |−i basis. What is the probability of each outcome and what is the state afterwards (depending on the outcome of the measurement)?
4. Consider a two-qubit system. Suppose that the second qubit is measured.
(a) Give an expression for the projector P0 which projects onto the subspace spanned by the states in which the outcome of the measurement is 0. Do the same for P1 which projects onto the subspace spanned by the states in which the outcome of the measurement is 1.
(b) For the state
|φi = 1 + i 3 |00i + √ 2 3 |01i + −i √ 2 3 |10i + −1 √ 3 |11i,
give the probability of each outcome of the measurement and the state afterwards.
5. Express each of the following linear opeartors as a 4 × 4 matrix.
(a) X ⊗ I
(b) I ⊗ X
(c) Z ⊗ X 1 2
6. Give the result of each operator applied to the state |Φi = √ 1 2 |00i + √ 1 2 |11i.
(a) X ⊗ I
(b) I ⊗ X
(c) Z ⊗ X
7. Consider an n-qubit system.
(a) Let I2,3,...,n denote the identity operator applied to qubits 2 through n. What are the dimensions of I2,3,...,n in matrix form?
(b) Give a schematic representation of H ⊗ I2,3,...,n.
(c) Give a schematic representation of I1,2,...,n−1 ⊗ H, where I1,2,...,n−1 is the identity operator applied to qubits 1 through n − 1.
8. If A is a linear operator, then the matrix representation of A† is obtained by taking the conjugate-transpose of the matrix representation of A. To get the conjugate-transpose of a matrix, take the transpose of the matrix and then take the complex conjugate of each entry in the matrix.
(a) What is |vi † (the conjugate-transpose of |vi)?
(b) For any two matrices A and B, if the number of columns of A is equal to the number of rows of B, then (AB) † = B†A† . Use this fact to show that (A|vi) † = hv|A† .
(c) Give an expression for (|vihw|) † that does not use the † operation.
(d) Prove that Xn j=1 cjAj !† = Xn j=1 c ∗ jA † j . You can use the fact that (A + B) † = A† + B† .
(e) The outer-bracket representation of A in basis |φ1i, . . . , |φN i is X N j=1 X N k=1 hφj |A|φki|φj ihφk|. Give an expression for A† in outer-bracket notation using the same basis. Your expression should not use the † operation.
(f) Let |v1i, . . . , |vN i set of eigenvectors for A with eigenvalues λ1, . . . , λN . In class we saw that the uter-bracket representation of A in basis |v1i, . . . , |vN i is X N j=1 λj |φj ihφj |.
Give an expression for A† in outer-bracket notation using the λ’s and the |vi’s. Your expression should not use the † operation.
For graduate students: skip problems 3-6, and do the following problem:
9. Prove that the eigenvalues of a unitary operator can be written in the form e iθ for some real θ.
10. Prove that two eigenvectors of a Hermitian matrix with different eigenvalues are orthogonal. A Hermetian matrix has real eigenvalues. You can also used the fact that if A is Hermetian, then A† = A.
11. The Hadamard operator on one qubit may be written as H = 1 √ 2 [(|0i + |1i)h0| + (|0i − |1i)h1|] .
Give a closed form expression for H⊗n using outer-bracket notation in the standard basis. (Hint: You will need to use the dot product of two strings. If x and y are two n-bit strings, then x · y = Pn j=1 xj · yj , where xj is the j th bit of x and yj is the j th bit of y.)