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

Boolean Functions, Projection Operators andQuantum Error Correcting Codes

Vaneet AggarwalDepartment of Electrical Engineering

Princeton UniversityEmail: vaggarwa@princeton.edu

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: calderbk@math.princeton.edu

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

1-4244-1429-6/07/$25.00 2007 IEEE 2091

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

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),,

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

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

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...