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QUANTUM COMPUTING

COMP 4114

October 1, 2024

Solve the problems below. Your answers should be detailed, complete, precise, concise and clear. Write the solutions on your own and acknowl- edge your sources in case you used“library”material. Look in the course webpage on how to avoid plagiarism and for submission details. Exercises marked with a (★) are usually more challenging and may involve mate-  rial not covered in class and some of the exercises may require material  that will be covered in forthcoming lectures. It is preferred that you type your work using your favourite software package but you submit only in pdf. At the discretion of the TA a 5% bonus will be added for carefully  typed and rigorously explained work.  Two excellent and free packages are LATEX for typesetting mathematics (add in the preamble the packages “braket”and“kpfonts”), Ipe (for drawing pictures), and qcircuit to plot quantum circuits.

Assignment AQ2

1    [10 pts] October 1, 2024

Consider the two Pauli matrices  calculate the tensor products σ1 σ2  and σ2 σ1 . Give details of your answers.

2    [10 pts] October 1, 2024

Consider the three Pauli matrices 

Show that for all i,j ∈ {1, 2, 3}, where δi,j = 1, if i = j, and = 0,otherwise.

3    [10 pts] October 1, 2024

Give a pair among the set of three vectors  which form an orthonormal basis.  How many such pairs exist?  Show details and justify your work.

4    [5 pts] October 1, 2024 

Calculate the commutators [σ1, σ2], [σ2, σ3], [σ3, σ1]. Show your work.

5    [5 pts] October 1, 2024

Consider the matrices  and compute their eigen- values using the characteristic polynomial equation det(M - zI) = 0, for M = A and M = B separately.

6    [10 pts] (★) October 1, 2024

Prove or disprove: Is it true that if [A, B] = [A, C] = 0 then [B, C] = 0, for 2 × 2 Hermitian matrices A,B,C, where 0 is the 2 × 2 zero matrix?  Hint: Use as A the identity matrix.