Quiz on Quantum Computing

Quiz – PH534 Spring 2022
Date: 27 Jan 2023

Name: ____________________ LDAP: _____________________ Dept: _______________

0 1 0 −𝑖 1 0 1 0
For all calculations use, !𝜎! , 𝜎" , 𝜎# $ = &' *,' *,' *- and 𝕀 = ' *.
1 0 𝑖 0 0 −1 0 1
!
1. Consider the density matrix: 𝜌 = (𝕀 − 𝑟⃗. 𝜎⃗), where 𝑟⃗ = ,𝑟# , 𝑟$ , 𝑟% .. For what value of 𝑟⃗
"
does the density matrix represent a pure quantum state. (1 marks)
! ! ! ! ! ! !
A) 𝑟# = , 𝑟$ = ' , 𝑟% = " B) 𝑟# = , 𝑟$ = 𝑟% = ( C) 𝑟# = , 𝑟$ = 𝑟% = "
√" √" √"
! ! !
D) 𝑟# = 𝑟$ = 𝑟% = E) 𝑟# = , 𝑟$ = 0, 𝑟% =
" " "




! !
2. For 𝑟# = , 𝑟$ = 0, 𝑟% = , what is the value of Tr[𝜌] and Tr[𝜌$ ]? (2 marks).
" "
1 1 5 1 1
A) Tr[𝜌] = 1; Tr[𝜌$ ] = 4
B) Tr[𝜌] = 2 ; Tr[𝜌$ ] = 16 C) Tr[𝜌] = 2 ; Tr[𝜌$ ] = 4
10 5
D) Tr[𝜌] = 1; Tr[𝜌$ ] = E) Tr[𝜌] = 1; Tr[𝜌$ ] =
16 16




3. For |𝜓⟩ = 𝛼|0⟩ + 𝛽|1⟩, what are the expectation values of 6𝜓7𝜎! + 𝑖𝜎" 7𝜓9 and ⟨𝜓|𝜎# |𝜓⟩?
(2 marks).
A) 2𝛼 ∗ 𝛽; |α|" − |β|" B) 2𝛼 ∗ 𝛽; |α|" + |β|" C) |α|" + |β|" ; |α|" − |β|"
D) 2𝛽 ∗ 𝛼; |α|" + |β|" E) 2𝛽 ∗ 𝛼; |α|" − |β|"