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A QUANTUM COMPUTING PRIMER FOROPERATOR THEORISTS

DAVID W. KRIBS

Abstract. This is an exposition of some of the aspects of quan-tum computation and quantum information that have connectionswith operator theory. After a brief introduction, we discuss quan-tum algorithms. We outline basic properties of quantum channels,or equivalently, completely positive trace preserving maps. Themain theorems for quantum error detection and correction are pre-sented and we conclude with a description of a particular passivemethod of quantum error correction.

1. Introduction

The study of the underlying mathematics for quantum computationand quantum information is quickly becoming an interesting area ofresearch [61]. While these fields promise far reaching applications [12,41, 60], there are still many theoretical and experimental issues thatmust be overcome, and many involve deep mathematical problems.The main goal of this paper is to provide a primer on some of thebasic aspects of quantum computing for researchers with interests inoperator theory or operator algebras. However, we note that the onlyprerequisite for reading this article is a strong background in linearalgebra.This work should not be regarded as an extensive introduction to

the subject. Indeed, the reader with knowledge of quantum computingwill surely have complaints about the selection of material presented.Moreover, we do not consider here the increasingly diverse fields ofmathematics for which there are connections with quantum comput-ing (algebraic geometry, Fourier analysis, group theory, number theory,operator algebras, etc). Rather, our intention is to give a brief intro-duction and prove some specific results with the hope that this paperwill help stimulate interest within the operator community.

2000 Mathematics Subject Classification. 47L90, 47N50, 81P68.key words and phrases. quantum computation, quantum information, quantum

algorithms, completely positive maps, quantum channels, quantum error correction,noiseless subsystems.

1

2 D.W.KRIBS

The paper is organized as follows. The next section (§2) containsa discussion of the basic notions, notation and nomenclature used inquantum computing. In §3 we give a brief introduction to quantumalgorithms by describing some elementary examples and presenting asimple algorithm (Deutsch’s algorithm [19, 20]) that demonstrates thepower of quantum computation. In §4 we outline the mathematical for-malism for the evolution of information inside quantum systems. Thisis provided by quantum channels, which are represented by completelypositive trace preserving maps [15, 51, 52, 62, 63]. The penulti-mate section (§5) includes a discussion of quantum error correctionmethods. We present the fundamental theorems for quantum error de-tection and correction in the ‘standard model’ [28, 47, 64]. In thefinal section (§6) we describe a specific method of quantum error pre-vention [23, 26, 32, 47, 48, 57, 82] called the ‘noiseless subsystem vianoise commutant’ method. Finally, we have included a large collectionof references as an attempt to give the interested reader an entrancepoint into the quantum information literature.

2. Quantum Computing Basics

Let H be a (complex) Hilbert space. We shall use the Dirac nota-tion for vectors and vector duals in H: A typical vector in H will bewritten as a ‘ket’ |ψ〉, and the linear functional on H determined bythis vector is written as the ‘bra’ 〈ψ|. Notice that the products of a braand a ket yield the inner product, 〈ψ1||ψ2〉, and a rank one operator,|ψ2〉〈ψ1|. In particular, given |ψ〉 ∈ H, the rank one projection of Honto the subspace {λ|ψ〉 : λ ∈ C} is written |ψ〉〈ψ|. Further let B(H)be the collection of operators which act on H. We will use the physicsconvention U † for the adjoint of an operator U .The study of operators on Hilbert space is central to the theory of

quantum mechanics. For instance, consider the following formulationof the postulates of quantum mechanics [61, 77]:

(i) To every closed quantum system there is an associated Hilbertspace H. The state of the system at any given time is described by aunit vector |ψ〉 in H, or equivalently by a rank one projection |ψ〉〈ψ|.When the state of the system is not completely known, it is repre-sented by a density operator ρ on H, which is a positive operator withtrace equal to one. (This is the quantum analogue of a probabilitydistribution.)(ii) The notion of evolution in a closed quantum system is described

by unitary transformations. That is, there is a unitary operator U

QUANTUM COMPUTING 3

on the system Hilbert space such that the corresponding evolution isdescribed by the conjugation map ρ 7→ UρU †.(iii) A measurement of a quantum system on H is described by a set

of operators Mk, 1 ≤ k ≤ r, such that

r∑

k=1

M †kMk = 1l.

The measurement is projective if each of theMk is a projection, and thusthe Mk have mutually orthogonal ranges. (A ‘classical measurement’arises when all the projections are rank one.) The index k refers to thepossible measurement outcomes in an experiment. If the state of thesystem is |ψ〉 before the experiment, then the probability that event k

occurs is given by p(k) ≡ 〈ψ|M †kMk|ψ〉. Notice that {p(k)} determines

a probability distribution.(iv) Given Hilbert spaces H1, . . . ,Hm associated with m quantum

systems, there is a composite quantum system on the Hilbert spaceH1⊗· · ·⊗Hm. In particular, if the states of the individual systems are|ψi〉, then the state of the composite system is given by |ψ1〉⊗· · ·⊗|ψm〉.The Hilbert spaces of interest in quantum information theory are of

dimension N = dn for some positive integers n ≥ 1 and d ≥ 2. (Itis generally thought that extensions to infinite dimensional space willbe necessary in the future, but the current focus is mainly on finitedimensional aspects.) For brevity we shall focus on the d = 2 cases.Thus we letHN be the Hilbert space of dimension 2n given by the n-foldtensor product HN = C2 ⊗ . . .⊗ C2 = (C2)⊗n. We will drop referenceto N when convenient. Let {|0〉, |1〉} be a fixed orthonormal basis for2-dimensional Hilbert space H2 = C2. These vectors will correspond tothe classical base states in a given two level quantum system; such asthe ground and excited states of an electron in an atom, ‘spin-up’ and‘spin-down’ of an electron, two polarizations of a photon of light, etc.We shall make use of the abbreviated form from quantum mechanicsfor the associated standard orthonormal basis for H2n = (C2)⊗n ≃ C2n .For instance, the basis for H4 is given by {|ij〉 : i, j ∈ Z2}, where |ij〉is the vector tensor product |ij〉 ≡ |i〉|j〉 ≡ |i〉 ⊗ |j〉.A quantum bit of information, or a ‘qubit’, is given by a unit vector

|ψ〉 = a|0〉 + b|1〉 in H2. The cases a = 0 or b = 0 correspond to theclassical states, and otherwise |ψ〉 is said to be in a superposition of thestates |0〉 and |1〉. A ‘qudit’ is a unit vector in Cd. A vector state |ψ〉 inHN is said to be entangled if it cannot be written as a tensor product ofstates from its component systems, so that |ψ〉 does not decompose as|ψ〉 = |φ1〉|φ2〉 for some vectors |φi〉, i = 1, 2. As an example, consider

4 D.W.KRIBS

|ψ〉 = |00〉+|11〉√2

in H4, this is a vector from the so-called ‘EPR pairs’ or

‘Bell states’. Roughly speaking, the notion of decoherence in a quantumsystem corresponds to the vanishing of off-diagonal entries in matricesassociated with the system as it evolves.A number of specific unitary matrices arise in the discussions below.

The Pauli matrices are given by

X =

(

0 11 0

)

, Y =

(

0 −ii 0

)

, Z =

(

1 00 −1

)

.

Let 1l2 be the 2 × 2 identity matrix. We regard these as matrix repre-sentations for operators acting on the given basis forH2. In the n-qubitcase (so N = 2n) we may consider all ‘single qubit unitary gates’ deter-mined by the Pauli matrices. This is the set of all unitaries {Xk, Yk, Zk :

1 ≤ k ≤ n} where X1 = X ⊗ 1l⊗(n−1)2 , X2 = 1l2 ⊗X ⊗ 1l

⊗(n−2)2 , etc. Fur-

ther let UCN be the ‘controlled-NOT gate’ on H4. This is the unitarywhich acts on H4 by UCN : |i〉|j〉 7→ |i〉|(i+ j)mod 2〉. The CNOT gate

has natural extensions {U (k,l)CN : 1 ≤ k 6= l ≤ n} to unitary gates on HN ,

where the kth and lth tensor slots act, respectively, as the control and

target qubits. For instance, with this notation UCN = U(1,2)CN . Note that

U(k,l)CN only acts on the kth and lth qubits in HN = (C2)⊗n. The set

{Xk, Yk, Zk, U(k,l)CN : 1 ≤ k 6= l ≤ n} forms a set of universal quantum

gates for HN , meaning that this set generates the set of N ×N unitarymatrices U(N) as a group (up to complex phases). We shall also makeuse of the Hadamard gate

H =1√2

(

1 11 −1

)

,

and the spin-12Pauli matrices σk = 1/2K for k = x, y, z.

3. Quantum Algorithms

Simply put, a quantum algorithm consists of an ensemble of initialstates ρ which evolves under a unitary matrix U to a final density ma-trix UρU †. The famous factoring algorithm of Shor [67, 70] and searchalgorithm of Grover [31] have received tremendous attention over thelast decade. As a proper treatment of these algorithms is beyond thescope of this article, in this note we shall present the Deutsch algo-rithm [19] (and its generalization, the Deutsch-Josza algorithm [20])since it is easily accessible and gives a good illustration of the power ofquantum computation. In doing so, we shall give a description of two

QUANTUM COMPUTING 5

fundamental classes of quantum algorithms: the simulation of a clas-sical function on a quantum computer and the algorithm for quantumparallel computation.Before continuing, let us illustrate a simple example of how an op-

eration such as addition may be performed with a quantum algorithm.This will also allow us to establish some notation for the discussionwhich follows. We shall identify the standard basis vectors |i1 · · · in〉for HN with the integers {0, 1, . . . , 2n−1} via binary expansions. Thenwe may define a unitary U on HN ⊗HN by U |x〉|y〉 = |x〉|x⊕y〉 wherethe addition x ⊕ y is modulo N . The corresponding quantum algo-rithm implements addition modulo N . (Note that the CNOT gate isobtained in the case N = 2.)Towards the Deutsch algorithm, let Hn = H⊗n be the n-fold tensor

product of the Hadamard gate acting on HN . Observe that Hn|0〉⊗n isthe uniform superposition

Hn|0〉⊗n =1√2n

2n−1∑

x=0

|x〉.

Fix positive integers k,m ≥ 1. Let f : Zm2 → Zk

2 be a function.Consider the space Hm,k = H2m ⊗H2k . Then Hm,k has basis |x〉|y〉 =|x〉⊗ |y〉 where x ∈ Zm

2 and y ∈ Zk2. Define a unitary map Uf : Hm,k →

Hm,k byUf : |x〉|y〉 7→ |x〉|y ⊕ f(x)〉.

For a given x ∈ Zm2 , notice that the action of Uf is to permute the basis

vectors {|x〉|y〉 : y ∈ Zk2}.

Note 3.1. Observe that Uf |x〉|0〉 = |x〉|f(x)〉 for all x. Thus Uf simu-

lates f on a quantum computer, and in this sense any classical functioncan be performed on a quantum computer.

Next we compute

Uf

(

(Hm|0〉⊗m)⊗ |0〉⊗k)

= Uf

( 1√2m

2m−1∑

x=0

|x〉 ⊗ |0〉)

=1√2m

2m−1∑

x=0

Uf(|x〉 ⊗ |0〉)

=1√2m

2m−1∑

x=0

|x〉 ⊗ |f(x)〉

Hence an application of Hm⊗1l2k followed by Uf yields a simultaneousparallel computation of f on all possible values of x. The corresponding

6 D.W.KRIBS

‘circuit-gate’ diagram for quantum parallelism for f is given below. Insuch a diagram, the ‘circuits’ correspond to states from the componentsystems (in this case |0〉⊗m and |0〉⊗k from the systems H2m and H2k

respectively). The ‘gates’, drawn as boxes, indicate unitary operatorsapplied as the system evolves with a left-to-right convention in thediagram (so here Hm ⊗ 1l2k is applied first, then Uf is applied).

|0〉⊗k

|0〉⊗m

1√2m

∑2m−1x=0 |x〉 ⊗ |f(x)〉Uf

Hm

Let f : Z2 → Z2 be a function. Then call f constant if f(0) = f(1)and balanced if f(0) 6= f(1). The problem addressed by Deutsch’s al-

gorithm is the following: Given f : Z2 → Z2, determine if it is constantor balanced. Classically an answer to this problem requires two eval-uations of f . On the other hand, Deutsch’s algorithm shows how todo this with one quantum operation. The circuit-gate diagram for thealgorithm is given below.

|1〉

|0〉

Uf

H

H

H

The initial state is given by |01〉 = |0〉 ⊗ |1〉. The first stage of thealgorithm evolves this state to

(H ⊗H)(|0〉 ⊗ |1〉) = (H|0〉)⊗ (H|1〉) = |+〉 ⊗ |−〉,

where |+〉 = |0〉+|1〉√2

and |−〉 = |0〉−|1〉√2

. Observe that the action of Uf as

defined above yields

Uf (|x〉 ⊗ |−〉) = (−1)f(x)|x〉 ⊗ |−〉 for x ∈ Z2.

QUANTUM COMPUTING 7

Thus, after the second stage of the algorithm the system has evolvedto

Uf (|+〉 ⊗ |−〉) =1√2Uf

(

(|0〉+ |1〉)⊗ |−〉)

=((−1)f(0)|0〉+ (−1)f(1)|1〉√

2

)

⊗ |−〉

=

{

±|+〉 ⊗ |−〉 if f is constant±|−〉 ⊗ |−〉 if f is balanced

Finally, we apply a Hadamard gate H to the first qubit; that is, weapply H ⊗ 1l2 to the full system. Thus the system evolves to

{

(H ⊗ 1l2)(±|+〉 ⊗ |−〉) = ±|0〉 ⊗ |−〉 if f is constant(H ⊗ 1l2)(±|−〉 ⊗ |−〉) = ±|1〉 ⊗ |−〉 if f is balanced

In particular, if we measure the first qubit we get

{

±|0〉 if f is constant±|1〉 if f is balanced

Note that there is no uncertainty in the result: If we measure the firstqubit and obtain ±|0〉 (respectively ±|1〉), then we know f is constant(respectively balanced) with probability 1.

Remark 3.2. The Deutsch-Josza generalization [20] yields a moredramatic result. Let f : Zm

2 → Z2 be a function. Define f to beconstant if f(x) = f(y) for x, y ∈ Zm

2 , and balanced if |f−1(0)| =2m−1 = |f−1(1)|. Suppose we know f is either constant or balancedand that we wish to determine which one f is. Classically, this requires2m−1 + 1 evaluations to know with certainty what type f is. TheDeutsch-Josza algorithm shows how this may be accomplished witha single operation on a quantum computer. The algorithm acts onH2m ⊗H2 with initial state |0〉⊗m ⊗ |1〉. The circuit-gate diagram maybe obtained by adjusting the Deutsch diagram with |0〉⊗m⊗|1〉 in placeof |0〉 ⊗ |1〉, m circuits at the top instead of one, Hm in place of H inthe top circuits prior to Uf , and Hm in place of the final H . Thus,

|0〉⊗m ⊗ |1〉 7→ (Hm ⊗ 1l2)Uf(Hm ⊗H)(|0〉⊗m ⊗ |1〉).

Note 3.3. As a starting point for more recent work on quantum algo-rithms we mention [11, 59, 78]. Also see [7] for an extensive mathe-matical introduction to the study of quantum algorithms.

8 D.W.KRIBS

4. Quantum Channels

While evolution in a closed quantum system is unitary (see postulate(ii)), experimentally evolution occurs in an ‘open quantum system’. Insuch a system evolution is described mathematically by completelypositive trace preserving maps [61]. The physical motivation for thisis discussed below.A (linear) map E : B(H) → B(H) where H is a Hilbert space is

completely positive if for all k ≥ 1 the ‘ampliation map’

E (k) : Mk(B(H)) → Mk(B(H))

given by E (k)((ρij)) = (E(ρij)) is a positive map. (We could also writeE (k) = 1lk ⊗ E .) This is a rather strong condition; for instance, thetranspose map is the standard example of a positive map which is notcompletely positive. The study of completely positive maps has beenan active area of research in both the quantum physics and operatortheory communities for at least thirty years. (In fact, it seems thatmany general results on completely positive maps have been obtainedin the two fields without knowledge of the other.) See the texts ofKraus [51] and Paulsen [62, 63] for good treatments of the subjectfrom the two perspectives.Thus, in the general setting quantum information evolves through an

open quantum system via completely positive trace preserving maps.

Definition 4.1. A quantum channel E : B(H) → B(H) is a map whichis completely positive and trace preserving.

Trace preservation for a channel is equivalent to requiring the preser-vation of probabilities as states evolve through a quantum system. Achannel is positive since density operators must evolve to density oper-ators, and it is completely positive because this property must be pre-served when the initial system is tensored with other systems (as partof a composite system). Physically, an open system can be regarded aslying inside a larger closed quantum system where all evolution occursin a unitary manner. Thus, evolution in the open system can be re-garded as a ‘compression’ of the unitary evolution on the larger closedsystem. The mathematical formalism for this physical description isprovided by Stinespring’s dilation theorem [72].The following fundamental result for completely positive maps was

proved independently by Choi [15] and Kraus [52]. We present Choi’sshort operator proof.

QUANTUM COMPUTING 9

Theorem 4.2. Let E : B(HN ) → B(HN) be a completely positive map.

Then there are operators Ek ∈ B(HN ), 1 ≤ k ≤ N2, such that

E(ρ) =N2

∑

k=1

EkρE†k for all ρ ∈ B(HN ).(1)

Proof. Let eij = |i〉〈j| be the matrix units associated with the stan-dard basis for HN . Let R = E (N)((eij)). This matrix is positive bythe N -positivity of E . (In fact, Choi proved that the positivity ofR characterizes complete positivity of E .) Consider a decomposition

R =∑N2

k=1 |ak〉〈ak|, where |ak〉 ∈ CN2

are (appropriately normalized)eigenvectors for R. Let {Pi : 1 ≤ i ≤ N} be the family of rank

N projections on CN2

which have mutually orthogonal ranges andsatisfy PiRPj = E(eij). Then |ak〉 =

∑N

i=1 Pi|ak〉. Define operatorsEk : C

N → CN by Ek|i〉 ≡ Pi|ak〉, so that

R =∑

k

∑

i,j

Pi|ak〉〈ak|Pj =∑

i,j

Pi

(

∑

k

Ek|i〉〈j|E†k

)

Pj .

Hence,

E(eij) = E(|i〉〈j|) = PiRPj =

N2

∑

k=1

Ek|i〉〈j|E†k,

and equation (1) holds by the linearity of E . �

The decomposition (1) is referred to as the operator-sum representa-

tion of E in quantum information theory. The operators Ek are calledthe noise operators or errors of the channel. Note that trace preserva-tion of a channel is equivalent to its noise operators Ek satisfying

∑

k

E†kEk = 1l.

Remark 4.3. Choi’s proof of this theorem provided the impetus fora recent application of Leung [56] to ‘quantum process tomography’.This is a procedure by which an unknown quantum channel can befully recovered from experimental data. To accomplish this for a givenchannel, (1) shows that it is enough to recover the noise operatorsEi. As Leung observes, Choi’s proof shows the only experimental datarequired to fully reconstruct the Ei are the output states E(|i〉|j〉). Thisfact is the core of the tomography procedure outlined in [56].

The following result, which we state without proof, shows preciselyhow different sets of noise operators for the same channel are related.

10 D.W.KRIBS

Proposition 4.4. Let {E1, . . . , Er} and {E ′1, . . . , E

′r} be the noise op-

erators for channels E and E ′ respectively. Then E = E ′ if and only if

there is an r × r scalar unitary matrix U = (uij) such that E = UE ′

where Et = [E1 · · ·Er] and (E ′)t = [E ′1 · · ·E ′

r]. In other words,

Ei =

r∑

j=1

uijE′j for 1 ≤ i ≤ r.

Let A = Alg{Ei, E†i } be the algebra generated by the Ei and E†

i .

This is the set of polynomials in the Ei and E†i . An application of the

Cayley-Hamilton theorem on minimal polynomials from linear algebrashows that all such polynomials may be written as polynomials withdegree below some global bound. In quantum computing, A is calledthe interaction algebra for the channel. It is †-closed by definition,hence it is a finite dimensional C∗-algebra [3, 17, 73]. Observe that,as a direct consequence of the previous result, A can be seen to be a relicof the channel; in other words, it is independent of the choice of noiseoperators which satisfy (1) for the channel. This is most succinctlyseen in the case of unital channels, see § 6 for details.

Note 4.5. We mention that an interesting and highly active area ofcurrent research in quantum information theory revolves around thestudy of quantum channel capacities. Specifically, there are a num-ber of deep mathematical problems which are concerned with comput-ing the capacity of a quantum channel to carry classical or quantuminformation. The following references give a starting point into theliterature [8, 21, 34, 35, 36, 39, 40, 44, 58, 65, 66, 79].

Note that the quantum measurement process (postulate (iii)) nat-urally determines a quantum channel. Let us consider more specificexamples. (The final section includes further examples.)

Examples 4.6. (i) Let 0 < p < 1 and define operators on H2 by

E1 = (√

1− p)1l2, E2 = (√p)X,

with respect to {|0〉, |1〉}. The bit flip channel E(ρ) = E1ρE†1 + E2ρE

†2

flips the state |0〉 to |1〉 and vice versa with probability p. For in-stance, observe that the projection |0〉〈0| evolves to the superpositionE(|0〉〈0|) = (1− p)|0〉〈0|+ p|1〉〈1|.(ii) Let E1 = 1l2

2and define a channel by E(ρ) = E1ρE1 + σxρσx +

σyρσy + σzρσz. Observe that E(ρ) = 1l22

for every density operatorρ ∈ M2. This property, distinct density matrices evolving in a channelto the same density matrix, has recently been exploited as part of ascheme for quantum cryptography [5].

QUANTUM COMPUTING 11

(iii) Let 0 < r < 1 and define operators on H2 by

E1 =

(

1 00

√1− r

)

, E2 =

(

0√r

0 0

)

.

These noise operators define an amplitude damping channel E(ρ) =

E1ρE†1 + E2ρE

†2. Such channels characterize energy dissipation within

a quantum system.(iv) Let U1, . . . , Ud be unitaries which act on a common Hilbert space

and let r1, . . . , rd be positive scalars such that∑

i ri = 1. Then we may

define a channel by E(ρ) = ∑d

i=1 riUiρU†i . These are the prototypical

examples of ‘unital’ (E(1l) = 1l) channels (see § 6). In fact, every unitalchannel on M2 may be written as a convex sum of unitaries in thisway. This is not the case in higher dimensions however.(v) The class of entanglement breaking channels was introduced in

[35] and studied in [39]. These are quantum channels which can bewritten in the form

E(ρ) =∑

k

|ψk〉〈ψk|〈φk|ρ|φk〉,

for some vectors |ψk〉 and |φk〉. With this representation, trace preser-vation is equivalent to

∑

k |φk〉〈φk| = 1l, as Tr(ρ|φk〉〈φk|) = 〈φk|ρ|φk〉.Such channels derive their name from the fact that for d ≥ 1, a densityoperator E (d)(Γ) is never entangled, even if Γ was initially entangled.

5. Quantum Error Correction

In this section we present the central aspects of the ‘standard model’for quantum error detection and correction. For an extensive introduc-tion to the subject we point the reader to the articles [28, 47, 64].The general error correction problem in quantum computing is muchmore delicate when compared to error correction in classical computing.The possible errors that can occur include all possible unitary matrices,whereas in classical computing the only errors are bit flips. Nonethe-less, methods have been (and are being) developed for quantum errorcorrection.

5.1. Error Detection. Let H be the Hilbert space for a given quan-tum system. Then a quantum code C onH is a subspace ofH. Let PC bethe projection of H onto C. Then PC and P⊥

C describe a measurementof the system which can be used to determine if a given state |ψ〉 ∈ Hbelongs to the code. The basic idea of an error-detection scheme in thissetting is to first prepare an initial state in C, for brevity let us restrict

12 D.W.KRIBS

ourselves to unit vectors |ψ〉 in C. The state |ψ〉〈ψ| is then transmit-ted through the quantum channel E of interest, evolving to E(|ψ〉〈ψ|).Finally, the measurement PC, P

⊥C is performed on this final state. This

motivates the following definition.

Definition 5.1. Let C be a quantum code on H and let E be an error(noise) operator associated with a given quantum channel on H. ThenC detects the error E if the states which are accepted after E acts areunchanged, up to a scaling factor. In other words, there is a scalar λEsuch that

PCE|ψ〉 = λE|ψ〉 for all |ψ〉 ∈ C.Observe that the set of error operators E which are detectable for

a fixed quantum code C form a subspace of operators, or a so-calledoperator space [62]. There are a number of equivalent conditions fordetectable errors.

Theorem 5.2. Let C be a quantum code and let E be an error operator

associated with a given quantum channel. Then the following conditions

are equivalent:

(i) E is detectable by C, with scaling factor λE.(ii) PCEPC = λEPC.(iii) 〈ψ1|E|ψ2〉 = λE〈ψ1||ψ2〉 for all |ψi〉 ∈ C, i = 1, 2.(iv) For every pair of vectors |ψ1〉 and |ψ2〉 in C which are orthogo-

nal, the vectors E|ψ1〉 and |ψ2〉 are orthogonal.

Proof. We shall prove the implication (iv) ⇒ (iii). The other direc-tions are either trivial or easy to see. We may clearly assume thatdim C ≥ 2. Let {|ψ1〉, |ψ2〉, . . .} be an orthonormal basis for C. We firstclaim that (iv) implies

〈ψi|E|ψi〉 = 〈ψj |E|ψj〉 for all i, j.(2)

Indeed, to see this, fix i, j and define

|+〉 = |ψi〉+ |ψj〉 and |−〉 = |ψi〉 − |ψj〉.Then by (iv) we have

0 = 〈+|E|−〉 = 〈ψi|E|ψi〉 − 〈ψj|E|ψj〉.Thus define λE = 〈ψi|E|ψi〉, and note that this is independent of i.Now let |ψ〉 = α1|ψ1〉+ α2|ψ2〉+ . . . and |φ〉 = β1|ψ1〉+ β2|ψ2〉+ . . . bevectors in C. Then by (iv) and (2) we have

〈ψ|E|φ〉 =∑

i,j

αiβj〈ψi|E|ψj〉 =∑

i

αiβi〈ψi|E|ψi〉 = λE〈ψ||φ〉,

and this completes the proof. �

QUANTUM COMPUTING 13

Let us describe a matrix perspective for detectable errors. This canbe seen through a simple example.

Example 5.3. Let C be the quantum code given by

C = span{|000〉, |111〉} ⊆ H8.

The (unnormalized) error operators for the n-qubit depolarizing chan-nel [61, 44] are E = {1l2n , Z1, . . . , Zn}. In the 3-qubit case considerthe error operator E ≡ Z1. Observe that E|000〉 = |000〉 whereasE|111〉 = −|111〉. Thus if E was detectable by C we would have,

λE = 〈000|E|000〉 = 1 and λE = 〈111|E|111〉 = −1.

From this contradiction it follows that none of the depolarizing er-rors Zk are detectable by the code C. In fact, from Theorem 5.2 thedetectable errors for C may be realized in matrix form as the operatorspace

{

( λ1l2 ∗∗ ∗ ) : λ ∈ C

}

, with respect to an ordered orthonormal basisfor H8 of the form {|000〉, |111〉, . . .}, and where the ∗ entries indicatethat any choice is admissible.

More generally, the conditions of Theorem 5.2 show that the de-tectable errors for a given quantum code C form the subspace of oper-ators

{(

λ1l ∗∗ ∗

)

: λ ∈ C

}

,

where the matrix form is given with respect to the spatial decomposi-tion H = PCH⊕ P⊥

C H.

5.2. Error Correction. Let E = {Ei} be a set of errors that act asnoise operators for a given quantum channel. If C is a quantum code,then the basic error-correction problem for C is to determine whenthere is a decoding procedure for C such that all the errors in E arecorrected. The simplest possible case occurs when every error Ei isthe multiple of a unitary operator and the subspaces EiC are mutuallyorthogonal. The obvious decoding procedure in this situation is to firstmake a projective measurement to determine which of the subspacesEiC a given state |ψ〉 ∈ C has evolved to, then apply the inverse of theerror operator Ei. This particular case motivates the following generaldefinition in the standard model for quantum error correction.

Definition 5.4. Let E be a quantum channel and let C be a quantumcode with projection PC. Then C is correctable for E if there is aquantum channel R such that

R ◦ E(ρ) = ρ

for all ρ supported on C; that is, all ρ with ρ = PCρPC .

14 D.W.KRIBS

There are a number of useful characterizations of correctable codes.Note that in the proof of (iii) ⇒ (i) below, the error correction op-eration R is explicitly constructed and hence this gives a constructiveapproach for decoding within this error correction model.

Theorem 5.5. Let E be a quantum channel with errors E = {Ei}and let C be a quantum code with projection PC. Then the following

conditions are equivalent:

(i) C is correctable for E .(ii) The operators in the set E†

E = {E†1E2 : Ei ∈ E} are detectable

by C.(iii) There are scalars Λ = (λij) such that

PCE†iEjPC = λijPC for all i, j.(3)

(iv) There is a linear transformation Ei 7→ E ′i on E such that the

new error operators E ′i satisfy the following properties:

(a) The subspaces E ′iC are mutually orthogonal,

(b) The restriction of every E ′i to C is proportional to a restric-

tion to C of a unitary operator.

Proof. Observe that any matrix Λ which satisfies (3) must be positivesince this equation may be written as a matrix product A†A = (λijPC)where A = [E1PC E2PC · · · ] is a row matrix. Conditions (ii) and (iii)are equivalent by definition. We shall prove (i) is necessary and suf-ficient for (iii) and leave the connection with (iv) for the interestedreader.For (i) ⇒ (iii), let C be a quantum code with code projection PC.

Suppose E is a quantum channel with errors {Ei} and that C is cor-rectable for E via the error-correction operation R with noise operators{Rj}. Define a compressed channel by EC(ρ) ≡ E(PCρPC). Then by hy-pothesis R(EC(ρ)) = R(E(PCρPC)) = PCρPC. In particular,

∑

i,j

RjEiPCρPCE†iR

†j = PCρPC for all ρ.

Thus by Proposition 4.4 there are scalars αki such that

RkEiPC = αkiPC for all i, k.

Hence,

PCE†iR

†kRkEjPC = αkiαkjPC for all i, j, k.(4)

But R preserves traces, so that∑

k R†kRk = 1l, and thus when we sum

(4) over k we find

PCE†iEjPC = λijPC for all i, j,

QUANTUM COMPUTING 15

where λij =∑

k αkiαkj.Conversely, let us assume that E is a channel with errors {Ei} and C is

a code with projection PC such that (3) holds for a positive scalar matrixΛ = {λij}. Let U be a unitary operator such that D = U †ΛU = (dkl)is diagonal (so that D = diag(dkk)). Note that

∑

k dkk = 1 by tracepreservation of E . By Proposition 4.4 the operators Fk =

∑

i uikEi alsoimplement E . A simple computation shows that

PCF†kFlPC = dklPC for all k, l.

The polar decomposition of FkPC gives

FkPC = Uk

√

PCF†kFkPC =

√

dkUkPC

for some unitary Uk. Define projections Pk ≡ UkPCU†k . Then

PlPk = P †l Pk =

UlPCF†l FkPCU

†k√

dlldkk= 0 for k 6= l,

and hence the ranges of the Pk are mutually orthogonal.Without loss of generality assume that

∑

k Pk = 1l (otherwise justadd the projection onto the orthogonal complement and define Uk = 1l).The candidate error-correction operation is defined by

R(ρ) =∑

k

U †kPkρPkUk.

Observe that for all ρ with ρ = PCρPC ,

U †kPkFl

√ρ = U †

kP†kFl

√ρ =

U †kUkPCF

†kFlPC

√ρ√

dkk

= δkl√

dkkPCρ = δkl√

dkkρ.

Thus we have

R(E(ρ)) =∑

k,l

U †kPkFlρF

†l PkUk =

∑

k,l

δkldkkρ = ρ

for all ρ = PCρPC, and we have proved that (iii) ⇒ (i). �

Let us discuss a pair of illustrative applications of Theorem 5.5.

Examples 5.6. (i) With C = span{|000〉, |111〉} inside H8 as in Ex-ample 5.3, it is easy to see that C satisfies (3) for the errors E ={1l, X1, X2, X3}. Let P0 be the projection onto C, let P1 be the projec-tion onto the subspace X1C = span{|100〉, |011〉} and similarly defineprojections P2 and P3. Then the correction operation given by theproof above is R = {P0, X1P1, X2P2, X3P3}.

16 D.W.KRIBS

(ii) Shor’s 9-qubit code [69] is defined by two orthonormal vectorsin H29 given by

|0L〉 =(|000〉+ |111〉)(|000〉+ |111〉)(|000〉+ |111〉)

2√2

|1L〉 =(|000〉 − |111〉)(|000〉 − |111〉)(|000〉 − |111〉)

2√2

.

Let C = span{|0L〉, |1L〉}. Given 1 ≤ k ≤ 9, the code C is correctablefor the errors {Xk, Yk, Zk} as (3) is satisfied for this triple. Let Px,k bethe projection onto the subspace XkC and similarly define projectionsPy,k and Pz,k. Then the correction operation given by the theorem isR = {XkPx,k, YkPy,k, ZkPz,k}.Since the Pauli matrices, together with the identity operator 1l2, form

a linear basis for M2 that is closed under multiplication up to scalarmultiples, it follows that C is correctable for any set of errors E whichact on one of the nine possible qubits. (In fact the Shor code alsocorrects for errors on multiple qubits [61].)

Note 5.7. The quantum error correction conditions of Theorems 5.2and 5.5 were established independently by Bennett, DiVincenzo, Smolinand Wooters [9] and Knill and Laflamme [50]. As a collection of entrypoint references for particular methods of quantum error correction wemention [14, 29, 30, 45, 46, 54, 68, 71, 83].

6. Noiseless Subsystems via the Noise Commutant

In this section we describe a specific method of passive quantumerror correction, by which we mean no active intervention is requiredafter information is encoded. The basic idea in a ‘noiseless subsystemmethod’ of error correction, classical or quantum, is to encode infor-mation on subsystems of the system of interest in such a way that itremains immune to the effects of the channel that the information isevolving through. Let E : B(H) → B(H) be a quantum channel with

noise operators {Ei} and interaction algebra A = Alg{Ei, E†i }. Recall

that E is unital if E(1l) =∑

iEiE†i = 1l. The noise commutant for E is

the †-algebraA′ = {ρ ∈ B(H) : ρA = Aρ for A ∈ A}

= {ρ ∈ B(H) : [ρ, Ei] = 0 = [ρ, E†i ] for i = 1, . . . , n}.

The procedure in the noiseless subsystem via noise commutant methodof quantum error correction [23, 26, 32, 47, 48, 57, 81, 82] is touse the structure of A′ to produce noiseless subsystems (which are alsocalled ‘decoherence-free subspaces’ in special cases, see Remark 6.3).

QUANTUM COMPUTING 17

The illustrations of this method that appear in the literature allinvolve unital channels. The following discussion shows why this is thecase. Let Fix(E) = {ρ ∈ B(H) : E(ρ) = ρ}. Observe that Fix(E) isa †-closed subspace of B(H). Now consider a unital channel E . Let ρbelong to A′. Then ρEi = Eiρ for all i and hence

E(ρ) =∑

i

EiρE†i = ρE(1l) = ρ.

Thus ρ belongs to Fix(E). The converse inclusion was proved indepen-dently in [13] and [55] (see also [32] for another proof): The set Fix(E)coincides with the noise commutant in the case of a unital channel. Infact, it is not hard to show that the identity Fix(E) = A′ characterizesunital channels. Thus, by building on the proofs from [13, 32, 55] wemay state the following.

Theorem 6.1. Let E be a quantum channel. Then Fix(E) = A′ if andonly if E is unital. Moreover, in this case A is equal to the algebra A0

generated by the Ei. In other words, the operators E†i belong to A0.

Note 6.2. Notice how the algebra A = (A)′′ = Fix(E)′ is a relic of thechannel in the unital case (see the discussion after Proposition 4.4).

Remark 6.3. In the case of unital channels, Theorem 6.1 shows thatthe noiseless subsystem via noise commutant method can be presentedas follows: Let E be a unital quantum channel. Then the interactionalgebraA is generated by the Ei and A′ = Fix(E) is a finite dimensionalC∗-algebra. As such, A′ is unitarily equivalent to a unique direct sum ofampliated full matrix algebras, A′ ∼= ⊕k(1lmk

⊗Mnk). Hence, a density

operator ρ that encodes the initial states of a quantum system will beimmune to the noise of the channel as it evolves through, E(ρ) = ρ,provided that ρ is initially prepared on one of the ampliated matrixblocks 1lmk

⊗ Mnkinside the noise commutant A′ = Fix(E). The

decoherence-free subspace method may be regarded as the special caseof this method that occurs when matrix blocks 1lmk

⊗Mnkwith mk = 1

are utilized.

In the case of a general quantum channel, however, the full struc-ture of A′ cannot be used for error correction. This can be seen mostdramatically in the following simple case.

Proposition 6.4. Let E be a completely positive map such that AE ≡E(1l) is not invertible. Let PE be the projection onto the subspace HE ≡Ran(AE). Suppose that Ei = PEEiPE for all i. Then H⊥

E is non-zero

and every operator ρ ∈ B(H⊥E ) belongs to A′ and satisfies E(ρ) = 0.

18 D.W.KRIBS

Proof. The non-invertibility of AE implies H⊥E is non-zero. Let ρ ≥ 0

belong to B(H⊥E ), by which we mean ρ acts on H with ρ = P⊥

E ρP⊥E .

Then ρ trivially belongs to A′ since ρEi = 0 = Eiρ for all i. Further,E(ρ) = E(P⊥

E ρ) = 0. �

Remark 6.5. It would be interesting to know if the noise commutantcan be used to produce noiseless subsystems for classes of non-unitalchannels. For instance, a natural generalization of unital channels isthe class of channels for which the identity evolves to a multiple of aprojection. Notice that if E(1lH) = mP for some projection P , then mdivides the dimension of H by trace preservation of E . A simple exam-ple of this phenomena is given by the channel E with noise operatorsAi = |0〉〈i| for 1 ≤ i ≤ dimH ≡ d. Trace preservation of E may bereadily verified, and in this case

E(1ld) =d

∑

i=1

AiA†i =

d∑

i=1

(|0〉〈i|)(|i〉〈0|) = d|0〉〈0|.

Note 6.6. There are a number of other quantum error correctionschemes that have been investigated and some of these will be of inter-est to operator researchers (See [1, 23, 25, 43, 54, 57, 80, 81, 82]).

We finish by considering noiseless subsystems for some special casesof unital channels.

Examples 6.7. (i) Let 0 < p < 1 and let E1, E2 be operators on H2

defined on the standard basis by E1 = (√1− p)1l2 and E2 = (

√p)Z.

Then E1 and E2 are the noise operators for a unital channel E onM2 which is a variant on the bit flip channel discussed earlier. Thequantum operation corresponding to this channel is equivalent to thephase flip or phase damping operation on single qubits [61]. It is sonamed because, for instance, E(|+〉〈+|) = (1−p)|+〉〈+|+p|−〉〈−| andhence E flips the phase of |+〉〈+| and |−〉〈−| with probability p. It iseasy to see in this case that

Fix(Φ) = A′ = {E1, E2}′

=

{(

a 00 b

)

: a, b ∈ C

}

∼= C1⊕ C1,

and hence this channel has no non-trivial noiseless subsystems.(ii) Let 0 < p < 1 and let E1, E2 be operators on H4 defined on

the standard basis by E1 =√1− p(1l2 ⊗ 1l2) and E2 =

√p(Z ⊗ Z).

These noise operators determine a unital channel E on M4 which can

QUANTUM COMPUTING 19

be regarded as an ampliation of the phase flip channel. Compute

Fix(Φ) = A′ = {E1, E2}′

=

a11 0 0 a140 a22 a23 00 a32 a33 0a41 0 0 a44

: aij ∈ C

.

Thus A′ is unitarily equivalent to the direct sum A′ ∼= M2 ⊕M2 andthere is a pair of 2-dimensional noiseless subsystems.(iii) More generally, let E1 = U1, . . . , Ed = Ud be unitaries on a

Hilbert space H. If we are given scalars λi such that∑

i |λi|2 = 1,then the operators λiEi are the noise operators for a unital channelE with Fix(E) = A′ = {U1, . . . , Ud}′. In this case the automatic self-adjointness of A can be seen directly through an application of theCayley-Hamilton theorem. (The algebra generated by any unitary Ucan be seen to include U † via minimal polynomial considerations.)(iv) An important special case of such channels is the class of ‘col-

lective rotation channels’ [6, 10, 25, 26, 32, 43, 48, 75, 76, 80, 81].The n-qubit example has noise operators given by weighted expo-

nentiations of the operators Jk =∑n

m=1 J(m)k for k = x, y, z, where

J(1)k = σk ⊗ (1l2)

⊗(n−1), etc. There is an abundance of noiseless sub-systems for these channels and they can be computed directly [33] orthrough combinatorial techniques discussed below.(v) A generalization of the collective rotation class is presented in

[42] and noiseless subsystems for this class of ‘universal collective rota-tion (ucr) channels’ are computed using Young tableaux combinatorics.For each pair of positive integers d, n ≥ 2 there is a family of such chan-nels with the case d = 2 yielding the class from (iv). It is proved in [42]that every ucr-channel possesses an abundance of noiseless subsystems.In fact, it is shown that the noise commutant for every channel in thisclass (for fixed d and n) contains the algebra Aπ = Alg{π(σ) : σ ∈ Sn}where Sn is the symmetric group on n letters and π : Sn → U(dn) isthe unitary representation of Sn on Hdn given by

π(σ)(

h1 ⊗ . . .⊗ hn)

= hσ(1) ⊗ . . .⊗ hσ(n),

for all h1, . . . , hn ∈ Hd and σ ∈ Sn. In particular, the Young tableauxmachine can be used to explicitly compute the structure of Aπ andidentify noiseless subsystems for the corresponding ucr-channel. Fur-ther, a recent paper of Bacon, et al [4], has shown that the change ofbasis transformation from the standard basis to the Young tableauxbasis can be implemented efficiently with a quantum algorithm.

20 D.W.KRIBS

Acknowledgements. I am grateful to the referees for helpful comments.I would like to thank all the participants in the Quantum InformationTheory Learning Seminar organized by the author and John Holbrookat the University of Guelph during the fall of 2003. I am also gratefulfor enlightening conversations with Daniel Gottesman, John Holbrook,Peter Kim, Raymond Laflamme, Michele Mosca, Ashwin Nayak, EricPoisson, David Poulin, Martin Roetteler, Jean-Pierre Schoch, RobertSpekkens and Paolo Zanardi. Support from NSERC, the Universityof Guelph, the Institute for Quantum Computing and the PerimeterInstitute is kindly acknowledged.

References

[1] D. Aharonov, M. Ben-Or. Fault-tolerant quantum computation with constant

error, In Proc. 29th. Ann. ACM Symp. on Theory of Computing, page 176,New York, 1998, ACM. arxiv.org/quant-ph/9906129, quant-ph/9611025.

[2] A. Arias, A. Gheondea, S. Gudder, Fixed points of quantum operations, J.Math. Phys., 43 (2002), 5872-5881.

[3] W. Arveson, An invitation to C∗-algebras, Graduate Texts in Mathematics,No. 39, Springer-Verlag, New York-Heidelberg, 1976.

[4] D. Bacon, I.L. Chuang, A.W. Harrow, Efficient quantum circuits for Schur

and Clebsch-Gordon transforms, arxiv.org/quant-ph/0407082.[5] S.D. Bartlett, T. Rudolph, R.W. Spekkens, Decoherence-full subsystems and

the cryptographic power of a private shared reference fram, Phys. Rev. A 70,032307 (2004).

[6] S.D. Bartlett, T. Rudolph, R.W. Spekkens, Classical and quantum communi-

cation without a shared reference frame, Phys. Rev. Lett. 91, 027901 (2003).[7] M. Batty, S.L. Braunstein, A.J. Duncan, S. Rees, Quantum algorithms in group

theory, arxiv.org/quant-ph/0310133.[8] C.H. Bennett, P.W. Shor, J.A. Smolin, A.V. Thapliyal Entanglement-assisted

classical capacity of noisy quantum channels, Phys. Rev. Lett. 83 (1999), 3081.[9] C.H. Bennett, D.P. DiVincenzo, J.A. Smolin, W.K. Wooters, Mixed state en-

tanglement and quantum error correction, Phys. Rev. Lett. 78 (1996), 3824.[10] J.-C. Boileau, D. Gottesman, R. Laflamme, D. Poulin, R.W. Spekkens, Ro-

bust polarization-based quantum key distribution over collective-noise channel,Phys. Rev. Lett. 92, 17901 (2004).

[11] G. Brassard, P. Hoyer, M. Mosca, A. Tapp, Quantum amplitude amplification

and estimation, Quantum computation and information (Washington, DC,2000), 53-74, Contemp. Math., 305, Amer. Math. Soc., Providence, RI, 2002.

[12] M. Brooks (Ed.), Quantum computing and communications, Springer-Verlag,London, 1999.

[13] P. Busch, J. Singh, Luders theorem for unsharp quantum effects, Phys. Lett.A, 249 (1998), 10-24.

[14] J.I. Cirac, T. Pellizzari, P. Zoller, Enforcing coherent evolution in dissipative

quantum dynamics, Science, 273 (1996), 1207.[15] M.D. Choi, Completely positive linear maps on complex matrices, Lin. Alg.

Appl. 10 (1975), 285-290.

QUANTUM COMPUTING 21

[16] C. Cohen-Tannoudji, B. Diu, F. Laloe, Quantum Mechanics, Volume One &

Two, John Wiley & Sons, Toronto, 1977.[17] K.R. Davidson, C∗-algebras by example, Fields Institute Monographs, 6, Amer.

Math. Soc., Providence, 1996.[18] K.R. Davidson, D.W. Kribs, M.E. Shpigel, Isometric dilations of non-

commuting finite rank n-tuples, Can. J. Math. 53 (2001), 506-545.[19] D. Deutsch, Quantum theory, the Church-Turing principle and the universal

quantum computer, Proc. Royal Soc. of London A 400 (1985), 97-117.[20] D. Deutsch, R. Jozsa, Rapid solution of problems by quantum computation,

Proc. Royal Soc. of London A 459 (1992), 553-558.[21] I. Devetak, P. Shor, The capacity of a quantum channel for simultaneous trans-

mission of classical and quantum information, arxiv.org/quant-ph/0311131.[22] D.P. DiVincenzo, The physical implementation of quantum computation,

arxiv.org/quant-ph/0002077.[23] L.-M. Duan, G.-C. Guo, Preserving coherence in quantum computation by pair-

ing quantum bits, Phys. Rev. Lett. 79 (1997), 1953.[24] R. Feynman, Simulating physics with computers, International Journal of The-

oretical Physics 21 (1982), 467-488.[25] S. De Filippo, Quantum computation using decoherence-free states of the phys-

ical operator algebra, Phys. Rev. A 62, 052307 (2000).[26] E.M. Fortunato, L. Viola, M.A. Pravia, E. Knill, R. Laflamme, T.F. Havel,

D.G. Cory, Exploring noiseless subsystems via nuclear magnetic resonance,

Phys. Rev. A 67, 062303 (2003).[27] A. Galindo, M.A. Martin-Delgado, Information and computation: classical and

quantum aspects, Rev. Mod. Phys. 74 (2002), 347-423.[28] D. Gottesman, An introduction to quantum error correction, in Quantum Com-

putation: A Grand Mathematical Challenge for the Twenty-First Century andthe Millenium, ed. S.J. Lomonaco, Jr., 221-235 (American Mathematical So-ciety, Providence, Rhode Island, 2002), arxiv.org/quant-ph/0004072.

[29] D. Gottesman, Stabilizer codes and quantum error correction, PhD thesis, Cal-ifornia Insitute of Technology, 1997.

[30] D. Gottesman, Class of quantum error correcting codes saturating the quantum

Hamming bound, Phys. Rev. A 54 (1996), 1862.[31] L.K. Grover, A fast quantum mechanical algorithm for database search, Proc.

28th ACM Symp. Theory of Computing (1996), 212-219.[32] J.A. Holbrook, D.W. Kribs, R. Laflamme, Noiseless subsystems and the struc-

ture of the commutant in quantum error correction, Quantum Inf. Proc. 2(2004), 381-419.

[33] J.A. Holbrook, D.W. Kribs, R. Laflamme, D. Poulin, Noiseless subsystems for

collective rotation channels in quantum information theory, Integral Eqtns. &Operator Thy., to appear.

[34] A.S. Holevo, R.F. Werner, Counterexample to an additivity conjecture for out-

put purity of quantum channels, Quantum Information Theory. J. Math. Phys.,43 4353, 2002.

[35] A.S. Holevo, Coding theorems for quantum channels, Rev. Math. Phys. 53(1999), 1295-1331.

[36] A.S. Holevo, The capacity of the quantum channel with general signal states,IEEE Trans. Inform. Theory, 44 (1998), 269-273.

22 D.W.KRIBS

[37] A.S. Holevo, Problems in the mathematical theory of quantum communication

channels, Rep. Mathematical Phys., 12 (1977), 273-278.[38] A.S. Holevo, On the mathematical theory of quantum communication channels,

(Russian) Problemy Peredachi Informacii, 8 (1972), 62-71.[39] M. Horodecki, P. Shor, M.B. Ruskai, Entanglement breaking channels, Rev.

Math. Phys. 15 (2003), 629-641.[40] M. Horodecki, P. Horodecki, R. Horodecki, D. Leung, B. Terhal, Classical ca-

pacity of a noiseless quantum channel assisted by noisy entanglement, Quan-tum Information and Computation 1 (2001), 70-78.

[41] G. Johnson, A shortcut through time: The path to the quantum computer,

Alfred A. Knopf, New York, 2003.[42] M. Junge, P. Kim, D.W. Kribs, Quantum error correction and Young tableaux,

J. Math. Phys., to appear.[43] J. Kempe, D. Bacon, D.A. Lidar, K.B. Whaley, Theory of decoherence-free

fault-tolerant universal quantum computation, Phys. Rev. A 63, 042307 (2001).[44] C. King, The capacity of the quantum depolarizing channel, IEEE Trans. In-

form. Theory 49 (2003), 221-229.[45] A.Y. Kitaev, Quantum error correction with imperfect gates, in O. Hirota et

al., editor, Quantum Communication and Computing and Measurement, NewYork, 1997.

[46] A.Y. Kitaev, Quantum computations: algorithms and error correction, Russ.Math. Surv., 52 (1997), 1191-1249.

[47] E. Knill, R. Laflamme, A. Ashikhmin, H. Barnum, L. Viola, W.H. Zurek,Introduction to Quantum Error Correction, Los Alamos Science, November27, 2002.

[48] E. Knill, R. Laflamme, L. Viola, Theory of quantum error correction for general

noise, Phys. Rev. Lett. 84 (2000), 2525-2528.[49] E. Knill, R. Laflamme, W. H. Zurek, Resilient quantum computation: error

models and thresholds, Science 279 (1998), 342-345.[50] E. Knill, R. Laflamme, A theory of quantum error-correcting codes, Phys. Rev.

A 55 (1997), 900.[51] K. Kraus, States, Effects and Operations: Fundamental Notions of Quantum

Theory, Lecture Notes in Physics, vol. 190, Berlin: Springer-Verlag, 1983.[52] K. Kraus, General state changes in quantum theory, Ann. Physics 64 (1971),

311-335.[53] D. Kretschmann, R.F. Werner, Tema con variazioni: quantum channel capac-

ity, New J. Phys. 6 26 (2004).[54] D.W. Kribs, R. Laflamme, A unified operator algebra approach to quantum

error correction, in preparation, 2004.[55] D.W. Kribs, Quantum channels, wavelets, dilations, and representations of

On, Proc. Edin. Math. Soc., 46 (2003).[56] D.W. Leung, Choi’s proof and quantum process tomography, J. Math. Phys.,

44 (2003), 528-533.[57] D. A. Lidar, I. L. Chuang, K. B. Whaley, Decoherence free subspaces for quan-

tum computation, Phys. Rev. Lett. 81 (1998), 2594.[58] S. Lloyd, The capacity of a noisy quantum channel, Phys. Rev. A, 55 (1997),

1613.

QUANTUM COMPUTING 23

[59] M. Mosca, C. Zalka, Exact quantum Fourier transforms and discrete logarithm

algorithms, arxiv.org/0301093.[60] M.A. Nielsen, Rules for a complex quantum world, Scientific American, No-

vember 2002.[61] M.A. Nielsen, I.L. Chuang, Quantum computation and quantum information,

Cambridge University Press, 2000.[62] V. Paulsen, Completely bounded maps and dilations, Pitman Res. Notes Math.

146, Longman Sci. Tech. Harlow, 1986.[63] V. Paulsen, Completely bounded maps and operator algebras, Cambridge Uni-

versity Press, Cambridge, United Kingdom, 2002.[64] J. Preskill, Reliable quantum computers, Proc. R. Soc. Lond. A, 454 (1998),

385-410.[65] B. Schumacher, M.D. Westmoreland, Sending classical information via noisy

quantum channels, Phys. Rev. A, 56 (1997), 131-138.[66] P.W. Shor, Equivalence of additivity questions in quantum information theory,

Commun. Math. Phys. 246 (2004), 453-472.[67] P.W. Shor, Polynomial-time algorithms for prime factorization and discrete

logarithms on a quantum computer, SIAM Journal on Computing 26 (1997),1484-1509.

[68] P.W. Shor, Fault-tolerant quantum communication, Proc. 37th Annual Sym-posium on Fundamentals of Computer Science, IEEE Press, 1996, 56-65.

[69] P.W. Shor, Scheme for reducing decoherence in quantum computing memory,Phys. Rev. A, 52 2493, 1995.

[70] P.W. Shor, Algorithms for quantum computation: discrete logs and factoring,Proc. of the 35th Symposium on the Foundations of Computer Science (1994),124-134.

[71] A.M. Steane, Error correcting codes in quantum theory, Phys. Rev. Lett. 77(1996), 793.

[72] W.F. Stinespring, Positive functions on C∗-algebras, Proc. Amer. Math. Soc.6 (1955), 211-216.

[73] M. Takesaki, Theory of operator algebras I, Springer-Verlag, New York-Heidelberg, 1979.

[74] L. Viola, E.M. Fortunato, M.A. Pravia, E. Knill, R. Laflamme, D.G. Cory,Experimental realization of noiseless subsystems for quantum information pro-

cessing, Science 293 (2001), 2059.[75] L. Viola, E. Knill, R. Laflamme, Constructing qubits in physical systems, J.

Phys. A 34, 7067 (2001).[76] L. Viola, E.M. Fortunato, M.A. Pravia, E. Knill, R. Laflamme, D.G. Cory,

Experimental realization of noiseless subsystems for quantum information pro-

cessing, Science 293, 2059 (2001).[77] J. von Neumann, Mathematical foundations of quantum mechanics, Princeton

University Press, Princeton (1955).[78] J. Watrous, Quantum algorithms for solvable groups, Proc. of the 33rd ACM

Symposium on the Theory of Computing (2001), 60-67.[79] A. Winter, Quantum and classical message identification via quantum chan-

nels, arxiv.org/0401060.[80] P. Zanardi, S. Lloyd, Topological protection and quantum noiseless subsystems,

Phys. Rev. Lett. 90, 067902 (2003).

24 D.W.KRIBS

[81] P. Zanardi, Stabilizing quantum information, Phys. Rev. A 63, 012301 (2001).[82] P. Zanardi, M. Rasetti, Noiseless quantum codes, Phys. Rev. Lett. 79 (1997),

3306.[83] W.H. Zurek, R. Laflamme, Quantum logical operations on encoded qubits, Phys.

Rev. Lett., 77 (1996), 4683-4686.

Department of Mathematics and Statistics, University of Guelph,

Guelph, ON, CANADA N1G 2W1.

Institute for Quantum Computing, University of Waterloo, Water-

loo, ON, CANADA N2L 3G1.

Perimeter Institute for Theoretical Physics, 35 King St. North,

Waterloo, ON, CANADA N2J 2W9.

E-mail address : dkribs@uoguelph.ca