MASSACHUSETTS INSTITUTE OF TECHNOLOGY
2.111J/18.435J/ESD.79
Quantum Computation
Problem 1. For any unit vector j = ( jx , jy , jz ) we can define the following operator
σ j = jx σX + jy σY + jz σZ
which corresponds to a π-radian rotation about j-axis.
(a) show that σ 2j = I .
(b) σ j has two eigenvalues: +1 and –1.
(c) Find the eigenvectors +
+1 and –1.
j
and −
j
respectively corresponding to eigenvalues
Problem 2. Find an approximation to
e iθσY / 2e iθσX / 2e −iθσY / 2e −iθσX / 2
up to the second order of θ for θ 1 .
Problem 3. Rewrite
{+
A
1
( 0 A ⊗ 0 B + 1 A ⊗ 1 B ) in the following basis
2
⊗ + B , + A ⊗ − B , − A ⊗ + B , − A ⊗ − B }.
What are Pr(++) , Pr(+−) , Pr(−+) , and Pr(−−) ?
Problem 4. For any two qubits ψ
following property:
U ψ
A
⊗ φ
and φ , find a unitary operator U with the
B
= φ
A
⊗ ψ
B.
Write down U in the basis { 0 A ⊗ 0 B , 0 A ⊗ 1 B , 1 A ⊗ 0 B , 1 A ⊗ 1 B } .
Problem 5. Write out
in the following basis:
1
(0
2
A
⊗ 1C + 1 A ⊗ 0C )⊗ + B
{ 000
ABC
, 001 ABC , 010 ABC , 011 ABC , 100 ABC , 101 ABC , 110 ABC , 111 ABC } .
Problem 6. Simplify the following expression:
A
+(0
A
⊗ 1B + 1A ⊗ 0
B
)/ 2 .