MULT20015 Elements of Quantum Computing
Assignment 1
Due: 5pm, 20th August 2021
Instructions: Work on your own, attempt all questions. Submit your completed written work electronically as a pdf (no other formats accepted) to LMS, with name and student number on the front, on or before the due date. Please show all working. Instructions on LMS submission to follow.
Total marks = 20. Number of questions = 3. 1. [1 + [0.5+0.5+0.5+2.0] = 4.5 marks]
(a) Consider the following single-qubit operators in matrix form, corresponding to rotations by angle 𝜃! about X, Y or Z axes:
cos’! −𝑖sin’! cos’! −sin’! 𝑅”(𝜃!)=𝑒#$/&’ & & 0, 𝑅((𝜃!)=𝑒#$/&’ & & 0,
−𝑖sin’! cos’! sin’! cos’! && &&
cos’! −𝑖sin’! 0 𝑅)(𝜃!) = 𝑒#$/& ‘ & & ‘!
0
Explain how these operators are related to the familiar Pauli matrices, X, Y and Z given in lectures. (NB. For future reference, these expressions explicitly include a global phase of 𝜋/2)
(b) A particular single qubit state |𝜓⟩ = 𝑎*|0⟩ + 𝑎+|1⟩, is obtained by rotating about the x+z axis from an initial state |0⟩ (zero global phase). The amplitudes are given by QUI as:
(i) Plot the amplitudes 𝑎* and 𝑎+ in the complex plane.
MULT20015 Assignment 1, © L. Hollenberg et al 2021 1
‘!0. cos& +𝑖sin&
(ii) Convert the state to QUI polar notation form: |𝜓⟩ = |𝑎*|𝑒#'”|0; + |𝑎+|𝑒#’#|1⟩ with the basis state phase angles 𝜃* and 𝜃+ in radians (expressed as multiples of 𝜋), and degrees.
(iii) Now convert to “Bloch sphere” form |𝜓⟩ = 𝑒#’$%&'(%