5.5 Quantum Synchronizable Codes
119
problem, synchronization can be achieved using a classical system external to the
quantum system, but such a solution does not take advantage of the benefits that
quantum processing can provide.
In his seminal paper [42], Fujiwara provided a framework for quantum block
synchronization. The approach is to employ QSCs, which allow the identification
of codeword boundaries without destroying the quantum states. More precisely, an
(a l , a r ) − [[n, k]] QSC is an binary [[n, k]] code that encodes k logical qubits into
a physical qubit, and corrects misalignments by up to al qubits to the left and up to
ar qubits to the right. These quantum codes may correct more phase errors than bit
errors. This is an advantage because, as shown by Ioffe and Mzard [70], in physical
systems the noise is typically asymmetric in the sense that bit errors (can) occur less
frequently than phase errors. In this light, QSCs can be considered as asymmetric
quantum codes.
Several constructions of QSCs have been presented in the literature [44, 164, 167].
These constructions utilize BCH codes, cyclic codes related to finite geometries,
punctured ReedMuller codes, quadratic residue codes and duadic codes. According to the authors in [44]: “One of the main hurdles in the theoretical study of
quantum synchronizable codes is that it is quite difficult to find suitable classical
error-correcting codes because the required algebraic constraints are very severe and
difficult to analyze.”
In order to proceed further we need a result due to Fujiwara.
Theorem 5.5.1 ([42, Theorem 1]) Let C be a dual-containing [n, k 1 , d 1 ] cyclic code
and let D be a C-containing cyclic code with parameters [n, k 2 , d 2 ], with k 1 < k 2 .
Then, for any pair of nonnegative integers (a l , a r ) satisfying a l + a r < k 2 − k 1 , there
exists an (a l , a r ) − [[n + a l + a r , 2k 1 − n]] QSC that corrects up to at least
d 1 −1
2
phase errors and up to at least
d 2 −1
2
bit errors.
5.5.1 Synchronizable Codes from Cyclic Codes
At this point we already have tools to develop the results. We start by presenting
two constructions of QSCs from cyclic codes. The first result is based on the sum of
cyclic codes.
Theorem 5.5.2 Let n ≥ 3 be an integer such that gcd(n, 2) = 1 and assume that
m = ord n (2). Let C 1 be an [n, k 1 , d 1 ] dual-containing cyclic code and C 2 be an
[n, k 2 , d 2 ] C 1 -containing cyclic code. Let also C 3 be a cyclic code with parameters [n, k 3 , d 3 ] and C 4 be an [n, k 4 , d 4 ] C 3 -containing cyclic code such that
deg(gcd(g 2 (x), g 4 (x))) < deg(gcd(g 1 (x), g 3 (x))), where g i (x) is the generator polynomial of C i , i = 1, 2, 3, 4. Then, for any pair of nonnegative integers (a l , a r ) satisfying a l + a r < deg(gcd(g 1 (x), g 3 (x))) − deg(gcd(g 2 (x), g 4 (x))), there exists an
(a l , a r ) − [[n + a l + a r , n − 2 deg(gcd(g 1 (x), g 3 (x)))]] QSC that corrects up to at
least
d−1
2
phase errors and up to at least
d
∗ −1
2
bit errors, where d is the minimum
distance of the code C 1 + C 3 and d
∗ is the minimum distance of C 2 + C 4 .
119
problem, synchronization can be achieved using a classical system external to the
quantum system, but such a solution does not take advantage of the benefits that
quantum processing can provide.
In his seminal paper [42], Fujiwara provided a framework for quantum block
synchronization. The approach is to employ QSCs, which allow the identification
of codeword boundaries without destroying the quantum states. More precisely, an
(a l , a r ) − [[n, k]] QSC is an binary [[n, k]] code that encodes k logical qubits into
a physical qubit, and corrects misalignments by up to al qubits to the left and up to
ar qubits to the right. These quantum codes may correct more phase errors than bit
errors. This is an advantage because, as shown by Ioffe and Mzard [70], in physical
systems the noise is typically asymmetric in the sense that bit errors (can) occur less
frequently than phase errors. In this light, QSCs can be considered as asymmetric
quantum codes.
Several constructions of QSCs have been presented in the literature [44, 164, 167].
These constructions utilize BCH codes, cyclic codes related to finite geometries,
punctured ReedMuller codes, quadratic residue codes and duadic codes. According to the authors in [44]: “One of the main hurdles in the theoretical study of
quantum synchronizable codes is that it is quite difficult to find suitable classical
error-correcting codes because the required algebraic constraints are very severe and
difficult to analyze.”
In order to proceed further we need a result due to Fujiwara.
Theorem 5.5.1 ([42, Theorem 1]) Let C be a dual-containing [n, k 1 , d 1 ] cyclic code
and let D be a C-containing cyclic code with parameters [n, k 2 , d 2 ], with k 1 < k 2 .
Then, for any pair of nonnegative integers (a l , a r ) satisfying a l + a r < k 2 − k 1 , there
exists an (a l , a r ) − [[n + a l + a r , 2k 1 − n]] QSC that corrects up to at least
d 1 −1
2
phase errors and up to at least
d 2 −1
2
bit errors.
5.5.1 Synchronizable Codes from Cyclic Codes
At this point we already have tools to develop the results. We start by presenting
two constructions of QSCs from cyclic codes. The first result is based on the sum of
cyclic codes.
Theorem 5.5.2 Let n ≥ 3 be an integer such that gcd(n, 2) = 1 and assume that
m = ord n (2). Let C 1 be an [n, k 1 , d 1 ] dual-containing cyclic code and C 2 be an
[n, k 2 , d 2 ] C 1 -containing cyclic code. Let also C 3 be a cyclic code with parameters [n, k 3 , d 3 ] and C 4 be an [n, k 4 , d 4 ] C 3 -containing cyclic code such that
deg(gcd(g 2 (x), g 4 (x))) < deg(gcd(g 1 (x), g 3 (x))), where g i (x) is the generator polynomial of C i , i = 1, 2, 3, 4. Then, for any pair of nonnegative integers (a l , a r ) satisfying a l + a r < deg(gcd(g 1 (x), g 3 (x))) − deg(gcd(g 2 (x), g 4 (x))), there exists an
(a l , a r ) − [[n + a l + a r , n − 2 deg(gcd(g 1 (x), g 3 (x)))]] QSC that corrects up to at
least
d−1
2
phase errors and up to at least
d
∗ −1
2
bit errors, where d is the minimum
distance of the code C 1 + C 3 and d
∗ is the minimum distance of C 2 + C 4 .