Chapter 3
Quantum Error-Correcting Codes
3.1 Introduction
In this chapter, we introduce the concept of quantum error-correcting codes and show
several examples chronologically. This chapter is fundamental for the development
of this work, since the main aim of this book is to present several types of different
constructions of quantum block and convolutional codes, as well as constructions of
asymmetric quantum codes.
In order to proceed further, we need to recall the concept of pure quantum state.
Definition 3.1.1 If a state |v in a quantum system is known exactly, we say that the
system is in a pure state.
There exist some apparent difficulties to construct quantum error-correcting codes.
The first situation is the impossibility of copying qubits, as we see in the well-known
No-cloning theorem.
Theorem 3.1.1 (No-cloning theorem) Assume that |v is a pure (unknown) state.
Then there is no unitary operator taking |v ⊗ |s to |v ⊗ |v (quantum copy of
|v) for all quantum states |v, where |s is a standard pure state. More precisely,
given two particular pure quantum states |v and |w to be copied, it follows that the
quantum copy process is possible only if |v = |w or if |v and |w are orthogonal.
Therefore, based on Theorem 3.1.1, it is not possible to construct a repetition quantum
code due to the impossibility of copying arbitrary qubits.
The second apparent difficulty is that the set of errors is continuous. Thus, at a
first glance, it seems that the quantum code must correct an infinity of different types
of errors.
The third difficulty is that the measurements of qubits destroy the quantum information.
Fortunately, (to guarantee our jobs) all these situations can be solved. In the
sequence, we present the first quantum code displayed in the literature which is
capable of correcting one arbitrary quantum error in any single qubit, the Shor code.
© Springer Nature Switzerland AG 2020
G. G. La Guardia, Quantum Error Correction, Quantum Science and Technology,
https://doi.org/10.1007/978-3-030-48551-1_3
25
Quantum Error-Correcting Codes
3.1 Introduction
In this chapter, we introduce the concept of quantum error-correcting codes and show
several examples chronologically. This chapter is fundamental for the development
of this work, since the main aim of this book is to present several types of different
constructions of quantum block and convolutional codes, as well as constructions of
asymmetric quantum codes.
In order to proceed further, we need to recall the concept of pure quantum state.
Definition 3.1.1 If a state |v in a quantum system is known exactly, we say that the
system is in a pure state.
There exist some apparent difficulties to construct quantum error-correcting codes.
The first situation is the impossibility of copying qubits, as we see in the well-known
No-cloning theorem.
Theorem 3.1.1 (No-cloning theorem) Assume that |v is a pure (unknown) state.
Then there is no unitary operator taking |v ⊗ |s to |v ⊗ |v (quantum copy of
|v) for all quantum states |v, where |s is a standard pure state. More precisely,
given two particular pure quantum states |v and |w to be copied, it follows that the
quantum copy process is possible only if |v = |w or if |v and |w are orthogonal.
Therefore, based on Theorem 3.1.1, it is not possible to construct a repetition quantum
code due to the impossibility of copying arbitrary qubits.
The second apparent difficulty is that the set of errors is continuous. Thus, at a
first glance, it seems that the quantum code must correct an infinity of different types
of errors.
The third difficulty is that the measurements of qubits destroy the quantum information.
Fortunately, (to guarantee our jobs) all these situations can be solved. In the
sequence, we present the first quantum code displayed in the literature which is
capable of correcting one arbitrary quantum error in any single qubit, the Shor code.
© Springer Nature Switzerland AG 2020
G. G. La Guardia, Quantum Error Correction, Quantum Science and Technology,
https://doi.org/10.1007/978-3-030-48551-1_3
25