Chapter 7
Quantum Error Correction
7.1
A Quantum ErrorCorrecting Code
In our study of quantum algorithms, we have found persuasive evidence that
a quantum computer would have extraordinary power.
But will quantum
computers really work? Will we ever be able to build and operate them?
To do so, we must rise to the challenge of protecting quantum information
from errors.
As we have already noted in Chapter 1, there are several as
pects to this challenge. A quantum computer will inevitably interact with its
surroundings, resulting in decoherence and hence in the decay of the quan
tum information stored in the device.
Unless we can successfully combat
decoherence, our computer is sure to fail. And even if we were able to pre
vent decoherence by perfectly isolating the computer from the environment,
errors would still pose grave difficulties. Quantum gates (in contrast to clas
sical gates) are unitary transformations chosen from a continuum of possible
values. Thus quantum gates cannot be implemented with perfect accuracy;
the effects of small imperfections in the gates will accumulate, eventually
leading to a serious failure in the computation. Any effective strategem to
prevent errors in a quantum computer must protect against small unitary
errors in a quantum circuit, as well as against decoherence.
In this and the next chapter we will see how clever encoding of quan
tum information can protect against errors (in principle). This chapter will
present the theory of quantum errorcorrecting codes.
We will learn that
quantum information, suitably encoded, can be deposited in a quantum mem
ory, exposed to the ravages of a noisy environment, and recovered without
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CHAPTER 7.
QUANTUM ERROR CORRECTION
damage (if the noise is not too severe). Then in Chapter 8, we will extend the
theory in two important ways. We will see that the recovery procedure can
work effectively even if occasional errors occur during recovery. And we will
learn how to
process
encoded information, so that a quantum
computation
can be executed successfully despite the debilitating effects of decoherence
and faulty quantum gates.
A quantum errorcorrecting code (QECC) can be viewed as a mapping
of
k
qubits (a Hilbert space of dimension 2
k
) into
n
qubits (a Hilbert space
of dimension 2
n
), where
n > k
.
The
k
qubits are the “logical qubits” or
“encoded qubits” that we wish to protect from error. The additional
n

k
qubits allow us to store the
k
logical qubits in a redundant fashion, so that
the encoded information is not easily damaged.
We can better understand the concept of a QECC by revisiting an ex
ample that was introduced in Chapter 1, Shor’s code with
n
= 9 and
k
= 1.
We can characterize the code by specifying two basis states for the code sub
space; we will refer to these basis states as

¯
0
)
, the “logical zero” and

¯
1
)
, the
“logical one.” They are

¯
0
)
= [
1
√
2
(

000
)
+

111
)
)]
⊗
3
,

¯
1
)
= [
1
√
2
(

000
)  
111
)
)]
⊗
3
;
(7.1)
each basis state is a 3qubit cat state, repeated three times.
As you will
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 Fall '11
 JohnPreskill
 Power, Work, Coding theory, Hamming Code, Error detection and correction, Quantum information science, Quantum error correction

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