Standard Form of Qudit Stabilizer Groups

0.16 MB·English

by  Vlad Gheorghiu

Standard form of qudit stabilizer groups
Vlad Gheorghiu1,∗
1Department of Physics, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA
(Dated: Version of January 10, 2011)
We investigate stabilizer codes with carrier qudits of equal dimension D, an arbitrary integer
greater than 1. Weprovethat thereis a direct relation between thedimension of a quditstabilizer
codeandthesizeofitscorrespondingstabilizer,andthisimpliesthatthecodeanditsstabilizerare
dual to each other. We also show that any qudit stabilizer can be put in a standard, or canonical,
form using a series of Clifford gates, and we provide an explicit efficient algorithm for doing this.
Ourworkgeneralizes knownresultsthatwerevalidonlyforprimedimensionalsystemsandmaybe
useful in constructing efficient encoding/decoding quantum circuits for qudit stabilizer codes and
betterqudit quantumerror correcting codes.
1
1
PACSnumbers: 03.67.Mn,03.67.Pp
0
2

n I. INTRODUCTION stabilizer codes over prime-power finite fields [12]), and
a theproofthatsuchaformexistsisoneofthemainresults
J
Quantum error correction is an important part of of the current article.
7
various schemes for quantum computation and quan- The reminder of the paper is organized as follows.

] tumcommunication,andhencequantumerrorcorrecting Sec.IIcontainsdefinitionsofthegeneralizedPauligroup
h codes, first introduced about a decade ago [1–3] have re- and some quantum gates used later in the paper. It
p ceivedagreatdealofattention. Foradetaileddiscussion also defines rigorously qudit stabilizers and their corre-
t- seeCh.10of[4]. Mostoftheearlyworkdealtwithcodes sponding stabilized subspaces (or codes), together with
n for qubits, with a Hilbert space of dimension D =2, but an alternative algebraic notation that we employ later.
a
u qudit codes with D > 2 have also been studied [5–15]. Sec. III contains our main results: a “size”theorem that
q They areof intrinsic interestand couldturn out to be of relatesthesizeofthestabilizergrouptothedimensionof
[ some practical value. itsstabilizedsubspace,followedbya“structure”theorem
The stabilizer formalism introduced by Gottesman in that shows that any qudit stabilizer can be brought to
1
[16] for D = 2 (qubits) provides a compact and pow- a standardformthrougha seriesof elementaryquantum
v
9 erful way of generating quantum error correcting codes gates. Finally, Sec. IV contains a summary, conclusions,
1 andextendsthenotionoflinearclassicalerrorcorrecting and some open questions.
5 codes [17] to the quantum domain. The stabilizer for-
1 malism has been generalized to cases where D is prime
. II. PRELIMINARY REMARKS AND
1 or a prime power, see e.g. [6, 12, 18, 19]. For composite
DEFINITIONS
0 D things are more complicated and there is no imme-
1 diate and natural way of generalizing the notions. Our
1 A. The generalized Pauli group on n qudits
approachis to use generalizedPaulioperatorsandstabi-
:
v lizers defined in the same way as in the prime case, see
i e.g. [13, 15]. This has the virtue that many (although We generalize Pauli operators to higher dimensional
X
not all) results that are valid in the prime dimensional systems of arbitrary dimension D in the following way.
r case can be extended without too much difficulty to the The X and Z operators acting on a single qudit are de-
a
more general composite case. fined as
Animportantprobleminthetheoryofstabilizercodes D−1 D−1
is what is their structure. Is there any “canonical” way Z = ωj j j , X = j j+1 , (1)
| ih | | ih |
of representing an arbitrary stabilizer code? If yes, can j=0 j=0
X X
oneusethisfactforimplementingvariousquantumerror-
and satisfy
correcting tasks? For prime D it turns out that there is
such a standard form, see e.g. Ch. 10.5.7 of [4], and this XD =ZD =I, XZ =ωZX, ω =e2πi/D, (2)
allows for a better understanding of the error-correcting
where the addition of integers is modulo D, as will be
capabilities of the stabilizer code and also provides an
assumedfromnowon. Foracollectionofnquditsweuse
efficient way of constructing encoding/decoding circuits
subscripts to identify the correspondingPauli operators:
for such stabilizer codes. For composite D we are not
thus Z and X operate on the space of qudit i. The
aware of any such standard form (except for the case of i i
Hilbert space of a single qudit is denoted by , and the
H
Hilbert space of n qudits by , respectively. Operators
n
H
of the form
∗ [email protected] ωλXxZz :=ωλXx1Zz1 Xx2Zz2 XxnZzn (3)
1 1 ⊗ 2 2 ⊗···⊗ n n
2
willbereferredtoasPauliproducts,whereλisaninteger Pauli operator Sq F
inZD andxandzaren-tuplesinZnD,theadditivegroup Z Zq X
of n-tuple integers mod D. For a fixed n the collection
X Xq¯ ZD−1
of all possible Pauli products (3) form a group under
operator multiplication, the Pauli group . If p is a
Pauli product, then pD = I is the identityPnoperator on TgaAteBsLFEaIn.dTShqe(cq¯onisjutghaetmionultoifplPicaautliiveopinevreartsoersofbqymonoed-qDud).it
, and hence the order of any element of is either
n n
H P
D or else an integer that divides D. While is not
n
Abelian, it has the property that two elementsPcommute Pauli product CNOTab SWAPab CPab
up to a phase: Ia⊗Zb Za⊗Zb Za⊗Ib Ia⊗Zb
Z ⊗I Z ⊗I I ⊗Z Z ⊗I
a b a b a b a b
p1p2 =ωλ12p2p1, (4) I ⊗X I ⊗X X ⊗I ZD−1⊗X
a b a b a b a b
X ⊗I X ⊗XD−1 I ⊗X X ⊗ZD−1
with λ an integer in Z that depends on p and p . a b a b a b a b
12 D 1 2
TABLEII. TheconjugationofPauliproductsonquditsaand
b by two-qudit gates CNOT, SWAP and CP. For the CNOT
B. Generalization of qubit quantum gates to gate, the first qudit a is the control and the second qudit b
higher dimensions thetarget.
In this subsection we define some one and two qudit
gates generalizing various qubit gates. The qudit gener- with
alization of the Hadamard gate is the Fourier gate
(CNOT )† =(CNOT )D−1 =(I F2)CNOT (I F2).
ba ba a⊗ b ba a⊗ b
D−1 (10)
1
F:= ωjk j k . (5) FinallywedefinethegeneralizedControlled-phaseorCP
√D j=0 | ih | gate as
X
For an invertible integer q Z (i.e. integer for which D−1 D−1
D
there exists q¯∈ ZD such th∈at qq¯≡ 1modD), we define CPab = |jihj|a⊗Zbj = ωjk|jihj|a⊗|kihk|b.
a multiplicative gate j=0 j,k=0
X X
(11)
D−1 The CP and CNOT gates are related by a local Fourier
S := j jq , (6) gate, similar to the qubit case
q
| ih |
j=0
X CNOT =(I F )CP (I F )†, (12)
ab a b ab a b
⊗ ⊗
where qj means multiplication mod D. The requirement
thatqbeinvertibleensuresthatS isunitary;foraqubit since F maps Z into X under conjugation (see Table I).
q
S is just the identity. The gates F, S , SWAP, CNOT and CP are unitary
q q
For two distinct qudits a and b we define the CNOT operators that map Pauli operators to Pauli operators
gate as under conjugation, as can be seen from Tables I and II.
They are elements of the so called Clifford group on n
D−1 D−1 qudits [20, 21], the group of n-qudit unitary operators
CNOTab := j=0 |jihj|a⊗Xbj =j,k=0|jihj|a⊗|kihk+j|b, tahCatliffleoarvdesopPenraitnovra,ritahnetnunpder con,juOgpaOtio†n, i.e..ifFOromis
X X n n
(7) Tables I and II one can ea∀sily∈dPeduce the r∈esuPlt of con-
the obvious generalizationof the qubit Controlled-NOT, jugation by F, S , SWAP, CNOT and CP on any Pauli
q
where a labels the control qudit and b labels the target product.
qudit. Next the SWAP gate is defined as
D−1
C. Qudit stabilizer codes
SWAP := k j j k . (8)
ab | ih |a⊗| ih |b
j,k=0
X Relative to this group we define a stabilizer code to
C
ItiseasytocheckthatSWAPgateishermitiananddoes be a K 1-dimensional subspace of the Hilbert space
≥
indeed swap qudits a and b. Unlike the qubit case, the satisfying three conditions:
quditSWAP gate is nota productofthree CNOTgates,
butcanbeexpressedintermsofCNOTgatesandFourier C1: There is a subgroup of n such that for every s
S P
gates as in and every ψ in
S | i C
SWAP =CNOT (CNOT )†CNOT (F2 I ), (9) sψ = ψ (13)
ab ab ba ab a⊗ b | i | i
3
C2: The subgroup is maximal in the sense that every Gate X-part Z-part
S
s in for which (13) is satisfied for all ψ
Pn | i ∈ C SWAPab Interchangecolumns Interchangecolumns
belongs to .
S a and b a+n and b+n
C3: Thecodingspace ismaximalinthesensethatany Sq,a Multiply column a by Multiply column a+n
ket ψ that satisCfies (13) for every s lies in . invertible integer q−1 byinvertible integer q
| i ∈S C
(CNOTab)m Substract m times Addm times column
If these conditions are fulfilled we call the stabilizer
S column a from column b b+n to column a+n
ofthecode . ThatitisAbelianfollowsfromthecommu-
C
tationrelation(4),sinceforK >0thereissomenonzero
TABLEIII. Conjugation bytheabovequantumgates corre-
ψ satisfying (13). spondtoelementarycolumnoperationsontheX andZ parts
| i
Note that one can always find a subgroup of n oftheparity-checkmatrixofastabilizercode. FortheCNOT
S P
satisfying C1 and C2 for any subspace of the Hilbert gate, the first qudit a is the control and the second qudit b
C
space, but it might consist of nothing but the identity. thetarget. Theintegerexponentmmeans CNOTappliedm
ThusitisconditionC3thatdistinguishesstabilizercodes times (or, equivalently,them-th power of theCNOT gate).
from nonadditive codes. A stabilizer code is uniquely
determined by as well as by , since determines
through C3, soSin a sense the coCde and iSts stabilizer arCe analysis can be done using the Theorem 1 of Sec. III,
dual to each other. which implies for this example that the size of the stabi-
lizergroupmustbeequalto4,henceitmustbegenerated
by a single generator of order 4 or two generators each
D. Stabilizer generators and equivalent algebraic of order 2. By inspection it is easy to rule out the first
descriptions of qudit stabilizer codes case, so indeed one must use 2 generators.
A conjugation of a stabilizer group by a Clifford op-
Anystabilizergroupcanbecompactlydescribedusing eration will change the stabilizer group to an isomor-
a set of group generators. A generator corresponds to a phic group. This will correspond to a column opera-
specific Pauli product and can be completely specified, tion ontheparity-checkmatrixofthestabilizer,together
see (3), by a phase λ and two n-tuples in Zn, x and z. with a transformation of the phase vector. On the other
D
A collectionof k generatorscan thereforebe represented hand, the generator description of a stabilizer group is
by a k-component phase vector over Z (that contains notunique: onecanmultiply ageneratorbyanotherone
D
all k phases) and a k 2n parity-check matrix over Z andstill getthe same group. This kind ofoperationcor-
D
withrowscorrespondin×gtothestabilizergenerators. For responds to a row operation on the parity-check matrix,
example, the stabilizer again keeping in mind that in general the phase vector
will modify. From now on for the sake of simplicity we
= ω2X3Z2,X2 (14) willignorethephasevector,althoughinrealapplications
S h 1 2 2i
one has to keep track of the phases.
corresponds to the phase vector (2,0) and parity-check The following represent what we call elemen-
matrix tary row/column operations: a) interchanging of
rows/columns, b) multiplication of a row/column by
S= 3 0 0 2 . (15) an invertible integer, c) addition of any multiple of a
0 2 0 0! row/columnto a different row/column. The column op-
erations can be realized by conjugations of the stabilizer
The angular brackets in (14) means “group generated by the Clifford operations in Table III, and the row op-
by”, i.e. the group obtained by all possible products of erationsjustensurethatthestabilizergroupremainsthe
the group generators. We call the left k n block of the same, i.e. the new set of generators generate the same
×
parity-checkmatrixthe X-block,andthe rightk n the stabilizer group and not a smaller one.
×
Z-block, since they describe the X and Z parts of the
stabilizer generators,respectively.
Note thatifD is aprimenumber,anystabilizergroup III. SIZE-STRUCTURE THEOREMS
canbe describedusing nomorethann generators. How-
ever, in composite dimensions one can have more that n
The following theorem generalizes to composite D a
generatorsbutno morethan 2n. Forexample, inD =4,
well-knownresultforprimeD thatrelatesthesizeofthe
the n = 1 qudit stabilizer = X2,Z2 is generated by
stabilizer group to the dimension of its stabilizer sub-
S h i
2 (and not 1) elements and specify the stabilizer state
space. Although the composite D result of our next the-
(0 + 2 )/√2. Thereisnowayofrepresentingthis state orem may have been known by the community (see e.g.
u|siing |onily 1 generator; Z2 by itself stabilizes both 0 the claim near the end of Sec. 3.6 of [16]), we have not
| i
and 2 , hence everything in their span, so the condition yet seen a proof of it.
| i
C3 is not satisfied, i.e. the coding space is not maximal.
ThesamekindofanalysisholdsforX2. Amorerigorous Theorem 1 (Size). Let be an n-qudit stabilizer code
C
4
with stabilizer . Then the 1-qudit stabilizer generated by = Z2 in D = 4.
S S h i
It is obvious that stabilizes a K =2-dimensional code
K =Dn, (16) =span 0 , 2 ,aSndthe size ofthe stabilizeris =2.
×|S| C {| i | i} |S|
WheneverD =2(qubits)itwasshownin[16](seealso
where K is the dimension of , is the size (or order)
C |S| Ch. 10.5.7of[4]foradetaileddiscussion)thatanystabi-
of the stabilizer group and D is the dimension of the
S lizer code can be put into a “standardform” or “canoni-
Hilbert space of one carrier qudit.
calform”, andthis is very useful for constructing encod-
ing/decoding quantum circuits for stabilizer codes. This
Proof. Define
result can be generalized at once to prime D. However,
1 for composite dimensions, it is not so obvious how to do
P := s. (17)
the generalization, and the main technical difficulty is
|S|sX∈S that Z is a ring (and not a field) and therefore some
D
We will first show that P is the projector onto . integers are not invertible. However, in the following
It follows at once that C Theorem we show that one can still apply a technique
similar to a Gaussian elimination over Z and put any
D
P =P† =P2, (18) composite D stabilizercode into a standardformsimilar
to the one of prime D case.
wheretheequalitiesfollowfromthegrouppropertyof ,
so P is an orthogonal projector. Let ψ be an arbitraSry Theorem 2 (Standard form). Let be a K dimen-
vectorthatbelongstothestabilizerco|dei . Thensψ = sional n-qudit stabilizer code with stabCilizer generated
ψ s (see the condition C1 (13) that aCstabilizer|coide byk 62ngeneratorsandwithcorrespondingpSarity-check
|muis∀t satisfy), which together with (17) yields matrix S of size k 2n. Then is isomorphic through a
× S
conjugation by a Clifford operation to another stabilizer
P ψ = ψ . (19) ′, with parity check matrix S’ in standard form
| i | i S
Thereforethesubspace ontowhichP projectsincludes r n−r r n−r
W
, . Let us now choose an arbitrary φ . r M 0 Z Z
CTheCn⊂ W | i ∈ W S’= k r{ z}0|{ z}0|{ zZ}|12{ zZ}|34{ !, (24)
− {
1
φ =P φ = sφ . (20)
where the dimensions of the block matrices are indicated
| i | i | i
|S|sX∈S by curly brackets.
HereM=diag(m ,...,m ),with16r 6n,isadiag-
Multiply (20)ontheleftbysomearbitraryt anduse 1 r
∈S onal matrix with all m =0 divisors of D. The matrices
the group property of S to get Z and Z satisfy j 6
1 2
1 1
tφ =tP φ = tsφ = sφ Z M=MZT mod D, (25)
| i | i | i | i 1 1
|S|sX∈S |S| Xs∈S Z M=0 mod D, (26)
2
=P φ = φ . (21)
| i | i
where T in the exponent denotes the transpose, and the
Since t was arbitrary we arrived at the conclusion that matrixZ isadiagonal rectangularmatrix,withdiagonal
4
elements divisors of D. The notation 0 denotes the zero-
tφ = φ , t , (22)
| i | i ∀ ∈S block matrix.
which proves that φ belongs to the stabilizer subspace
| i Proof. The key ingredient of the proof is the Smith nor-
,andthisimpliesthat . HenceP projectsstrictly
C W ⊂C malform: throughasequenceofelementaryrow/column
onto (and not on some larger subspace that includes
C operations mod D (see the discussion at the end of
). Its trace is just the dimension K of ,
C C Sec. III, a matrix M over ZD can be converted to the
1 Smithnormalform[22,23](seealsoSec. IV.Bof[15]for
Tr(P)=K = Dn. (23) an example)
|S|
where we have used that fact that all Pauli products in M’=V M W, (27)
· ·
(17)aretracelessexcepttheidentity(thatmustbelongto
thesum,since isagroup),oftraceDn. Thisconcludes where V and W are invertible (in the mod D sense)
S square matrices, and M’ is a diagonal rectangular ma-
the proof.
trix, with diagonal elements divisors of D. The matrix
When D is a composite integer the dimension of the V represents the row operations and W the column op-
stabilizer code does not have to be a power of D any erations.
more (as was the case in the prime D case), but can be Note that in our case all necessary column operations
any divisor of Dn. As an illustrative example, consider can be realized by the corresponding gates in Table III,
5
and,moreimportant,theydonotmix theX andZ parts Also for prime D it was proven in [12] that any stabi-
oftheparity-checkmatrix. Therefore,withoutbeingcon- lizer group is Clifford equivalent to another stabilizer
cerned with what happens to the Z part of the parity- ′ generatedSonly by Z’s, ′ = Z ,Z ,...,Z 2. This
1 2 k
check matrix S, we can put its X part in the Smith nor- rSesult is not true for compoSsite Dh ’s, and one eaisy to see
mal form (again we stress that this can be done because counterexample is the 1-qudit stabilizer = X2,Z2 in
S h i
the Z part to not interfere with the X part), and ar- D =4, mentioned before in Sec. IID.
rive at a parity-check matrix of the form (24). Next by IV. CONCLUSIONS AND OPEN QUESTIONS
another series of Clifford gates acting only on the last
(n r) qudits one can further put the Z4 matrix in its Westudiedstabilizercodeswithcarrierquditsofcom-
−
Smith normal form, without modifying the X part of posite dimension D. We proved a size theorem that re-
the parity-check matrix (which is already in Smith nor- lates the size of the stabilizer group to the dimension of
mal form), since there are only zeros on the last n r its stabilized code. Furthermore, we have shown that
−
columns of the X part; note that the row operations are any stabilizer code can be put in a standard (canonical)
done onthe lastk r rows,andagaindonotmodify the formthroughaseriesofCliffordgates,andweprovideda
−
X part of the parity-checkmatrix. Since the elementary constructive algorithm. Our result generalizes what was
row operations do not change the stabilizer group and known in the prime D case and may be useful in con-
the elementarycolumnoperationscorrespondto Clifford structing efficient encoding/decoding quantum circuits,
gates, see Table III, our whole transformationfrom to following the procedures outlined in [16].
′ is a conjugation by a Clifford operation. S OurapproachwasbasedonthegeneralizedPauligroup
S
Finally note that conjugation by Clifford operations introduced by (1) and (2). However, for composite di-
do not change the commutation relations. It is easy to mensions, this is not the only way of introducing Pauli
deduce from (3) that two Pauli products described by operators. Analternativewayistosplitthedimensionin
(xz) and (x’z’) commute if and only if its prime-power factors which will then induce a natural
| |
splitting of the carrier qudits in subsystems of prime-
x z’=z x’ mod D, (28) power dimensions. In each of these subsystems then one
· ·
can define Pauli operators using finite fields (any finite
where the dot represents the usual inner product of two field is isomorphic to a prime-power canonical represen-
vectorsinZD, e.g. the sumofthe products ofindividual tation), as done e.g. in [12]. Although this is the scope
components. Using(28)weobserveatoncethatthefinal offuture work,wethink itmaybe usefulsince inasense
set of generators commute if and only if (25) and (26) “decouples” the stabilizer into prime-power subsystems,
hold, and this concludes the proof. and the latter can be put into standard forms as done
in [12]. One can then use previously known results for
It is proved in [23] that a M N matrix can be re- stabilizersoverfinite fields to study variouspropertiesof
duced to the Smith form in only× (Mθ−1N) operations compositeDstabilizers,andthismayhelpbuildingmore
from Z , where θ is the exponentOfor matrix multiplica- efficient quantum error correcting codes.
D
tion overthe ring Z , i.e. two M M matrices overZ Finally onemay askif thereexistalternativestandard
D D
can be multiplied in (Mθ) opera×tions from Z . Using forms of qudit stabilizer codes, perhaps more useful that
D
O
standard matrix multiplication θ = 3, but better algo- the one presented here. We do not know if such forms
rithms [24] allow for θ = 2.38. This ensures that our exist, and searching for them may be worthwhile.
procedure outlined above is computationally efficient.
Whenever D is prime its only non-zero divisor is 1,
hence M is just the identity matrix and Z has only 0’s ACKNOWLEDGMENTS
4
and 1’s on the diagonal. From (25) and (26), Z must
1
besymmetricandZ mustbethezeromatrix,henceour The researchdescribedherereceivedsupportfromthe
2
standard form reduces to the one known for prime D’s Office of Naval Research and from the National Science
[16]. 1 Foundation through Grant No. PHY-0757251.
[1] P.W. Shor, Phys. Rev.A 52, R2493 (1995). [2] E.KnillandR.Laflamme,Phys.Rev.A55,900(1997).
[3] A. M. Steane, Phys. Rev.Lett. 77, 793 (1996).
[4] M. A. Nielsen and I. L. Chuang, Quantum Computation
and Quantum Information, 5th ed. (Cambridge Univer-
1 ExceptthefactthatwealsoperformcolumnoperationsbyClif- sity Press, Cambridge, 2000).
[5] E. M. Rains, IEEE Trans. Inf.Theory 45, 1827 (1999).
fordconjugations, whichfurther simplifiesthe standard formof
[16]. [6] A.AshikhminandE.Knill,IEEETrans.Inf.Theory47,
2 Their result is more general and holds for any finite field, pro- 3065 (2001).
videdoneredefines thePaulioperators in(1)accordingly. [7] D. Schlingemann and R. F. Werner, Phys. Rev. A 65,
6
012308 (2001). [17] F. J. MacWilliams and N. J. A. Sloane, The Theory
[8] Dirk Schlingemann, “Stabilizer codes can be realized as ofError-Correcting Codes (North-HollandMathematical
graph codes,” E-printarXiv:quant-ph/0111080. Library, Amsterdam, 1977).
[9] D. Schlingemann, “Logical network implementation for [18] A. Ketkar, A. Klappenecker, S. Kumar, and P. K.
cluster states and graph codes,” E-print arXiv:quant- Sarvepalli, IEEE Trans. Inf.Theory 51, 4892 (2006).
ph/0202007. [19] X. Chen, B. Zeng, and I. L. Chuang, Phys. Rev. A 78,
[10] M. Grassl, T. Beth, and M. Roetteler, Int. J. Quantum 062315 (2008).
Inf.2, 55 (2004). [20] D. Gottesman, “Fault-Tolerant Quantum Computation
[11] V. Arvind, P. P. Kurur, and K. R. Parthasarathy, withHigher-DimensionalSystems,”E-printarXiv:quant-
“Non-stabilizer quantum codes from abelian subgroups ph/9802007.
of theerror group,” E-print arXiv:quant-ph/0210097. [21] E. Hostens, J. Dehaene, and B. D. Moor, Phys. Rev. A
[12] M. Grassl, M. Roetteler, and T. Beth, “Efficient Quan- 71, 042315 (2005).
tum Circuits for Non-Qubit Quantum Error-Correcting [22] M. Newman, Integral Matrices (Academic Press, New
Codes,” E-print arXiv:quant-ph/0211014. York, 1972).
[13] S. Y. Looi, L. Yu, V. Gheorghiu, and R. B. Griffiths, [23] A. Storjohann, in ISSAC ’96: Proceedings of the 1996
Phys.Rev.A 78, 042303 (2008). international symposium onSymbolicandalgebraiccom-
[14] D. Hu, W. Tang, M. Zhao, Q. Chen, S. Yu, and C. H. putation (ACM,NewYork,NY,USA,1996)pp.267–274.
Oh,Phys. Rev.A 78, 012306 (2008). [24] D. Coppersmith and S. Winograd, in STOC ’87: Pro-
[15] V.Gheorghiu,S.Y.Looi, andR.B.Griffiths,Phys.Rev. ceedings of the nineteenth annual ACM symposium on
A 81, 032326 (2010). Theory of computing (ACM,NewYork,NY,USA,1987)
[16] D.Gottesman, “Stabilizer codesandquantumerrorcor- pp. 1–6.
rection,” E-printarXiv:quant-ph/9705052.