MASSACHUSETTS INSTITUTE OF TECHNOLOGY
2.111J/18.435J/ESD.79
Quantum Computation
QUIZ 1 SOLUTION
Problem 1. In NMR quantum computing, a Hadamard gate is implemented by rotating around
the axis (x + z )/ 2 . Compute the matrix obtained by rotation around this axis by π radians,
and compare to a Hadamard gate.
Solution:
If we denote the rotation by angle θ about (x + z )/ 2 by R(θ) , we have
R(θ) = exp[−i(θ / 2)(σX + σZ )/ 2}
θ
θ
= cos I − i sin (σX + σZ )/ 2
2
2
⇒
R(π) = −i(σX + σZ
)/ 2
−i   0 1   1 0  

=
+


2   1 0   0 −1  
−i  1 1 
=
2  1 −1 
= −iH
where H is the Hadamard gate.
Problem 2. Let
1
H = (σX ⊗ σX + σY ⊗ σY + σZ ⊗ σZ + I ⊗ I )
2
be an operator on two qubits.
a)
b)
c)
d)
Find H 2 and write it in a simple form.
Using (a), find exp(−i πH / 4) and exp(−i πH / 2) . Find the eigenvalues of H . Find a set of orthonormal eigenstates of H . Solution:
a) We have
1
H 2 = (σX ⊗ σX + σY ⊗ σY + σZ ⊗ σZ + I ⊗ I )H .
2
Note that
1
1
(σX ⊗ σX )H = (σX ⊗ σX )(σX ⊗ σX + σY ⊗ σY + σZ ⊗ σZ + I ⊗ I )
2
4
1
= (σX σX ⊗ σX σX + σX σY ⊗ σX σY + σX σZ ⊗ σX σZ + σX ⊗ σX )
4
1
= (I ⊗ I + i σZ ⊗ iσZ + (−i )σY ⊗ (−i )σY + σX ⊗ σX )
4
1
= (I ⊗ I − σZ ⊗ σZ − σY ⊗ σY + σX ⊗ σX ) .
4
Similarly,
1
1
(σY ⊗ σY )H = (−σX ⊗ σX + σY ⊗ σY − σZ ⊗ σZ + I ⊗ I )
2
4
1
1
(σ ⊗ σZ )H = (−σX ⊗ σX − σY ⊗ σY + σZ ⊗ σZ + I ⊗ I )
2 Z
4
1
H
(I ⊗ I )H = .
2
2
Adding up these four relations, one can obtain
H2 = I ⊗ I .
b) Using equation (4.7) of N&C, we have
exp(i θH ) = cos(θ)I ⊗ I + i sin(θ)H
⇒
exp(−i πH / 4) =
and
2I ⊗ I / 2 − i 2H / 2
exp(−i πH / 2) = −iH .
c) Using Problem 1(b) in Problem Set 2, it can be seen that the only possible values for the
eigenvalues are +1 and –1.
d) You can easily verify that the Bell states, described in the first problem of Problem Set 3, are
one possible set of eigenstates. (In fact, H = Ι2AB − I ⊗ I .) The first state in that problem, the
singlet sate, has eigenvalue –1 and the other three have eigenvalues +1.
Problem 3. Let N be an integer larger than 5. Consider the following state:
1
ψ =
N
N −1
∑
x mod N
x =0
A
⊗ 3x mod N
B
⊗ 5x mod N
C
.
Find the output state if we take a quantum Fourier transform modulus N on each of the registers
A, B, and C. That is, if we denote the corresponding QFT operators to each system by U A , U B ,
and UC , find U A ⊗ U B ⊗ UC ψ . Write your answer in the basis { 0 , 1 ,…, N − 1 }⊗3 , and
show that it is the superposition of equally probable states. What is this probability?
Solution:
U A ⊗ U B ⊗ UC
N −1
1
ψ =
N
=
=
=
=
=
∑U
( )
1
N
1
N2
1
N2
1
N2
1
N
x mod N
A
x =0
4 N −1 N −1
∑ ∑e
A
⊗ U B 3x mod N
k
A
⊗
∑e
∑∑∑
k
∑∑∑
k
k =0 m =0 n =0
N −1 N −1 N −1
∑∑
m
B
⊗
∑ e2πi(5x )n
n =0
m= 0
x = 0 k = 0 m = 0 n = 0
N −1 N −1
N −1
k
A
m
B
nC
N −1
A
m
A
m
B
nC
∑ e2πi(k +3m +5n )x
x =0
B
n C N δk ,−3m −5n mod N
−3m − 5n mod N
A
C
N −1
2πi (3x )m
∑ ∑ ∑ ∑ e2πi(k +3m +5n )x
m =0 n =0
⊗ UC 5x mod N
N −1
2πixk
x =0 k =0
N −1 N −1 N −1 N −1
k = 0 m = 0 n = 0
N −1
N −1
B
m
B
nC.
This is the superposition of N 2 states each with probability of occurrence 1/ N 2 .
nC