Introduction to Quantum Physics and Information Processing

Problems 141

(a) HXH = Z

(b) HY H = −Y

(c) HZH = X

7.3. Show the following relations concerning rotation matrices:

(a) R

n

(θ

1

)R

n

(θ

2

) = R

n

(θ

1

+ θ

2

)

(b) XR

n

(θ)X = R

n

(−θ)

7.4. The “SWAP” gate S interchanges two inputs, defined by

S|xyi = |yxi.

(a) Give the matrix representing this gate.

(b) Show that it can be implemented by 3 CNOT gates as

S

12

= C

12

C

21

C

12

.

(c) Show that the matrix is equivalent to

S

12

=

1

2

( + X

1

X

2

+ Y

1

Y

2

+ Z

1

Z

2

)

7.5. The controlled phase-flip gate takes |11i to −|11i while leaving the other

basis states unchanged. It is sometimes represented as follows, since its

action is symmetric in the inputs:

|xi

•

|xi

|yi

•

(−1)

xy

|yi

(a) Construct the matrix for this