One of the main results of the theory of quantum error correction is that
any general quantum error can be composed only of discrete errors represented
by the Pauli operators X, Z, and Y = −iXZ. Errors are induced on our qubit
system due to effects of everything outside this system, which we will call
the environment. The environment interacts weakly with the system to cause
a change in the amplitudes of the basis states, a process called decoherence
of the initial state. Initially, let’s assume the system is created in a definite
state |ψi. The environment has been excluded experimentally, so that the
combined environment-qubit system is in a product state: |ei|ψi. Subsequent
interaction between the two results in a change of this state. In order to
model this evolution, we represent the transformation of the computational
basis states by
|ei|0i −→ |e
1
i|0i + |e
2
i|1i; (10.18a)
|ei|1i −→ |e
3
i|0i + |e
4
i|1i. (10.18b)
Here the kets |e
i
i are (un-normalized) environment states that can be ex-
pressed as fractions of |ei: |e
i
i = a
i
|ei. For example, the bit-flip error can
be modelled with a
1
= 0 = a
4
, a
2
= 1 = a
3
and the phase flip by
a
2
= 0 = a
3
, a
1
= 1 = −a
4
. Now we want to be able to recognize the ef-
fect of such an evolution on the superposition state |ψi = α|0i + β|1i, as an
operation on the qubit system alone. In order to separate the effects on |0i
and |1i we’ll now write a general error in the terms of the projectors
P
0
= |0ih0|, and P
1
= |1ih1|. (10.19)
So we can write Equations 10.18 as
|ei|0i −→
|e
1
iP
0
+ |e
2
iXP
0
|0i (10.20a
|ei|1i −→
|e
3
iXP
1
+ |e
4
iP
1
|1i. (10.20b)
The error acting on |ψi can be written as
|ei|ψi −→
h
|e
1
i + |e
2
iX
P
0
+
|e
3
iX + |e
4
i
P
1
i
|ψi. (10.21)
Now the projection operators can be written in terms of the Pauli matrices:
Z = |0ih0| − |1ih1|; = |0ih0| + |1ih1|;
=⇒ P
0
=
+ Z
2
; P
1
=
− Z
2
. (10.22)
Also, using XZ = iY , we get
|ei|ψi −→
|E
1
i + |E
2
i
ˆ
X + |E
3
i
ˆ
Y + |E
4
i
ˆ
Z
|ψi, (10.23)
where we have appropriately regrouped the environment states |e
i
i to obtain
the new environment states |E
i
i. (We do not care about the exact form of
these states since we are not going to observe them.) We thus see that the
generic error can be expressed as a linear combination of the discrete errors
corresponding to the action of the Pauli matrices.
If we encode using n qubits for error-correction, then a generic state would
become
|ei|
˜
ψi
n
−→
|di +
n
X
i=1
(|a
i
i
ˆ
X
i
+ |b
i
i
ˆ
Y
i
+ |c
i
i
ˆ
Z
i
)
!
|
˜
ψi
n
. (10.24)
In order to diagnose the syndromes, the 2
n
-d Hilbert space must admit at
least 1 + 3n 2-d subspaces:
2
n−1
≥ 1 + 3n,
so n = 5, 7, 9…
Thus the minimum codeword size is 5 qubits. We can see now that the
9-qubit Shor code is not efficient; we can make do with fewer qubits
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