Generalized Concatenated Quantum Codes

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by  Markus Grassl

Generalized Concatenated Quantum Codes
Markus Grassl,1 Peter Shor,2 Graeme Smith,3 John Smolin,3 and Bei Zeng3,4
1Institute for Quantum Optics and Quantum Information,
Austrian Academy of Sciences, Technikerstraße 21a, 6020 Innsbruck, Austria
2Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
3IBM T.J. Watson Research Center, Yorktown Heights, NY 10598, USA
4Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
(Dated: January 9, 2009)
Weintroducetheconcept of generalized concatenated quantumcodes. This generalized concate-
nation method provides a systematical way for constructing good quantum codes, both stabilizer
codesandnonadditivecodes. Usingthismethod,weconstructfamiliesofnewsingle-error-correcting
nonadditive quantum codes, in both binary and nonbinary cases, which not only outperform any
9
0 stabilizercodesforfiniteblocklength,butalsoasymptoticallyachievethequantumHammingbound
0 for large block length.
2
PACSnumbers: 03.67.Lx

n
a
J
Quantum error-correcting codes (QECCs) play a vi- provetheparametersofconventionalconcatenatedcodes

9 tal role in reliable quantum information transmission as for short block lengths [17] as well as their asymptotic

wellas fault-tolerantquantumcomputation (FTQC). So performance [18]. Many good codes, linear and nonlin-
]
h far, most good quantum codes constructed are stabi- ear, can be constructed from this method. One may ex-
p lizer codes, which correspond to classicaladditive codes. pect that moving to the quantum scenario, the GCQC
-
Thereisarichtheoryofstabilizercodes,andathorough method should be also a powerful one in making good
t
n
understanding of their properties [1, 2]. However, these codes, which we show is the case.
a
u codes are suboptimal in certain cases—there exist non- We demonstratethe powerofthis newGCQC method
q additive codes which encode a larger logical space than by showing that some good stabilizer quantum codes,
[ any stabilizer code of the same length that is capable of such as some quantum Hamming codes, can be con-

tolerating the same number of errors [3, 4, 5]. structed this way. We then further construct families of
1
v The recently introduced codeword stabilized (CWS) nonadditivesingle-error-correctingCWSquantumcodes,
9 quantum codes [6, 7, 8] framework, followed by the idea in both binary and nonbinary cases, which outperform
1 of union of stabilizer codes construction [9, 10], provides anystabilizercodes. Thisisthe firstknownsystematical
3
a unifying way of constructing a large class of quan- construction of these good nonadditive codes, while pre-
1
. tum codes, both stabilizer codes and nonadditive codes. vious codes were found by exhaustive or random numer-
1
The CWS frameworknaturally allowsto searchfor good icalsearchwithnostructuretogeneralizeto othercases.
0
9 quantum codes, and some good nonadditive codes that We also show that these families of nonadditive codes
0 outperform any stabilizer codes have been found. How- asymptotically achieve the quantum Hamming bound.
: ever, this search algorithm is very inefficient [7], which Basic Principle A general quantum code Q of n
v
i prevents us from searching for good quantum codes of q-dimensional systems, encoding K levels, is a K-
X lengthn≥10inthebinarycaseandevensmallerlengths dimensional subspace of the Hilbert space H⊗n. We say
q
r in the nonbinary case. Q is of distance d if all d−1 errors (i.e., operators act-
a
ing nontrivially on less than d individual H s) can be
This letter introduces the concept of generalized con- q
detected or have no effect on Q, and we denote the pa-
catenated quantum codes (GCQCs), which is a system-
rameters of Q by ((n,K,d)) .
atical way of constructing good QECCs, both stabilizer q
codesandnonadditivecodes. Comparedtotheusualcon- Recall that concatenated quantum codes are con-
catenatedquantumcodeconstruction,theroleoftheba- structed from two quantum codes, an outer code A and
sisvectorsoftheinnerquantumcodeistakenonbysub- an inner code B. If B is an ((n,K,d))q code with basis
spacesoftheinnercode. Theideaofconcatenatedcodes, vectors {|ϕii}Ki=−01, then the outer code A is taken to be
originallydescribedby Forney in a seminalbook in 1966 an ((n′,K′,d′)) code, i.e., a subspace A ⊂ H⊗n′. The
K K
[11],wasintroducedtoquantumcomputationcommunity concatenatedcodeQcisconstructedinthefollowingway:
threedecadeslater[1,12,13,14,15]. Theseconcatenated for any codeword |φi = Pi1...in′ αi1...in′|i1...in′i in A,
quantum codes play a central role in FTQC, as well as replaceeachbasisvector|i i(wherei =0,...,K−1for
j j
the study of constructing good degenerate QECCs. j =1,...,n′) by a basis vector |ϕ i in B, i.e.,
ij
The classical counterpart of GCQCs, i.e., general-
ized concatenated codes, was introduced by Blokh and |φi7→|φ˜i= X αi1...in′|ϕi1i...|ϕin′i, (1)
Zyablov [16], followed by Zinoviev [17]. These codes im-
i1...in′
2
so the resulting code Q is an ((nn′,K′,δ)) code, and For constructing a nondegenerate CWS code, we require
c q
the distance δ of Q is at least dd′, for examples, see that the distance of the code be ≤ d . Then any quan-
c G
[1, 12]. tum error E acting on Q can be transformed into a
CWS
In its simplest version, a generalized concatenated classical error by a mapping Cl (E) whose image is an
G
quantum code is also constructed from two quantum n-bit string. The nondegenerate code Q detects the
CWS
codes, an outer code A and an inner code B which is error set E if and only if C detects Cl (E) [6, 7, 8].
G
an ((n,K,d)) code. The inner code B is further par- We take the inner code B to be an ((n,K,d)) non-
q q
titioned into r mutually orthogonal subcodes {B }r−1, degenerate CWS code, constructed by a graph G and
i i=0
i.e. a classical code B. Furthermore, we decompose B as
B = r−1B such that each B is an ((n,K ,d ))
r−1 Li=0 i i i i q
CWScodeconstructedfromG. Thebasisvectorsofeach
B =MBi, (2)
B can be represented by classical codewords of a code
i
i=0 B = {b }Ki . Then consequently, the classical code B
i i,j j=1
and each Bi is an ((n,Ki,di))q code, with basis vectors has a partition B =Sri=−01Bi.
{|ϕi,ji}jK=i0−1, and i=0,...,r−1. Now we take the outer code A to be an ((n′,K′,d′ =
Now choose the outer code A to be an ((n′,K′,d′))r 1))r codeinthe HilbertspaceHr⊗n,whichis constructed
quantumcode in the HilbertspaceH⊗n′. While for con- from a classical (n′,K′,d ) code A over an alphabet
r c r
catenatedquantumcodeseachbasisstate|iiofthespace of size r, of length n′, size K′, and distance d in the
c
Hr is replacedby a basis state |ϕii of the inner code, for followingway: the basisvector|ψi1...in′i of A is givenby
a generalized concatenated quantum code Q the ba-
sis state |ii is mapped to the subcode Bi ofgcthe inner |ψi1...in′i=|i1...in′i, ∀(i1...in′)∈An′. (4)
code. For simplicity we assume that all subcodes B are
i Denote the generalized concatenated code obtained
ofequaldimension,i.e.,K =K =...=K =R. Then
1 2 r fromA and B by Q . It is straightforwardto see Q is
the dimension of the resulting code Q is K = K′Rn′, gc gc
gc alsoaCWScode,wherethecorrespondinggraphisgiven
i.e., for each of the n′ coordinates of the outer code, the by n′ disjoint copies of the graph G. The corresponding
dimension K is increased by the factor R. For a code-
classical code C is a classical generalized concatenated
gc
word |φi = Pi1...in′ αi1...in′|i1...in′i of the outer code code with inner code B = Sri=−01Bi and outer code A.
and a basis state |j ...j i (where j = 0,...,R−1 for
1 n′ l The minimum distance of Q is at least min{d,d ,d }.
l=1,...,n′)ofthespaceHR⊗n′,the encodingisgivenby However, the following stategmcent provides an impirovGed
the following mapping: lower bound.
Main Result: The minimum distance of Q is given
gc
|φi|j1...jn′i7→ X αi1...in′|ϕi1,j1i...|ϕin′,jn′i. (3) by min{ddc,di,dG}.
i1...in′ We will not give a technical detailed proof of this re-
sulthere. Instead, sincethe proofidea canbe illustrated
Note that the special case when R=1 corresponds to
clearly with a simple example, we will analyze such an
concatenatedquantumcodes. TheresultingcodeQ has
gc example, which also illustrates a systematical method of
parameters ((nn′,K,δ)) where the distance δ is at least
q constructing good nonadditive quantum codes that out-
min{dd′,d }. If some of the K s differ, the calculationof
i i perform the best stabilizer codes.
the dimension is more involved.
Good Nonadditive Codes We start taking the subcode
CWS-GCQC Fromnowonwerestrictourselvesincon-
B of the inner code B to be the well-known ((5,2,3))
0 2
structing some special kind of quantum codes, namely,
code,theshortestone-error-correctingquantumcode. As
CWS codes. CWS codes include all the stabilizer codes
aCWScode,thiscodecanbeconstructedbyapentagon
andmanygoodnonadditivecodes[6],soitisalargeclass
graph as well as a classical code B = {00000,11111}.
0
of quantum codes. The advantage of the CWS frame-
Further details canbe found in [6], here we just focus on
work is that the problem of constructing quantum codes
the classical error patterns given by the mapping Cl .
G
isreducedtotheconstructionofsomeclassicalcodescor-
Since the pentagon has graph distance 3, the CWS code
rectingcertainerrorpatternsinducedbyagraph. Sothe
B has distance at least 3 if B detects up to two errors
0 0
point of view of constructing these codes could be fully
with the error patterns induced by the pentagon. The
classical. For simplicity we only consider nondegenerate
induced error patterns are given by the following strings
codes here.
Anondegenerate((n,K,d)) CWScodesQ isfully Z : {10000,01000,00100,00010,00001},
q CWS
characterizedbyagraphG andaclassicalcodeC [6,7,8], X : {01001,10100,01010,00101,10010},
andfor simplicity we only consider q a prime power. For
Y : {11001,11100,01110,00111,10011}. (5)
anygraphG ofnvertices,thereexistsauniquestabilizer
code ((n,1,d )) defined by G (called the graph state of It is straightforwardto check that B indeed detects two
G 0
G). We call the distance d the graph distance of G. of these errors.
G
3
The classical code B0 is linear, so we can choose 15 23 codewords in Cg{c15} obtained by concatenating three
disjoint proper cosets, e.g., B1 = {00001,11110} and codewords of Bi. The size of Cg{c15} is then 23×16=27.
B15 = {01111,10000}. Combining these classical codes We now show that the distance of Q{15} is 3. To
gc
with the pentagon gives us the CWS codes B , each
i see this, we only need to show that C{15} detects up to
of which is a ((5,2,3)) quantum code. The union gc
2
two errors of the error patterns induced by three pen-
15
B = B of all cosets is a classical (5,32,1) code
Si=0 i 2 tagons. This is clear via the following two observation:
which consists of all 5-bit strings. Combining B with a
i)c ,c ∈C{15} correspondto differentcodewordsofthe
pentagongivesustheCWSquantuminnercodeB which 1 2 gc
outercodeA: sincethepentagonsaredisjoint,andAhas
is a ((5,32,1)) quantum code. It can be decomposedas
2
distance3,atleast3stringsintheinducederrorpatterns
15
B = B .
Li=0 i are needed to transform c to c . ii) c ,c ∈ C{15} cor-
For the outer code we take a quantum code A which 1 2 1 2 gc
respondtosamecodewordsofthe outercodeA: sinceat
corresponds to a classical code A = (3,16,3) , i.e., a
16
least 3 strings in the induced error patterns are needed
distance three code over GF(16) of length 3. Hence the
to transform codewords in B , at least 3 strings in the
basis of A is given by |i i i i where (i i i ) is one of i
1 2 3 1 2 3 induced error are needed to transform c to c .
the 16 codewords of {000,111,...,aaa,...,fff} of A. 1 2
Now one can generalize the construction of Q{15} to
Here we use the hexadecimal notation to denote the 16 gc
the case of more than three pentagons. Suppose we use
symbols of the alphabet GF(16).
n′ pentagons to construct single-error-correcting CWS
codes, then we observe the following
3 8 13
Fact 1 Choose the inner code as B = 15 B with
Si=0 i
2 4 7 9 12 14 each B a ((5,2,3)) quantum code, and the outer code
i 2
A corresponding to the classical code A with param-
1 5 6 10 11 15 eters (n′,K′,3)16, then the resulting GCQC Qgc is a
((5n′,2n′K′,3)) binary quantum code.
2
FIG. 1: Three pentagons: graph with 15 vertices.
This indicates that if we have a good classical code
over GF(16) of distance 3, then we may systematically
Now we construct the GCQC Q{15} of length 15 from
gc construct good quantum codes via the generalized con-
A and B in the following way: first, due to the prod-
catenation method described above.
uct state form of the basis of A, we choose the corre-
sponding graph G{15} to be by three disjoint pentagons, Example 1 Using the quantum code correspond-
as shown in FIG. 1. We denote this graph by G{15}. ing to the classical Hamming code with parameters
The distance of the graph state corresponding to G{15} (17,1615,3) as the outer code, then by Fact 1 we get
16
is still 3. So from these three pentagons we can obtain a quantum code with parameters ((85,277,3)) , which is
2
a nondegenerate CWS quantum code whose distance is a quantum Hamming code [2]. If we properly choose the
at most 3. The error patterns induced by the mapping labeling of the subcodes B by elements of GF(16), the
i
ClG{15} given by this 15 vertex graph are simply the correponding classical code is linear [19], and hence this
strings from Eq. (5) on the coordinates 1–5 (or 6–10, quantum code is a stabilizer code [6].
or 11–15) and zeros on the other coordinates. For in-
stance, 10000 in Eq. (5) gives rise to three strings of If we take a quantum code corresponding to a good
length15whichare100000000000000,000001000000000, nonlinear classical code as the outer code, then we can
and 000000000010000. In total there are 45 strings in constructagoodnonadditivequantumcode[6]. Herewe
the induced error set of three pentagons corresponding give examples of such a good quantum codes which are
to the 45 single-qubit errors on 15 qubits. constructedusingagoodnonlinearclassicalcodes. Those
nonlinear codes are obtained via the following classical
Now we need to figure out what the corresponding
classical code C{15} is. We know that it is the gener- construction, called ‘subcode over subalphabet’ (see [19,
gc
15 Lemma 3.1]).
alized concatenated code with inner code B = B
Si=0 i
and outer code A. To see how this works explicitly, con- Fact 2 If there exists an (n,K,d) code, then for any
q
sider the first codeword a0 = 000 of A. Each of the s < q, there exists an (n′,K′,d)s code with size at least
three zeros is replacedby the code B0 ={00000,11111}, K(s/q)n.
i.e, (a ,j ,j ,j ) (where j = 0,1) will be mapped to
0 1 2 3 l
one of the 8 codewords of C{15}, which are strings of Example 2 It is known that there is a classical Ham-
gc
length 15, given by 000000000000000,000000000011111, ming code with parameters (18,1716,3)17. Therefore, us-
000001111100000, 000001111111111, 111110000000000, ing Fact 2 there is a (18,⌈1618⌉,3) code. Then the
172 16
111110000011111, 111111111100000, 111111111111111. resulting quantum code has parameters ((90,281.825,3)) .
2
Similarly, anyother codeworda of A will be mapped to For a binary quantum code with n = 90 and d = 3, the
i
4
quantum Hamming bound (K ≤ qn/((q2−1)n+1), see eralizingtheconcatenationofEq. (3)istoputsomecon-
[2]) gives K < 281.918, and the linear programing bound straints on the additional degrees of freedom |j ...j i
1 n′
(see [2]) gives K < 281.879. So the best stabilizer quan- by using a second outer code. Additionally, one can re-
tum code can only be ((90,281,3)) . Hence our simple cursively decompose the codes B in the decomposition
2 i
construction gives a nonadditive single-error-correcting (2) of the inner code, which leads to a more generalcon-
quantum code which outperforms any possible stabilizer structionofGCQCwithwhichmoregoodquantumcodes
codes. This is the first such example given by construc- can be constructed (see [21]). While the nonadditive
tion, not by numerical search. codes of this letter tighten the gap between lower and
upper bounds for the dimension of the codes, we believe
Example 3 The similar CWS-GCQC idea works also thatingeneraltheGCQCconstructiongivesapromising
for the nonbinary case using the nonbinary CWS con- wayforfurther constructingnewquantumcodesofgood
struction[8]. Taketheinnercodetobeaunionof81mu- performance,and we hope that this generalizedconcate-
tually orthogonal ((10,729,3))3 codes that is constructed nation technique will also shed light on improvements of
fromagraphthatisaringoftenvertices[20]. Choosethe fault-tolerant protocols.
outercodeasthequantumcodecorrespondingtotheclas-
sical (84,⌈8184⌉,3) , which is obtained from the Ham-
832 81
ming code (84,8382,3) . Then the resulting quantum
83
code has parameters ((840,3831.955,3)) . For a ternary
3
quantum code with n = 840 and d = 3, the Hamming [1] D. Gottesman, Ph.D. Thesis, Caltech, 1997. arXiv:
bound gives K < 3831.978, and the linear programing quant-ph/9705052.
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only be ((840,3831,3)) . This is the first known nonbi-
3 [3] E. M. Rains, R. H. Hardin, P. W. Shor, and N. J. A.
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q Inf. Theory, 55, 433 (2009).
prime power q. For this, we take the inner code B as
0 [7] I. Chuang, A. Cross, G. Smith, J. Smolin, and B. Zeng,
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0
code is then corresponding to a classical code over an Inform. Theory,pp. 300–304 (2008). arXiv: 0801.2150.
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[11] G. D. Forney, Jr. Concatenated Codes, Cambridge, MA:
[L ,L − i,3] where P is the least prime power ex-
i i P M.I.T. Press, 1966.
ceeding Q, and Li = (Pi − 1)/(P − 1). The result [12] E. Knill, and R. Laflamme, arXiv: quant-ph/9608012.
is the code Vsi = ((Nsi,Msi,3))q with length Nsi = [13] E. Knill, R. Laflamme, and W. Zurek, arXiv:
L n = (Pi −1)(Q−1)/(q2−1)(P −1) and dimension quant-ph/9610011 (1996); E. Knill, R. Laflamme, and
i s
Msi ≥qNsi/Pi. W. Zurek, arXiv: quant-ph/9702058 (1997).
[14] C. Zalka, arXiv: quant-ph/9612028 (1996).
The number of different errorswe wantto dealwith is
[15] D.AharonovandM.Ben-Or,Prof.29thAnn.ACMSym-
(q2−1)N +1>Qi =qsi for P >Q and i>1. By the
si posium on Theory of Computing, pp. 176-188 (1997).
quantumHamming boundK ≤qNsi/((q2−1)Nsi+1)< arXiv: quant-ph/9611025.
qNsi/Qi, the dimension of any stabilizer code (including [16] E.L.BlokhandV.V.Zyablov,Probl.PeredachiInform.
degenerate codes) is upper bounded by K ≤ qNsi−2si−1. 10, 45 (1974).
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[19] I.Dumer,ConcatenatedCodesandTheirMultilevelGen-
have qNsi/Pi ≤ Msi ≤ qNsi/Qi. Since Q/P → 1 for eralizations, Chapter 23, pp. 1911–1988. In Handbook of
s → ∞ [19], these families of nonadditive codes asymp-
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Discussion WehaveintroducedtheconceptofGCQC, [20] S. Y. Looi, L. Yu, V. Gheorghiu, and R. B. Griffiths,
whichisasystematicconstructionofgoodQECCs,both Phys. Rev.A78, 042303 (2008).
stabilizer codes and nonadditive codes. One way of gen- [21] M. Grassl, P. W. Shor, and B. Zeng, in preparation.