2.3 Universal Quantum Gates
23
Theorem 2.3.2 An arbitrary multiple qubit gate can be composed into CNOT and
single qubit gates.
Exercise 2.3.2 Show Theorem 2.3.2.
2.3.3 Quantum Channels
There exist several types of quantum noises. The more common are the bit flip channel, which will be studied in Sect. 3.2.1 (see Definition 3.2.1), the phase shift channel, which will be presented in Sect. 3.2.2 (see Definition 3.2.2) and the depolarizing
channel, presented here.
We show how the depolarizing channel corrupts the quantum message. We start
with a single qubit which has probability p to suffer depolarization through the
channel and probability 1 − p to left unchanged. After passing through the channel,
the state is corrupted by the noise and becomes
ε(ρ) =
pI
2
+ (1 − p)ρ.
(2.1)
Since
I
2
=
ρ + Xρ X + YρY + Zρ Z
4
,
replacing such equation in Eq. (2.1) we have
ε(ρ) =
1 −
3 p
4
ρ +
p
4
(Xρ X + YρY + Zρ Z ).
If we want to make explicit the state ρ with its corresponding probability to remain
unchanged 1 − p, we write
ε(ρ) = (1 − p)ρ +
p
3
(Xρ X + YρY + Zρ Z ).
(2.2)
Notice that in Eq. (2.2), the Pauli matrices X , Y and Z act with probability p/3.