boolean functions, projection operators and quantum error

ISIT2007, Nice, France, June 24 - June 29, 2007

Boolean Functions, Projection Operators and

Quantum Error Correcting Codes

Vaneet AggarwalDepartment of Electrical Engineering

Princeton UniversityEmail: [email protected]

Abstract-This paper describes a common mathematicalframework for the design of additive and non-additive QuantumError Correcting Codes. It is based on a correspondence betweenboolean functions and projection operators. The new frameworkextends to operator quantum error correcting codes.

I. INTRODUCTION

The additive or stabilizer construction of Quantum ErrorCorrecting Codes (QECC) takes a classical binary code thatis self-orthogonal with respect to a certain symplectic innerproduct, and produces a quantum code, with minimum dis-tance determined by the classical code (For more details see[6], [7] and [11]). In [16], Rains et al. presented the firstnon-additive quantum error-correcting code. This code wasconstructed numerically by building a projection operator witha given weight distribution. Grassl and Beth [10] generalizedthis construction by introducing union quantum codes, wherethe codes are formed by taking the sum of subspaces generatedby two quantum codes. Roychowdhury and Vatan [18] gavesome sufficient conditions for the existence of nonadditivecodes, and Arvind et al. [4] developed a theory of non-additivecodes based on the Weyl commutation relations. Followedby this, Kribs et al. [13] introduced Operator Quantum ErrorCorrection (OQEC) which is a unifying approach that incor-porates the standard error correction model, the method ofdecoherence-free subspaces, and the noiseless subsystems asspecial cases.We will describe a mathematical framework for code design

that encompasses both additive and non-additive quantum errorcorrecting codes. It is based on a correspondence betweenboolean functions and projection operators in Hilbert spacethat is described in Sections II, III and V. This type ofcorrespondence was first given in [2] where this relation wasused to construct space-time codes. We will give sufficientconditions for the existence of QECC in terms of the existenceof the boolean function satisfying a few properties in SectionVII. Hence, we convert the problem of finding a quantumcode into a problem of finding boolean function satisfyingsome properties. For some parameters of Quantum code,we have given examples of the boolean functions satisfyingthese properties. We focus on non-degenerate codes which isreasonable given that we know of no parameters k, M and dfor which there exists a ((k, M, d)) degenerate QECC but not

A. Robert CalderbankDepartment of Electrical Engineering

Princeton UniversityEmail: [email protected]

a ((k, M, d)) non-degenerate QECC. Further, in Section VIII,we will see how this scheme fits into a general framework ofoperator quantum error correcting codes.

II. BOOLEAN FUNCTION

A boolean function is defined as a mapping f: {0, I}m{0, 1}. To each m-tuple representing an assignment of valuesfor the variables v (vi, ..., vUr), vi C {0, 1}, an integer vfrom the set {0, 1., 2n 1} can be assigned by the mapping

mv = E vi2-1. This value of v is called the decimal index

i=lfor a given m-tuple.An m-variable boolean function f can be specified by listing

the values at all decimal indices. The binary-valued vectorof function values Y = [yo, Yl, Y,Y2-1] is called the truthvector for f.An m-variable boolean function f (vl, ..., vUn) can be repre-

sented as E (i) V2c(i) ....vm (i) where yj is the valuei=O

of the boolean function at the decimal index j and co(j),cl(j) .... (j) {0, 1} are the coordinates in the binaryrepresentation for j (with cmn1 as the most significant bit andco as the least significant bit) with vjvi and vj= vi.

Definition 1: The Hamming weight of a boolean functionis defined as the number of nonzero elements in Y.

Definition 2: The autocorrelation function of a booleanfunction f (v) at a is the inner product of f with a shift of f by

2m1a. More precisely, r(a) = 1(_l)f(v)f(v Ea) where a e

v=Om

{fO,1, ..., 2n 1}, a = E ai2'- . An autocorrelation functioni=l

is represented as a vector B = [r(0),r(1).r(2m -1)]Definition 3: The complementary set of a boolean function

2T 1f (v) is defined by Csetf {a , f (v)f (v D a) = O}

v=O

This means that for any element a in the Csetf, f(v)f(v Da) = 0 for any choice of v e {0,1, ..., 2n _ 1}. Thecomplementary set links distinguishability in the quantumworld (orthogonality of subspaces) with properties of booleanfunctions. The quantity f(v D a) plays the counterpart in the

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Quantum world of the Quantum subspace after the error hasoccurred, which is to be orthogonal to the original subspacecorresponding to f (v).Lemma 1: If the Hamming weight of the boolean function

f is M, and M < 2'm1, then the Csetf = {a r(a) = 2m4M}

III. BOOLEAN FUNCTIONS AND A LOGIC OF PROJECTIONOPERATORS

In [2], the authors introduced boolean logic for projectionoperators derived from the Heisenberg-Weyl group. In thissection, we generalize results from [2] for a larger class ofprojection operators.

Let B(H) be the set of bounded linear operators on a Hilbertspace H. An operator P e TB(H) is called a projection operator(sometimes we will use the terms orthogonal projection oper-ator and self-adjoint projection operator) on H iff P = PPt.We denote the set of projection operators on H by P(H) andthe set of all subspaces in H by L(H).

Definition 4: 1) If S C H, the span of S is defined asVS = n{K K is a subspace in H with S C K}. Itis easy to see that VS is the smallest subspace in Hcontaining S.

2) If S C H, the orthogonal complement of S is definedasS' {Cx HxlsVs CS}.

3) If S is a collection of subsets of H, we write VSES =

V(UsEES).Definition 5: Let P e P(H) and let K = image(P) =

{Px x e H}. We call P the projection of H onto K. Twoprojections P and Q onto K and L are orthogonal (denotedPIQ) if PQ = 0. It is easy to verify that PQ = OKIL X QP = 0. (Theorem 5B.9, [8])

Definition 6: Let P, Q C P(H) with K = image(P) andL image(Q). Then we define

P < Q iff Kc L (K#7 L)P V Q is the projection of H onto K V LP A Q is the projection of H onto K n L.

* P is the projection of H onto K' ( We will alsosometimes use P in place of P).

The structure (P(H),,<I) is a logic with unit IH (identitymap on H) and zero ZH (ZH(X) = 0 Vx e H) (Theorem5B.18, [8]). This logic is called Projection Logic.

Lemma 2: (Theorem 5B.18, [8]) The map P -* image(P)from P(H) to L(H) is a bijection that preserves order,orthogonality, meet(A) and join(V).

Lemma 3: If P and Q are commutative operators, then thedistributive law holds (and this law fails to hold for non-commutative operators). Also, in this case,

P A Q PQP D QA(P A Q) V (P A Q) = P + Q -2PQP = -P+QPQPVQ =P+Q-PQ

Now we define projection function along the lines of [2].Definition 7: Given an arbitrary boolean function

f (vi. ....UnV), we define the projection function f(Pi, ..., Pm)in which vi in the boolean function is replaced by Pi,multiplication in the boolean logic is replaced by the meetoperation in the projection logic, summation in the booleanlogic (or the or function) is replaced by the join operation inthe projection logic and the not operation in boolean logic bythe tilde (P) operation in the projection logic.

As is standard when writing boolean functions, we use xorin place of or, hence by above definition, we will replace thexor in the boolean logic by the xor operation in the projectionlogic.

Theorem 1: If (Pi, ..., Pm) are pairwise commutative pro-jection operators, each of dimension 2m- 1, such thatPlP2. Pm, PlP2. Pm, ... PlP2..Pm are all one-dimensionalprojection operators and H is of dimension 2m, then Pf =

f(Pi. Pm) is an orthogonal projection on the subspace ofdimension Tr(Pf) = wt(f), where wt(f) is the Hammingweight of the boolean function f.

Theorem 1 is a generalization of the Theorem 1 of [2]because we consider any pairwise commutative projection op-erators, while in [2], a special case of commutative projectionoperators using Heisenberg-Weyl group was used. This specialcase is described in Section V. Hence, to prove Theorem 1,we use projection logic [8] rather than the properties of aparticular commutative subgroup.

IV. THE HEISENBERG-WEYL GROUP

Let or,, o y, and ur, be the Pauli matrices, given by

and consider linear operators E of the form E el X ... 9em,where ej C {I2,(JX,(7Y,(Z}. We form the Heisenberg-Weylgroup (sometimes we will use the terms extra-special 2-groupor Pauli group) Em of order 4m±1, which is realized as a groupof linear operators acE, a = ±1, ±i. (For a detailed descriptionof extra-special group and its use to construct quantum codessee [6], [7].)

Next we define the symplectic product of two vectors.

Definition 8: The symplectic inner product of vectors(a,b), (a', b') e IF2,n is given by

(a,b) * (a',b') = a b' D a' b.

The center of the group Em is {±I2, ±JiI2m} and thequotient group Em is isomorphic to the binary vector spaceIF2n. We associate with binary vectors (a, b) e IF2, operatorsE(a,b) defined by

E(a,b) = ..1(8 ... (8 em,

where ei =(7

x,I7Cz

(7y:

°,bi1,bi°,bi1,bi

0O0,1,

1.

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V. THE CONSTRUCTION OF COMMUTATIVE PROJECTIONOPERATORS FROM THE HEISENBERG-WEYL GROUP

have symplectic product zero and that all the rows arelinearly independent.

We will now describe how to construct commutative pro-jection operators along the lines of [2]. Take m linearlyindependent vectors X1, X2, ...m of length 2m bits with theproperty that the symplectic product between any pair is equalto zero. If we take Pi = (I + ExJ, then Pp, ... Pm satisfyall the properties of Theorem 1, and hence, f (Pi, Pm) isan orthogonal projection operator [2]. This technique has beenused earlier to construct grasmannian packings associated withbinary Reed-Muller and Nordstrom-Robinson codes in [2] and[1] respectively.

VI. FUNDAMENTALS OF QUANTUM ERROR CORRECTION

A ((k, M)) quantum error correcting code is an M-dimensional subspace of C2k. The parameter k is the code-length and the parameter M is the dimension or the sizeof the code. Let Q be the quantum code, and P be thecorresponding orthogonal projection operator on Q. (For adetailed description, see [3].)

Definition 9: An error operator E is called detectable iffPEP = CEP, where CE is a constant that depends only onE.

Following [9], we confine ourselves to the errors in theHeisenberg-Weyl group. Next, we define the minimum dis-tance of the code.

Definition 10: The minimum distance of Q is the maximuminteger d such that any error E, with symplectic weight at mostd-1 is detectable.

The parameters of the quantum error correcting code arewritten ((k, M, d)) where the additional parameter d is theminimum distance of Q. We say that ((k, M, d)) Quantum errorcorrecting code exist if there exists a ((k, M)) Quantum errorcorrecting code with minimum distance > d. In this paper,we focus on non-degenerate ((k, M, d)) codes, for whichPEP = 0 for all errors E of symplectic weight < d- 1, whichis a sufficient condition for existence of the quantum code.The assumption of non-degeneracy is reasonable since we arenot aware of any case when degenerate code performs betterthan a non-degenerate code. A quantum code is additive iff itcan be constructed by the stabilizer framework of [7][11]. Aquantum code is non-additive if it is not additive.

VII. QUANTUM ERROR CORRECTING CODES WITHMINIMUM DISTANCE d

Theorem 2: A ((k, M, d))-QECC is determined by aboolean function f with the following properties

1)2)

f is a function of k variables and has weight M.The complementary set associated with f contains theset {[X1,X2, ....x2k] X WTI W is a 2k bit vector ofsymplectic weight < d 1} and the matrix Af =

[X1X2. X2klkx2k has the property that any two rows

The projection operator corresponding to the QECC is ob-tained as follows:

1) Construct the matrix Af as above.2) Define k projection operators each of the form 2 (I+Ev)

where v is a row of the matrix Af, with Pk correspond-ing to the I" row, Pk-1 corresponding to the 2nd rowand so on, so that P1 corresponds to the last row (asdescribed in Section V).

3) Transform the boolean function f into the projectionoperator Pf using Definition 7 where the commutativeprojection operators P1 .... Pk are determined by thematrix Af.

Example 1: For m > 2, we construct a ((2m,4m-1,2))additive QECC as an example of the above approach. Note thatRains [17] has shown that M < 4m- 1 for any ((2m, M, 2))quantum code and this example meets the upper bound. Takef (v) = V2mV2m -. It is a function of k = 2m variables withHamming weight 4m-1 and the corresponding complementaryset is {(010..0), (010...01), ....(111...1)} (or {4m 1, 4m 1 +1: ...., 4m 1I} in decimal notation). This complementary setcontains the set {X1, X2...X2k, 1+Xk+1...Xk +X2k} where x1= x2 = ... =Xk = (O I 0 .. 0) (or 4-1 ), Xk+1=(1 0 1.. 1),Xk+2 = ( 0 1 0 - 0), Xk+3 (O0 0 1 0 ±.. ), ..,X2k-1 (0 0 .. 0 1) and X2k = (I 0 0 .. 0). The matrix Af is given by

Af=

010

000

010

000

101

111

101

000

100

000

100

010

100

001

100

000

We see that the symplectic inner product of any two rows iszero. Hence, we have constructed a ((2m, 4m- 1, 2)) QECC.Tracing through the construction of the projection operatorPf we find that Pf = P2mP2m- , where Pi = (I + Ev)and vi is the (2m + 1-i)th row of the matrix Af. Hence,P2m = (I + Eoo..o0 11..1) and P2m- 1 = '(I+ Ell..1100..o)Example 2: For m > 2, we construct a ((2m,4m-1,2))

non-additive QECC as an example of the aboveapproach. Consider the boolean function f(v)V2mV2m-1 V2m-2 +V2mV2m-1 V2m-2(V2m-3 +V2m-3V2m-4+V2m-3V2m-4V2m-5 + *** + 14V2m 3...v2 ) +

V2mV2mTlV2mT2...vl. It is a function of k = 2m variableswith weight 4m-1, and the corresponding complementary setis {(011..1), (100...0), (100...1), ....(111 ...1)} (or {22m-1 1,22m-1 ....4 1} in decimal notation). This complementarycontains the set {X, x2, ...X2k, 1 + Xk+1...Xk +X2k} whereX1= 2 = ... = Xk = (OI I .. 1) (or 22--1), Xk+1 = (10 1.. 1), Xk+2 =(1 0 1 0 .. 0), Xk+3 = (1 0 0 1 0 .. 0), .. .

X2k-1 = (I 00 *-0 ) and X2k = (I 00 .. 0). The matrix Af

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is given by

Af=

011

111

011

111

101

1i1

101

000

100

000

100

010

100

001

VIII. OPERATOR QUANTUM ERROR CORRECTION(OQEC)1

00

000

Hence, we can also see that the second property is satisfied.Hence, we have constructed a ((2n, 4m1, 2)) QECC that isnon-additive. Note that this construction has ((4, 4, 2))-QECCas a special case, which was mentioned as an open questionin [17].

Example 3: The ((5,6,2)Y)-QECC constructed by Rains etal. [16] is also a special case of the above procedure. Takethe boolean function f(v) = V1V2V3 + V3V4V5 + V2V3V4 +V1V2V5 + V1V4V5 + V2V3V4V5. It is a function of 5 variableswith weight 6, and the corresponding complementary setis {1,3,4,6,8,11,12,14,17,19,21,22,24,26,28,31}. Take(xl, , x10 ) to be (6, 12, 24, 17, 3, 14, 31, 28, 26, 22)and form the matrix

/0 00 1

Af= I II 0

kO O

11000

10001

00011

01110

11111

11

00

11010

10110

The symplectic inner product of any two rows is zero and thecorresponding projection operator Pf coincides with the onedetermined by ((5,6,2))-QECC in [16].

Lemma 4: . If there exists a ((k, M, 2)) QECC, thenthere exists a ((k+2, 4M, 2)) QECC determined by samef(v) and Aft = (X1,x2, .. ,Xk-1,Xk,Xk,Xk,Xk+1,Xk+2, ...., X2k-1, 2k+1 +2k +X2k, 2 +X2k, 2k+1 +X2k)

If there exists a ((k, M, 2)) QECC, then there exists a((k,M -1, 2)) QECC determined by same Af and f'(v)having support (support is the set of inputs of the booleanfunction at which the output is 1) a subset of f(v).

Example 4: We can extend the Rains code to get ((2m +1, 3 x 22m-3,2))-QECC for m > 2 using the above lemma.

Example 5: The perfect ((5,2,3)) code of R. Laflamme etal. [14] can be obtained by the above approach. Take f (v) =V5V4V3V2. The corresponding complementary set is {2, 3, ...31}. The matrix Af is given by

Af=

00010

10000

11001

01100

00110

10101

01010

00100

10010

01001 I

and it is easy to see that all rows are linearly independent, andthat the symplectic inner product of any two rows is zero.

The theory of Operator Quantum error correction [13] usesthe framework of noiseless subsystems to improve the perfor-mance of decoding algorithms which might help improve thethreshold for fault-tolerant quantum computation. It requires afixed partition of the systems Hilbert space H = A XB D C'1 .Information is encoded on the A subsystem; the logicalquantum state PA C BA is encoded as PA X PB 0OC withan arbitrary PB C BB (where BA and BB are the sets ofall endomorphisms on subsystems A and B respectively). Wesay that the error E is correctable on subsystem A whenthere exists a physical map R that reverses its action, up to atransformation on the B subsystem. In other words, this errorcorrecting procedure may induce some nontrivial action on theB subsystem in the process of restoring information encodedin the A subsystem. In the case of classical quantum errorcorrecting codes, the dimension of B is 1.

Given a ((k, MN, d))-QECC as above, we take MN ba-sis vectors, say gl, 92, ... YMN for the MN-dimensionalvector space. Consider a sector of this subspace formed byPA X PB where PA C BA and PB C BB. We can encodeinformation on subsystem A, giving an ((k, M, N, d))-OQEC,where M is the dimension of the subsystem on which weencode the information (called the logical subsystem), and Nis the dimension of the subsystem that is allowed to suffer atransformation on the occurrence of error (called the Gaugesubsystem). We also see that given such an ((k,M,N,d))-OQEC, we can define a ((k, M, d))-QECC in which the M-dimensional subspace is formed by PA X IB. This is becausethis M dimensional subspace is a subspace of the above MNdimensional subspace, and we know that any subspace ofthe quantum code is also a quantum code. In other words,if we fix PB (for example IB above), we get the classicalerror correcting code. Thus, we have a general method ofconstructing non-additive and additive OQEC. In the classicalquantum error correcting codes, N = 1. In standard quan-tum error correcting codes, one requires the ability to applya procedure which exactly reverses on the error-correctingsubspace any correctable error. In contrast, for operator error-correcting subsystems, the correction procedure need not undothe error which has occurred, but instead one must performcorrections only modulo the subsystem structure (subsystemB). This leads to recovery routines which explicitly make useof the subsystem structure [5].

Example 6: Consider the ((5,6,2))-QECC code as in Ex-ample 3. In the 6-dimensional space, we take 6 basis vectors91, 92, --, 96-

Let'a b c'

PA d Z fg h s p

is an endomorphism on 3-dimensional subspace.

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a O b 0 c 0O a O b O c

PdO eO f O

PA XIB 10 d 0 e 0 fg O h i 0j0O g O h O i,

is an endomorphism in the 6-dimensional space which forms((5, 3, 2))-QECC.

Also, quantum state PA is encoded as PA X PB OC =

ajaldjdlgigl

akamdkdmgkgm

bjblejelhjhl

bkbmekemhkhm

ciclfifl

iI

ckcmfkfmikim/

for arbitrary j, k , I and m. This operator is w.r.t. basis formedby gl 92,..9 , 96. At the receiver, from the corrupted version ofPA X PB OC , we can recover PA (we need just PA since ourinformation is only encoded on the subsystem A) by takingprojection onto U = {Pi p2 P1 C IBA, P2 C TBB} followedby taking the trace over the subsystem B.

Example 7: The stabilizer framework for OQEC is givenin [15] which provides a method of constructing the stabilizerOQEC. We denote by Xj the matrix X (the Pauli matrix)acting on the jth qubit, and similarly for Yj and Zj. ThePauli group P, =< i,X1,Z1,...,X,Zr, >. The first stepin constructing a stabilizer code is to choose a set of 2noperators {XX, Z }~ 1,.. , from Pr, that is Clifford isomorphicto the set of single-qubit Pauli operators {Xj, Zj}j=} ,..,n inthe sense that the primed and unprimed operators obey thesame commutation relations among themselves. The operators{XJ, Z }j= 1,..,n generate Pn and behave as single-qubit Paulioperators. We can think of them as acting on n virtual qubits.

Suppose there exists a ((k, 2s, d))-additive QECC corre-sponding to a 25 dimensional subspace, say C by the aboveframework of boolean functions. This means that for f(v) =Vs+lvs+2±...Vk, there exists a matrix Af such that all its rowsare linearly independent and have pairwise symplectic productzero. The first k -s rows correspond to the stabilizers of thecode. Form Z', ..., Z' corresponding to the rows of matrixAf. (The image of the first row in the Pauli group gives Z{and so on.) Given all the ZX, we can easily find XI whichhave symplectic product of 1 with X} and symplectic productof 0 with all other XIl, l#j.

Hence, the stabilizer group is given by S = < ZI, ZI,.Z/ >. If we want to construct a ((k, 2t, 2s-t, d))-OQEC,then we need to find a subsystem of dimension 2t in theabove subspace C of dimension 2s. It is easy to see that if wetake the Gauge group (corresponding to the Gauge subsystemdefined before) G = < S, XI_S+1~ Z/ *X-t: Z/ -t >and the logical group L = < XI -I XK,Z >'the action of any I e L and g e G restricted to the code

subspace C is given by

gPIP -

= IA (9 9BlA ('IB

for some IA 9B in BA and BB respectively, where A and Bare the required subsystems [15][19].

IX. CONCLUSION

We have described a new mathematical framework thatunifies the construction of additive and non-additive quantumcodes. It is based on a correspondence between booleanfunctions and projection operators. We have given sufficientconditions for the existence of QECC in terms of existenceof a boolean function satisfying certain properties. Examplesof boolean functions have been presented that satisfy theseproperties. Using these boolean functions, we have presenteda construction of additive and non-additive ((2m, 4`-1, 2))codes, the original ((5,6,2)) code constructed by Rains et. al.,the extension of this code to ((2m+1, 3x22m-3, 2)) codes,and the perfect ((5,2,3)) code. Finally we have shown how thenew framework can be integrated with operator quantum error

correcting codes.

REFERENCES

[1] V. Aggarwal, A. Ashikhmin and A.R. Calderbank, "A grassmannian pack-ing based on the nordstrom-robinson code," IEEE Information TheoryWorkshop, pp. 1-5, Chengdu, China, Oct. 2006.

[2] A. Ashikhmin and A.R. Calderbank, "Space-time reed-muller codes fornoncoherent MIMO transmission," IEEE International Symposium onInformation Theory, pp. 1952-1956, Adelaide, Australia, Sept. 2005.

[3] A. Ashikhmin and S. Litsyn, "Foundations of quantum error correction,"Recent Trend in Coding Theory and its Applications, 2007.

[4] V. Arvind, P.P. Kurur and K.R. Parthasarathy, "Nonstabilizer quantumcodes from abelian subgroups of the error group," quant-ph/0210097.

[5] D. Bacon, "Operator quantum error-correcting subsystems for self-correcting quantum memories," Phys. Rev. A 73, 012340, 2006.

[6] A.R. Calderbank, E.M. Rains, P.M. Shor and N.J.A. Sloane, "Quantumerror correction via codes over GF(4)," IEEE Trans. on Inf Th., Jul 1998.

[7] A. R. Calderbank, E. M. Rains, P. W. Shor, and N. J. A. Sloane, "Quantumerror correction and orthogonal geometry", Phys. Rev. Lett, vol. 78, pp.405-409, 1997.

[8] D.W. Cohen, "An introduction to hilbert space and quantum logic,"Springer-Verlag, 1989.

[9] A. Ekert and C. Macchiavello, "Quantum error correction for communi-cation," Phys. Rev. Lett. 77, pp. 2585-2588, Sept. 1996.

[10] M. Grassl and T. Beth, "A note on non-additive quantum codes," quant-ph/9703016, March 1997.

[11] D. Gottesman, "Stabilizer codes and quantum error correction," PhDThesis, quant-ph/9705052.

[12] A. Ketkar, A. Klappenecker, S. Kumar and P. K. Sarvepalli, "Nonbinarystabilizer codes over finite fields," IEEE Transactions on InformationTheory, pp. 4892-4914, Nov. 2006.

[13] David Kribs, Raymond Laflamme and David Poulin, "Unified andgeneralized approach to quantum error correction," Phys. Rev. Lett. 94,180501, 2005.

[14] R. Laflamme, C. Miquel, J. P. Paz, and W. H. Zurek, "Perfect quantumerror correcting code," Phys. Rev. Lett. 77, pp. 198201, 1996.

[15] D. Poulin, "Stabilizer formalism for operator quantum error correction,"quant-ph/0508131 Jun 2006.

[16] E. M. Rains, R. H. Hardin, P. W. Shor, and N. J. A. Sloane, "Anonadditive quantum code," Phys. Rev. Lett. 79, pp. 953954, 1997.

[17] E.M. Rains, "Quantum codes of minimum distance two," IEEE Trans-actions on Information Theory, pp. 266-271, Jan 1999.

[18] V. P. Roychowdhury and F. Vatan, "On the existence of nonadditivequantum codes", Lecture notes in computer science, Springer, 1998.

[19] P. Zanardi, D. A. Lidar, and S. Lloyd, "Quantum tensor productstructures are observable induced," Phys. Rev. Lett. 92, 060402, 2004.

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boolean functions, projection operators and quantum error

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