Chapter

Quantum Error Correction

AQuantum ErrorCorrecting Co de

In our study of quantum algorithms wehave found p ersuasive evidence that

a quantum computer would have extraordinary p ower But will quantum

computers really work Will weever b e able to build and op erate them

To do so wemust rise to the challenge of protecting quantum information

from errors As wehave already noted in Chapter there are several as

p ects to this challenge A quantum computer will inevitably interact with its

surroundings resulting in decoherence and hence in the decay of the quan

tum information stored in the device Unless we can successfully combat

decoherence our computer is sure to fail And even if wewere able to pre

vent decoherence by p erfectly isolating the computer from the environment

errors would still p ose grave diculties Quantum gates in contrast to clas

sical gates are unitary transformations chosen from a continuum of p ossible

values Thus quantum gates cannot b e implemented with p erfect accuracy

the eects of small imp erfections in the gates will accumulate eventually

leading to a serious failure in the computation Any eective strategem to

prevent errors in a quantum computer must protect against small unitary

errors in a quantum circuit as well as against decoherence

In this and the next chapter we will see how clever enco ding of quan

tum information can protect against errors in principle This chapter will

present the theory of quan tum errorcorrecting co des We will learn that

quantum information suitably enco ded can b e dep osited in a quantum mem

ory exp osed to the ravages of a noisy environment and recovered without

CHAPTER QUANTUM ERROR CORRECTION

damage if the noise is not to o severe Then in Chapter we will extend the

theory in two imp ortantways We will see that the recovery pro cedure can

work eectively even if o ccasional errors o ccur during recovery And we will

learn howto process enco ded information so that a quantum computation

can b e executed successfully despite the debilitating eects of decoherence

and faulty quantum gates

Aquantum errorcorrecting co de QECC can b e viewed as a mapping

k

of k qubits a Hilb ert space of dimension into n qubits a Hilb ert space

n

of dimension where n k The k qubits are the logical qubits or

enco ded qubits that we wish to protect from error The additional n k

qubits allow us to store the k logical qubits in a redundant fashion so that

the enco ded information is not easily damaged

We can b etter understand the concept of a QECC by revisiting an ex

ample that was intro duced in Chapter Shors co de with n and k

Wecancharacterize the co de by sp ecifying two basis states for the co de sub

space we will refer to these basis states as ji the logical zero and ji the

logical one They are

p

ji ji ji

p

ji jiji

each basis state is a qubit cat state rep eated three times As you will

recall from the discussion of cat states in Chapter the two basis states

can b e distinguished by the qubit observable where

x x x

i

denotes the Pauli matrix acting on the ith qubit we will use the

x

x

notation X X X for this op erator There is an implicit I I I

acting on the remaining qubits that is suppressed in this notation The

states ji and ji are eigenstates of X X X with eigenvalues and

resp ectively But there is no way to distinguish ji from ji to gather any

information ab out the value of the logical qubit by observing an y one or two

of the qubits in the blo ck of nine In this sense the logical qubit is enco ded

nonlocal ly it is written in the nature of the entanglement among the qubits

in the blo ck This nonlo cal prop erty of the enco ded information provides

protection against noise if we assume that the noise is lo cal that it acts

indep endently or nearly so on the dierent qubits in the blo ck

Supp ose that an unknown quantum state has b een prepared and enco ded

as aji bjiNow an error o ccurs we are to diagnose the error and reverse

A QUANTUM ERRORCORRECTING CODE

it Howdowe pro ceed Let us supp ose to b egin with that a single bit ip

o ccurs acting on one of the rst three qubits Then as discussed in Chapter

the lo cation of the bit ip can b e determined by measuring the twoqubit

op erators

Z Z Z Z

The logical basis states ji and ji are eigenstates of these op erators with

eigenvalue But ipping any of the three qubits changes these eigenvalues

For example if Z Z and Z Z then we infer that the rst

qubit has ipp ed relative to the other two Wemay recover from the error

by ipping that qubit back

It is crucial that our measurement to diagnose the bit ip is a collective

measurementontwo qubits at once we learn the value of Z Z butwe

must not nd out ab out the separate values of Z and Z for to do so

would damage the enco ded state How can such a collective measurement

b e p erformed In fact we can carry out collectiv e measurements if wehave

a quantum computer that can execute controlledNOT gates We rst intro

duce an additional ancilla qubit prepared in the state ji then execute the

quantum circuit

Figure

and nally measure the ancilla qubit If the qubits and are in a state

with Z Z either ji ji or ji ji then the ancilla qubit will ip

once and the measurement outcome will b e ji But if qubits and are

in a state with Z Z either ji ji or ji ji then the ancilla qubit

will ip either twice or not at all and the measurement outcome will b e ji

Similarly the twoqubit op erators

Z Z Z Z

Z Z Z Z

can b e measured to diagnose bit ip errors in the other two clusters of three

qubits

A threequbit co de would suce to protect against a single bit ip The reason the qubit clusters are rep eated three times is to protect against

CHAPTER QUANTUM ERROR CORRECTION

phase errors as well Supp ose now that a phase error

j iZ j i

o ccurs acting on one of the nine qubits We can diagnose in which cluster

the phase error o ccurred by measuring the two sixqubit observables

X X X X X X

X X X X X X

The logical basis states ji and ji are b oth eigenstates with eigenvalue one

of these observables A phase error acting on any one of the qubits in a

particular cluster will change the value of XXX in that cluster relativeto

the other two the lo cation of the change can b e identied by measuring the

observables in eq Once the aected cluster is identied we can reverse

the error by applying Z to one of the qubits in that cluster

X Notice Howdowe measure the sixqubit observable X X X X X

p

that if its control qubit is initially in the state ji ji and its target is

an eigenstate of X that is NOT then a controlledNOT acts according to

x

p p

ji ji jxi ji ji jxi CNOT

it acts trivially if the target is the X x state and it ips the

control if the target is the X x state To measure a pro duct of

X s then we execute the circuit

Figure

p

and then measure the ancilla in the jijibasis

We see that a single error acting on any one of the nine qubits in the blo ck

will cause no irrevo cable damage But if two bit ips o ccur in a single cluster

of three qubits then the enco ded information wil l b e damaged For example

if the rst two qubits in a cluster b oth ip we will misdiagnose the error and

attempt to recover by ipping the third In all the errors together with our

CRITERIA FOR QUANTUM ERROR CORRECTION

mistaken recovery attempt apply the op erator X X X to the co de blo ck

Since j i and ji are eigenstates of X X X with distinct eigenvalues the

eect of two bit ips in a single cluster is a phase error in the enco ded qubit

X X X aji bjiajibji

The enco ded information will also b e damaged if phase errors o ccur in two

dierent clusters Then wewillintro duce a phase error into the third cluster

in our misguided attempt at recovery so that altogether Z Z Z will have

b een applied which ips the enco ded qubit

aji bjiaji bji Z Z Z

If the likeliho o d of an error is small enough and if the errors acting on

distinct qubits are not strongly correlated then using the ninequbit co de

will allow us to preserve our unknown qubit more reliably than if we had not

b othered to enco de it at all Supp ose for example that the environment

acts on each of the nine qubits indep endently sub jecting it to the dep olar

izing channel describ ed in Chapter with error probability pThenabit

p and a phase ip with probability p The ip o ccurs with probability

probability that b oth o ccur is p We can see that the probabilityofaphase

error aecting the logical qubit is b ounded ab ovebyp and the probability

of a bit ip error is b ounded ab ovebyp The total error probabilityisno

worse than p this is an improvementover the error probability p for an

unprotected qubit provided that p

Of course in this analysis wehave implicitly assumed that enco ding

deco ding error syndrome measurement and recovery are all p erformed aw

lessly In Chapter we will examine the more realistic case in whicherrors

o ccur during these op erations

Criteria for Quantum Error Correction

In our discussion of error recovery using the ninequbit co de wehave assumed

that each qubit undergo es either a bitip error or a phaseip error or b oth

This is not a realistic mo del for the errors and wemust understand howto

implementquantum error correction under more general conditions

To b egin with consider a single qubit initially in a pure state that in

teracts with its environment in an arbitrary manner Weknow from Chapter

CHAPTER QUANTUM ERROR CORRECTION

that there is no loss or generalitywemay still represent the most gen

eral sup erop erator acting on our qubit if we assume that the initial state

of the environmentisapurestatewhichwe will denote as ji Then the

E

evolution of the qubit and its environment can b e describ ed by a unitary

transformation

U jiji jije i jije i

E E E

jiji jije i jije i

E E E

here the four je i are states of the environment that need not b e normalized

ij E

or mutually orthogonal though they do satisfy some constraints that follow

bji of from the unitarityof U Under U an arbitrary state j i aji

the qubit evolves as

U aji bjiji ajije i jije i

E E E

bjije i jije i

E E

aji bji je i je i

E E

je i je i ajibji

E E

aji bji je i je i

E E

ajibji je i je i

E E

I j i je i X j ije i Y j ije i

I E X E Y E

Z j ije i

Z E

The action of U can b e expanded in terms of the unitary Pauli op erators

fI X Y Z g simply b ecause these are a basis for the vector space of

matrices Heuristicallywemightinterpret this expansion bysaying that one

of four p ossible things happ ens to the qubit nothing I a bit ip X a

phase ip Z or b oth Y iXZ However this classication should not

i je i je ig of the en be taken literally b ecause unless the states fje i je

X Y Z I

vironmentareallmutually orthogonal there is no conceivable measurement

that could p erfectly distinguish among the four alternatives

CRITERIA FOR QUANTUM ERROR CORRECTION

n n

Similarly an arbitrary matrix acting on an nqubit Hilb ert space

n

can b e expanded in terms of the op erators

n

fI X Y Z g

that is each such op erator can b e expressed as a tensorpro duct string of

singlequbit op erators with each op erator in the string chosen from among

the identity and the three Pauli matrices X Y and Z Thus the action

of an arbitrary unitary op erator on n qubits plus their environmentcanbe

expanded as

X

j iji E j i je i

E a a E

a

n

here the index a ranges over values The fE g are the linearly inde

a

p endentPauli op erators acting on the n qubits and the fje i g are the

a E

corresp onding states of the environment which are not assumed to b e nor

malized or mutually orthogonal A crucial feature of this expansion for what

follows is that each E is a unitary op erator

a

Eq provides the conceptual foundation of quantum error correc

tion In devising a quantum errorcorrecting co de we identify a subset E of

all the Pauli op erators

n

EfE gfI X Y Z g

a

these are the errors that we wish to b e able to correct Our aim will b e

to p erform a collective measurementofthe n qubits in the co de blo ckthat

will enable us to diagnose which error E E o ccurred If j i is a state

a

in the co de subspace then for some but not all co des this measurement

will prepare a state E j ije i where the value of a is known from the

a a E

y

measurement outcome Since E is unitarywemay pro ceed to apply E

a

a

E to the co de blo ck thus recovering the undamaged state j i

a

EachPauli op erator can b e assigned a weightaninteger t with t n

the weight is the numb er of qubits acted on bya nontrivial Pauli matrix

X Y or Z Heuristically then wecaninterpret a term in the expansion

eq where E has weight t as an eventinwhich errors o ccur on t

a

qubits but again we cannot take this interpretation to o literally if the states

fje i g are not mutually orthogonal Typicallywe will take E to b e the set

a E

of all Pauli op erators of weight up to and including tthenifwe can recover

from any error sup erop erator with supp ort on the set E we will saythatthe

CHAPTER QUANTUM ERROR CORRECTION

co de can correct t errors In adopting such an error set we are implicitly

assuming that the errors aicting dierent qubits are only weakly correlated

with one another so that the amplitude for more than t errors on the n

qubits is relatively small

Given the set E of errors that are to b e corrected what are the necessary

and sucientconditionstobesatisedby the co de subspace in order that

recovery is p ossible Let us denote by fjiigan orthonormal basis for the

co de subspace We will refer to these basis elements as co dewords It

will clearly b e necessary that

y

hj jE E jii i j

a

b

where E E If this condition were not satised for some i j then

ab

errors would b e able to destroy the p erfect distinguishability of orthogonal

co dewords and enco ded quantum information could surely b e damaged A

more explicit derivation of this necessary condition will b e presented b elow

We can also easily see that a sucient condition is

y

hj jE E j ii

a ab ij

b

In this case the E s take the co de subspace to a set of mutually orthogonal

a

error subspaces

H E H

a a code

Supp ose then that an arbitrary state j i in the co de subspace is prepared

and sub jected to an error The resulting state of co de blo ckandenvironment

is

X

E j i je i

a a E

E E

a

where the sum is restricted to the errors in the set E Wemay then p erform

an orthogonal measurement that pro jects the co de blo ckonto one of the

spaces H so that the state b ecomes

a

E j ije i

a a E

y

We nally apply the unitary op erator E to the co de blo ck to complete the

a

recovery pro cedure

CRITERIA FOR QUANTUM ERROR CORRECTION

A co de that satises the condition eq is called a nondegenerate

co de This terminology signies that there is a measurement that can unam

biguously diagnose the error E E that o ccurred But the example of the

a

ninequbit co de has already taught us that more general co des are p ossible

The ninequbit co de is degenerate b ecause phase errors acting on dierent

qubits in the same cluster of three aect the co de subspace in precisely the

same wayeg Z j i Z j i Though no measurement can determine

which qubit suered the error this need not p ose an obstacle to successful

recovery

The necessary and sucient condition for recovery to b e p ossible is easily

stated

y

hj jE E jii C

a ba ij

b

y

E jii is an arbitrary Hermitian matrix The where E EandC hijE

a ab ba

b

eaker necessary nontrivial content of this condition that go es b eyond the w

y

condition eq is that hijE E jii do es not dep end on i The origin of

a

b

this condition is readily understo o d were it otherwise in identifying an

error subspace H wewould acquire some information ab out the enco ded

a

state and so would inevitably disturb that state

Toprove that the condition eq is necessary and sucient we

invoke the theory of sup erop erators develop ed in Chapter Errors acting

on the co de blo ck are describ ed by a sup erop erator and the issue is whether

another sup erop erator the recovery pro cedure can b e constructed that will

reverse the eect of the error In fact we learned in Chapter that the only

sup erop erators that can b e inverted are unitary op erators But nowwe are

demanding a bit less We are not required to b e able to reverse the action of

the error sup erop erator on any state in the nqubit co de blo ck rather it is

enough to b e able to reverse the errors when the initial state resides in the

k qubit enco ded subspace

An alternativeway to express the action of an error on one of the co de

basis states jii and the environment is

X

jiiji M jiiji

E E

w the states ji are elements of an orthonormal basis for the envi where no

E

ronment and the matrices M are linear combinations of the Pauli op erators

CHAPTER QUANTUM ERROR CORRECTION

E contained in E satisfying the op eratorsum normalization condition

a

X

y

M I M

The error can b e reversed by a recovery sup erop erator if there exist op erators

R such that

X

y

R R I

and

X

R M jiiji j i

E A

jiijstu i

EA

here the j i s are elements of an orthonormal basis for the Hilb ert space of

A

the ancil la that is employed to implementtherecovery op eration and the

state jstu i of environment and ancilla must not dep end on iItfollows

EA

that

jii jii R M

for each and the pro duct R M acting on the co de subspace is a multiple

of the identity Using the normalization condition satised bytheR s we

infer that

X X

y y

y

M jii jii M M jii M R R

y

M is likewise a multiple of the identity acting on the co de so that M

subspace In other words

y

hj jM M j ii C

ij

since each E in E is a linear combination of M s eq then follows

a

Another instructiveway to understand why eq is a necessary con

dition for error recovery is to note that if the co de blo ck is prepared in the

state j i and an error acts according to eq then the density matrix

for the environment that we obtain by tracing over the co de blo ckis

X

y

ji h jM M j i h j

E E E

CRITERIA FOR QUANTUM ERROR CORRECTION

Error recovery can pro ceed successfully only if there is no way to acquire

any information ab out the state j i by p erforming a measurementonthe

environment Therefore we require that b e indep endentof j iifj i is

E

any state in the co de subspace eq then follows

To see that eq is sucient for recovery as well as necessarywe

can explicitly construct the sup erop erator that reverses the error For this

purp ose it is convenienttocho ose our basis fji g for the environmentso

E

that the matrix C in eq is diagonalized

y

hj jM M jii C

ij

P

C follows from the op eratorsum normalization condition where

For each with C let

X

y

p

jiihi jM R

C

i

so that R acts according to

q

R M jii C jii

Then we easily see that

X

R M jiiji j i

E A

q

X

jii C j i j i

E A

the sup erop erator dened bytheR s do es indeed reverse the error It only

remains to checkthatthe R s satisfy the normalization condition Wehave

X X X

y y

M jiihijM R R

C

i

which is the orthogonal pro jection onto the space of states that can b e reached

by errors acting on co dewords Thus we can complete the sp ecication of

the recovery sup erop erator by adding one more element to the op erator sum

the pro jection onto the complementary subspace

In brief eq is a sucient condition for error recovery b ecause it is

p ossible to cho ose a basis for the error op erators not necessarily the Pauli

CHAPTER QUANTUM ERROR CORRECTION

op erator basis that diagonalizes the matrix C and in this basis wecan

ab

unambiguously diagnose the error by p erforming a suitable orthogonal mea

surement The eigenmo des of C with eigenvalue zero like Z Z in the

ab

case of the qubit co de corresp ond to errors that o ccur with probability

zero We see that once the set E of p ossible errors is sp ecied the recov

ery op eration is determined In particular no information is needed ab out

the states je i of the environment that are asso ciated with the errors E

a E a

Therefore the co de works equally eectively to control unitary errors or de

coherence errors as long as the amplitude for errors outside of the set E is

negligible Of course in the case of a nondegenerate co de C is already

ab

diagonal in the Pauli basis and we can express the recovery basis as

X

y

R jiihijE

a

a

i

there is an R corresp onding to each E in E

a a

Wehave describ ed error correction as a two step pro cedure rst a col

lective measurement is conducted to diagnose the error and secondly based

erse on the measurement outcome a unitary transformation is applied to rev

the error This p oint of view has many virtues In particular it is the quan

tum measurement pro cedure that seems to enable us to tame a continuum of

p ossible errors as the measurement pro jects the damaged state into one of a

discrete set of outcomes for eachofwhich there is a prescription for recov

ery But in fact measurement is not an essential ingredient of quantum error

correction The recovery sup erop erator of eq may of course b e viewed

as a unitary transformation acting on the co de blo ck and an ancilla This

sup erop erator can describ e a measurement followed by a unitary op erator if

we imagine that the ancilla is sub jected to an orthogonal measurement but

the measurement is not necessary

If there is no measurement we are led to a dierent p ersp ective on the

reversal of decoherence achieved in the recovery step When the co de blo ck

interacts with its environment it b ecomes entangled with the environment

and the Von Neumann entropyoftheenvironment increases as do es the

entropy of the co de blo ck If we are unable to control the environment that

increase in its entropy can never b e reversed how then is quantum error

correction p ossible The answer provided by eq is that wemay apply

a unitary transformation to the data and to an ancilla that we do control

If the criteria for quantum error correction are satised this unitary can b e

chosen to transform the entanglement of the data with the environmentinto

SOME GENERAL PROPERTIES OF QECCS

entanglement of ancilla with environment restoring the purity of the data in

the pro cess as in

Figure

While measurement is not a necessary part of error correction the ancilla

is absolutely essential The ancilla serves as a dep ository for the entropyin

serted into the co de blo ckby the errors it heats as the data co ols If

we are to continue to protect quantum information stored in quantum mem

ory for a long time a continuous supply of ancilla qubits should b e provided

that can b e discarded after use Alternatively if the ancilla is to b e recycled

it must rst b e erased As discussed in Chapter the erasure is dissipative

and requires the exp enditure of p ower Thus principles of thermo dynamics

dictate that we cannot implement quantum error correction for free Errors

cause entropy to seep into the data This entropy can b e transferred to the

ancilla by means of a reversible pro cess but work is needed to pump entropy

from the ancilla back to the environment

Some General Prop erties of QECCs

Distance

A quantum co de is said to b e binary if it can b e represented in terms of

k

qubits In a binary co de a co de subspace of dimension is emb edded in a

n

space of dimension where k and nk are integers There is actually no

need to require that the dimensions of these spaces b e p owers of two see the

exercises nevertheless we will mostly conne our attention here to binary

co ding which is the simplest case

umb er of enco ded qubits k In addition to the blo ck size n and the n

another imp ortant parameter characterizing a co de is its distance d The

distance d is the minimum weightofaPauli op erator E such that

hijE jj i C

a a ij

We will describ e a quantum co de with blo cksizen k enco ded qubits and

distance d as an n k d quantum co de We use the doublebracket no

CHAPTER QUANTUM ERROR CORRECTION

tation for quantum co des to distinguish from the n k d notation used for

classical co des

Wesay that an QECC can correct t errors if the set E of E s that allow

a

recovery includes all Pauli op erators of weigh t or less Our denition of

distance implies that the criterion for error correction

y

hijE E jj i C

b ab ij

a

will b e satised byallPauli op erators E of weight t or less provided that

ab

d t Therefore a QECC with distance d t can correct t errors

Lo cated errors

A distance d t co de can correct t errors irresp ective of the lo cation of

the errors in the co de blo ck But in some cases wemayknow that particular

qubits are esp ecially likely to have suered errors Perhaps wesaw a hammer

strike those qubits Or p erhaps you sentablockofn qubits to me but t n

of the qubits were lost and never received I am condent that the n t

qubits that did arrivewere well packaged and were received undamaged But

I replace the t missing qubits with the arbitrarily chosen state j i

realizing full well that these qubits are likely to b e in error

A given co de can protect against more errors if the errors o ccur at known

lo cations instead of unknown lo cations In fact a QECC with distance d

t can correct t errors at known lo cations In this case the set E of errors

to b e corrected is the set of all Pauli op erators with support at the t sp ecied

lo cations each E acts trivially on the other n t qubits But then for each

a

y

E also has weight at most t Therefore E and E in E the pro duct E

b a b

a

the error correction criterion is satised for all E E provided the co de

ab

has distance at least t

In particular a QECC that corrects t errors in arbitrary lo cations can

correct t errors in known lo cations

Error detection

In some cases wemay b e satised to detect whether an error has o ccurred

en if we are unable to fully diagnose or reverse the error A measurement ev

designed for error detection has two p ossible outcomes go o d and bad

SOME GENERAL PROPERTIES OF QECCS

If the go o d outcome o ccurs we are assured that the quantum state is un

damaged If the bad outcome o ccurs damage has b een sustained and the

state should b e discarded

If the error sup erop erator has its supp ort on the set E of all Pauli op

erators of weightupto t and it is p ossible to make a measurementthat

correctly diagnoses whether an error has o ccurred then it is said that wecan

detect t errors Error detection is easier than error correction so a given co de

can detect more errors than it can correct In fact a QECC with distance

d t can detect t errors

Such a co de has the prop ertythat

hijE jj i C

a a ij

for every Pauli op erator E of weight t or less or

a

i E jii C jii j

a a

ai

where j i is an unnormalized vector orthogonal to the co de subspace

ai

Therefore the action on a state j i in the co de subspace of an error su

is p erop erator with supp ort on E

X X

A

j i ji E j ije i j i C je i jorthog i

E a a E a a E

E E E E

a a

where jorthog i denotes a vector orthogonal to the co de subspace

Nowwe can p erform a fuzzy orthogonal measurement on the data with

two outcomes the state is pro jected onto either the co de subspace or the

complementary subspace If the rst outcome is obtained the undamaged

state j i is recovered If the second outcome is found an error has b een

detected We conclude that our QECC with distance d can detect d

errors In particular then a QECC that can correct t errors can detect t

errors

Quantum co des and entanglement

A QECC protects quantum information from error by enco ding it nonlo

cal ly that is by sharing it among many qubits in a blo ck Thus a quantum

co deword is a highly entangled state

CHAPTER QUANTUM ERROR CORRECTION

In fact a distance d t nondegenerate co de has the following prop erty

Cho ose any state j i in the co de subspace and any t qubits in the blo ck

Trace over the remaining n t qubits to obtain

t

tr j ih j

nt

the density matrix of the t qubits Then this density matrix is totally random

t

I

t

In any distancet co de we cannot acquire any information ab out the

t

enco ded data by observing any t qubits in the blo ck that is is a constant

indep endent of the co deword But only if the co de is nondegenerate will the

density matrix of the t qubitsbeamultiple of the identity

Toverify the prop erty eq we note that for a nondegenerate distance

t code

hijE jj i

a

for any E of nonzero weightupto t so that

a

t

tr E

a

t

for any tqubit Pauli op erator E other than the identityNow likeany

a

t t

Hermitian matrix can b e expanded in terms of Pauli op erators

X

t

I E

a a

t

E I

a

Since the E s satisfy

a

trE E

a b ab

t

t

we nd that each and we conclude that isamultiple of the

a

identity

PROBABILITY OF FAILURE

ProbabilityofFailure

Fidelity b ound

If the supp ort of the error sup erop erator contains only the Pauli op erators

in the set E that we knowhow to correct then we can recover the enco ded

quantum information with p erfect delity But in a realistic error mo del

there will b e a small but nonzero amplitude for errors that are not in E so

that the recovered state will not b e p erfect What can wesay ab out the

delityoftherecovered state

The Pauli op erator expansion of the error sup erop erator can b e divided

intoasumover the go o d op erators those in E and the bad ones those

not in E so that it acts on a state j i in the co de subspace according to

X

j iji E j i je i

E a a E

a

X X

i E j ije i E j i je

b E a a E b

E E E E

a

b

jGOODi jBADi

The recovery op eration a unitary acting on the data and the ancilla then

maps jGOODi to a state jGOOD i of data environment and ancilla and

jBADi to a state jBAD i so that after recovery we obtain the state

jGOOD i jBAD i

here since recovery works p erfectly acting on the go o d state

jGOOD i j ijsi

EA

where jsi is some state of the environment and ancilla

EA

Supp ose that the states jGOODi and jBADi are orthogonal This would

hold if in particular all of the go o d states of the environment are orthog

onal to all of the bad states that is if

he je i for E E E E

a b a b

Let denote the density matrix of the recovered state obtained bytracing

rec

out the environment and ancilla and let

F h j j i rec

CHAPTER QUANTUM ERROR CORRECTION

b e its delityNow since jBAD i is orthogonal to jGOOD i that is jBAD i

has no comp onent along j ijsi the delity will b e

EA

F h j j i h j j i

GOOD BAD

where

tr jGOOD ihGOOD j tr jBAD ihBAD j

EA EA

GOOD BAD

The delity of the recovered state therefore satises

kjsi k kjGOOD ik F h j j i

EA

GOOD

Furthermore since the recovery op eration is unitarywehave kjGOOD ik

kjGOODik and hence

X

F kjGOOD ik k E j ije i k

a a E

E E

a

In general though jBADi need not b e orthogonal to jGOOD i so that

jBAD i need not b e orthogonal to jGOOD i Then jBAD i mighthavea

comp onent along jGOOD i that interferes destructively with jGOOD i and

e can still obtain a lower b ound on the delityin so reduces the delityW

this more general case by resolving jBAD i into a comp onent along jGOOD i

and an orthogonal comp onent as

jBAD i jBAD i jBAD i

k

Then reasoning just as ab ovewe obtain

ik F kjGOOD i jBAD

k

Of course since b oth the error op eration and the recovery op eration are uni

tary acting on data environment and ancilla the complete state jGOOD i

jBAD i is normalized or

kjGOOD i jBAD ik kjBAD ik

k

and eq b ecomes

ik F kjBAD

PROBABILITY OF FAILURE

Finally the norm of jBAD i cannot exceed the norm of jBAD iandwe

conclude that

X

F kjBAD ik kjBADik k E j ije i k

b b E

E E

b

This is our general b ound on the failure probability of the recovery op er

ation The result eq then follows in the sp ecial case where jGOODi

and jBADi are orthogonal states

Uncorrelated errors

Lets now consider some implications of these results for the case where errors

acting on distinct qubits are completely uncorrelated In that case the error

sup erop erator is a tensor pro duct of singlequbit sup erop erators If in fact

the errors act on all the qubits in the same waywe can express the nqubit

sup erop erator as

h i

n

n

error error

where is a onequbit sup erop erator whose action in its unitary repre

error

sen tation has the form

j iji j ije i X j i je i Y j ije i

E I E X E Y E

Z j ije i

Z E

The eect of the errors on enco ded information is esp ecially easy to analyze

if we supp ose further that each of the three states of the environment je i

XYZ

is orthogonal to the state je i In that case a record of whether or not an

I

error o ccurred for each qubit is p ermanently imprinted on the environment

and it is sensible to sp eak of a probability of error p for each qubit where

error

he je i p

I I error

If our quantum co de can correct t errors then the go o d Pauli op erators

t greater than haveweightupto t and the bad Pauli op erators haveweigh

t recovery is certain to succeed unless at least t qubits are sub jected to

errors It follows that the delity ob eys the b ound

n

X

n n

ns

t s

p p p F

error

error error

t s

st

CHAPTER QUANTUM ERROR CORRECTION

n

For each of the ways of cho osing t lo cations the probabilitythat

t

t

errors o ccurs at every one of those lo cations is p where we disregard

error

whether additional errors o ccur at the remaining n t lo cations There

fore the nal expression in eq is an upp er b ound on the probability

that at least t errors o ccur in the blo ckof n qubits For p small and t

error

large the delity of the enco ded data is a substantial improvementover the

delity F O pmaintained by an unprotected qubit

For a general error sup erop erator acting on a single qubit there is no clear

notion of an error probability the state of the qubit and its environment

obtained when the Pauli op erator I acts is not orthogonal to and so cannot

b e p erfectly distinguished from the state obtained when the Pauli op erators

X Y and Z act In the extreme case there is no decoherence at all the

errors arise b ecause unknown unitary transformations act on the qubits

If the unitary transformation U acting on a qubit were known we could

y

recover from the error simply by applying U

qubits in the co de Consider uncorrelated unitary errors acting on the n

blo ck each of the form up to an irrelevant phase

q

p

p i p W U

where W is a traceless Hermitian linear combination of X Y and Z

satisfying W I If the state j i of the qubit is prepared and then the

unitary error eq o ccurs the delity of the resulting state is

F h jU j i p h jW j i p

If a unitary error of the form eq acts on eachofthen qubits in the

co de blo ck and the resulting state is expanded in terms of Pauli op erators

as in eq then the state jBADi which arises from terms in which W

p

t

acts on at least t qubits has a norm of order p and eq

b ecomes

t

F O p

We see that co ding provides an improvement in delity of the same order

irresp ective of whether the uncorrelated errors are due to decoherence or due

to unknown unitary transformations

CLASSICAL LINEAR CODES

Toavoid confusion let us emphasize the meaning of uncorrelated for

the purp ose of the ab ove discussion We consider a unitary error acting on

n qubits to b e uncorrelated if it is a tensor pro duct of singlequbit unitary

transformations irresp ectiveofhow the unitaries acting on distinct qubits

might b e related to one another For example an error whereby all qubits

rotate byanangle ab out a common axis is eectively dealt with by quantum

t

error correction after recovery the delitywillbe F O if the

co de can protect against t uncorrelated errors In contrast a unitary error

n

n

that would cause more trouble is one of the form U i E where

bad

n

E is an nqubit Pauli op erator whose weight is greater than t Then

bad

jBADi has a norm of order and the typical delity after recovery will b e

F O

Classical Linear Co des

Quantum errorcorrecting co des were rst invented less than four y ears ago

but classical errorcorrecting co des havea much longer historyOver the past

ftyyears a remarkably b eautiful and p owerful theory of classical co ding has

b een erected Much of this theory can b e exploited in the construction of

QECCs Here we will quickly review just a few elements of the classical

theory conning our attention to binary linear co des

In a binary co de k bits are enco ded in a binary string of length nThat

n

is from among the strings of length nwe designate a subset containing

k

strings the co dewords A k bit message is enco ded by selecting one of

k

these co dewords

In the sp ecial case of a binary linear co de the co dewords form a k

n

dimensional closed linear subspace C of the binary vector space F That is

the bitwise XOR of twocodewords is another co deword The space C of the

co de is spanned by a basis of k vectors v v v an arbitrary co deword

k

may b e expressed as a linear combination of these basis vectors

X

v v

i i k

i

h f g and addition is mo dulo Wemaysaythatthe where eac

i

lengthn vector v enco des the k bit message

k k

CHAPTER QUANTUM ERROR CORRECTION

The k basis vectors v v may b e assembled into a k n matrix

k

v

C B

C B

G

A

v

k

called the generator matrix of the co de Then in matrix notation eq

can b e rewritten as

v G

the matrix G acting to the left enco des the message

An alternativewaytocharacterize the k dimensional co de subspace of

n

F is to sp ecify n k linear constraints There is an n k n matrix H

suchthat

Hv

for all those and only those vectors v in the co de C This matrix H is called

H are n k linearly the paritycheck matrix of the co de C The rows of

indep endentvectors and the co de space is the space of vectors that are

orthogonal to all of these vectors Orthogonality is dened with resp ect to

the mo d bitwise inner pro duct two lengthn binary strings are orthogonal

is they collide b oth take the value at an even numb er of lo cations Note

that

T

HG

T

where G is the transp ose of Gtherows of G are orthogonal to the rows of

H

For a classical bit the only kind of error is a bit ip An error o ccurring

in an nbit string can b e characterized byan ncomp onentvector ewhere

the s in e mark the lo cations where errors o ccur When aicted by the

error e the string v b ecomes

v v e

Errors can b e detected by applying the paritycheck matrix If v is a co de

word then

H v eHv He He

CLASSICAL LINEAR CODES

He is called the syndrome of the error e Denote by E the set of errors

fe g that we wish to b e able to correct Error recovery will b e p ossible if

i

and only if all errors e have distinct syndromes If this is the case wecan

i

unambiguously diagnose the error given the syndrome Heandwemaythen

recover by ipping the bits sp ecied by e as in

v e v ee v

On the other hand if He He for e e then wemaymisinterpret an

e error as an e error our attempt at recovery then has the eect

v e v e e v

The recovered message v e e lies in the co de but it diers from the

intended message v the enco ded information has b een damaged

The distance d of a co de C is the minimum weightofanyvector v C

where the weight is the numb er of s in the string v A linear co de with

distance d t can correct t errors the co de assigns a distinct syndrome

to each e E where E contains all vectorsofweight t or less This is so

b ecause if He He then

He He H e e

and therefore e e C Butife and e are unequal and each has weight

no larger than t then the weightofe e is greater than zero and no larger

than t Since d t there is no suchvector in C Hence He and He

cannot b e equal

A useful concept in classical co ding theory is that of the dual codeWe

have seen that the k n generator matrix G and the n k n paritycheck

T

matrix H of a co de C are related by HG Taking the transp ose it

T T

follows that GH Thus wemay regard H as the generator and G as

the paritycheckofann k dimensional co de which is denoted C and

called the dual of C In other words C is the orthogonal complementof

n

C in F Avector is selforthogonal if it has even weight so it is p ossible

for C and C to intersect A co de contains its dual if all of its co dewords

haveeven weightandaremutually orthogonal If n k it is p ossible that

C C inwhichcaseC is said to b e selfdual

An identity relating the co de C and its dual C will prove useful in the

CHAPTER QUANTUM ERROR CORRECTION

following section

k

u C

X

v u

u C

v C

The nontrivial content of the identity is the statement that the sum vanishes

for u C This readily follows from the familiar identity

X

v w

w

k

v fg

where v and w are strings of length k We can express v G as

v G

where is a k vector Then

X X

u Gu v

k

v C

fg

for Gu Since G the generator matrix of C is the paritycheck matrix

for C we conclude that the sum vanishes for u C

CSS Co des

Principles from the theory of classical linear co des can b e adapted to the

construction of quantum errorcorrecting co des We will describ e here a

family of QECCs the CalderbankShorSteane or CSS co des that exploit

the concept of a dual co de

Let C b e a classical linear co de with n k n paritycheck matrix H

and let C be a subcode of C withn k n paritycheck H where k k

The rst n k rows of H coincide with those of H but there are k k

additional linearly indep endentrows thus eachword in C is contained in

C but the words in C also ob ey some additional linear constraints

The sub co de C denes an equivalence relation in C wesay that u v

C are equivalentu v if and only if there is a w in C suchthat u v w

The equivalence classes are the cosets of C in C

CSS CODES

A CSS co de is a k k k quantum co de that asso ciates a co deword

with each equivalence class Each element of a basis for the co de subspace

can b e expressed as

X

p

jv w i jw i

k

v C

an equally weighted sup erp osition of all the words in the coset represented by

k k k k

w There are cosets and hence linearly indep endentcodewords

The states jw i are evidently normalized and mutually orthogonal that is

hw jw i if w and w b elong to dierent cosets

Now consider what happ ens to the co deword jw i if we apply the bitwise

n

Hadamard transform H

X

n

p

jv w i H jw i

F

k

v C

X X

uv uw

p p

jw i jui

P

n

k

u

v C

X

uw

p

jui

nk

uC

we obtain a coherent sup erp osition weighted by phases of words in the dual

co de C in the last step wehave used the identity eq It is again

manifest in this last expression that the co deword dep ends only on the C

coset that w represents shifting w byanelementof C has no eect on

uw

if u is in the co de dual to C

has Now supp ose that the co de C has distance d and the co de C

distance d such that

d t

F

d t

P

Then we can see that the corresp onding CSS co de can correct t bit ips

F

ip

and t phase ips If e is a binary string of length nletE denote the

P

e

Pauli op erator with an X acting at each lo cation i where e itactson

i

the state jv i according to

ip

jv i jv ei E e

CHAPTER QUANTUM ERROR CORRECTION

phase

And let E denote the Pauli op erator with a Z acting where e its

i

e

action is

phase ve

E jv i jv i

e

which in the Hadamard rotated basis b ecomes

phase

E juiju ei

e

Now in the original basis the F or ip basis each basis state jw i of

F

the CSS co de is a sup erp osition of words in the co de C To diagnose bit ip

error we p erform on data and ancilla the unitary transformation

jv iji jv ijH v i

A A

and then measure the ancilla The measurement result H e is the bit ip

F

er wemay correctly infer from syndromeIfthenumb er of ips is t or few

F

this syndrome that bit ips have o ccurred at the lo cations lab eled by e We

F

recover by applying X to the qubits at those lo cations

To correct phase errors we rst p erform the bitwise Hadamard transfor

mation to rotate from the F basis to the P phase basis In the P basis

each basis state jw i of the CSS co de is a sup erp osition of words in the co de

P

To diagnose phase errors we p erform a unitary transformation C

jv iji jv ijG v i

A A

and measure the ancilla G the generator matrix of C is also the parity

checkmatrixof C The measurement result G e is the phase error syn

P

drome If the numb er of phase errors is t or fewer wemay correctly infer

P

from this syndrome that phase errors have o ccurred at lo cations lab eled by

e We reco ver by applying X in the P basis to the qubits at those lo

P

cations Finallywe apply the bitwise Hadamard transformation once more

to rotate the co dewords back to the original basis Equivalentlywemay

recover from the phase errors by applying Z to the aected qubits after the

rotation backtotheF basis

then If e has weight less than d and e has weight less than d

F P

phase ip

hw jE E jw i

e e

P F

unless e e AnyPauli op erator can b e expressed as a pro duct of

F P

a phase op erator and a ip op erator a Y error is merely a bit ip and

THE QUBIT CODE

phase error b oth aicting the same qubit So the distance d of a CSS co de

satises

d mind d

CSS co des have the sp ecial prop erty not shared by more general QECCs

that the recovery pro cedure can b e divided into two separate op erations one

to correct the bit ips and the other to correct the phase errors

The unitary transformation eq or eq can b e implemented

by executing a simple quantum circuit Asso ciated with eachofthe n k

rows of the paritycheck matrix H is a bit of the syndrome to b e extracted

To nd the ath bit of the syndrome we prepare an ancilla bit in the state

ji and for eachvalue of with H we execute a controlledNOT

Aa a

gate with the ancilla bit as the target and qubit in the data blo ckasthe

control When measured the ancilla qubit reveals the value of the parity

P

checkbit H v

a

Schematically the full error correction circuit for a CSS co de has the

form

Figure

Separate syndromes are measured to diagnose the bit ip errors and the phase

errors An imp ortant sp ecial case of the CSS construction arises when a co de

C contains its dual C Thenwemaycho ose C C and C C C the

C paritycheck is computed in b oth the F basis and the P basis to determine

the two syndromes

The Qubit Co de

The simplest of the CSS co des is the n k d quantum co de rst

formulated by Andrew Steane It is constructed from the classical bit

Hamming co de

The Hamming co de is an n k d classical co de with the

CHAPTER QUANTUM ERROR CORRECTION

paritycheck matrix

B C

H

A

To see that the distance of the co de is d rst note that the weight

string passes the paritycheck and is therefore in the co de Now

we need to show that there are no vectors of weight or in the co de If e

has weightthen He is one of the columns of H But no column of H is

trivial all zeros so e cannot b e in the co de Anyvector of weight can b e

expressed as e e where e and e are distinct vectors of weight But

He H e e He

b ecause all columns of H are distinct Therefore e e cannot b e in the

co de

The rows of H themselves pass the paritycheck and so are also in the

co de Contrary to ones usual linear algebra intuition a nonzero vector over

the nite eld F can b e orthogonal to itself The generator matrix G of

the Hamming co de can b e written as

B C

B C

G B C

A

the rst three rows coincide with the rows of H andtheweight co deword

is app ended as the fourth row

The dual of the Hamming co de is the co de generated by H In

this case the dual of the co de is actually contained in the co de in fact it

is the even subcode of the Hamming co de containing all those and only those

Hamming co dewords that haveeven weight The o dd co deword

is a representative of the nontrivial coset of the even sub co de For the CSS

construction wewillcho ose C to b e the Hamming co de and C to b e its

dual the even sub co de Therefore C C is again the Hamming co de

we will use the Hamming paritycheck b oth to detect bit ips in the F basis

and to detect phase ips in the P basis

THE QUBIT CODE

In the F basis the two orthonormal co dewords of this CSS co de each

asso ciated with a distinct coset of the even sub co de can b e expressed as

X

p

ji jv i

F

even v

 Hamming

X

p

ji jv i

F

odd v

 Hamming

Since b oth j i and ji are sup erp ositions of Hamming co dewords bit ips

can b e diagnosed in this basis by p erforming an H paritycheck In the

Hadamard rotated basis these co dewords b ecome

X

p

ji ji jv i H ji ji

F F F P

v Hamming

X

wtv

p

jv i ji ji ji ji

F F F P

v Hamming

In this basis as well the states are sup erp ositions of Hamming co dewords

so that bit ips in the P basis phase ips in the original basis can again

b e diagnosed with an H paritycheck We note in passing that for this

co de p erforming the bitwise Hadamard transformation also implements a

Hadamard rotation on the enco ded data a p oint that will b e relevanttoour

discussion of faulttolerantquantum computation in the next chapter

Steanes quantum co de can correct a single bit ip and a single phase

ip on any one of the seven qubits in the blo ck But recovery will fail if

two dierent qubits b oth undergo either bit ips or phase ips If e and e

are two distinct weightone strings then He He is a sum of two distinct

columns of H and hence a third column of H all seven of the nontrivial

strings of length app ear as columns of H Therefore there is another

weightone string e such that He He He or

H e e e

thus e e e is a weight word in the Hamming co de We will interpret

the syndrome He as an indication that the error v v e has arisen and

we will attempt to recover by applying the op eration v v e Altogether

CHAPTER QUANTUM ERROR CORRECTION

then the eect of the two bit ip errors and our faulty attempt at recovery

will b e to add e e e an o ddweight Hamming co deword to the data

which will induce a ip of the encoded qubit

j i ji

F F

Similarlytwo phase ips in the F basis are two bit ips in the P basis which

after the b otched recovery induce on the enco ded qubit

j i ji

P P

or equivalently

ji j i

F F

ji ji

F F

a phase ip of the enco ded qubit in the F basis If there is one bit ip and

one phase ip either on the same qubit or dierent qubits then recovery

will b e successful

Some Constraints on Co de Parameters

Shors co de protects one enco ded qubit from an error in any single one of

nine qubits in a blo ck and Steanes co de reduces the blo ck size from nine to

seven Can we do b etter still

The Quantum Hamming b ound

To understand howmuch b etter wemight do lets see if we can deriveany

b ounds on the distance d t of an n k d quantum co de for given

n and k At rst supp ose we limit our attention to nondegenerate co des

which assign a distinct syndrome to each p ossible error On a given qubit

there are three p ossible linearly indep endent errors X Y or Z Inablock

n

of n qubits there are ways to cho ose j qubits that are aected by errors

j

and three p ossible errors for each of these qubits therefore the total number

of p ossible errors of weightupto t is

t

X

n

j

N t

j

j

SOME CONSTRAINTS ON CODE PARAMETERS

k

If there are k enco ded qubits then there are linearly indep endent

co dewords If all E jj is are linearly indep endent where E is any error

a a

of weightupto t and jii is any element of a basis for the co dewords then

n

the dimension of the Hilb ert space of n qubits must b e large enough to

k

accommo date N t indep endentvectors hence

t

X

n

j nk

N t

j

j

This result is called the quantum Hamming b ound An analogous b ound

j

applies to classical blo ck co des but without the factor of since there is

only one typ e of error a ip that can aect a classical bit We also emphasize

that the quantum Hamming b ound applies only in the case of nondegenerate

co ding while the classical Hamming b ound applies in general However no

degenerate quantum co des that violate the quantum Hamming co de haveyet

b een constructed as of January

Inthespecialcaseofacodewithoneencodedqubitk that corrects

one error t the quantum Hamming b ound b ecomes

n

n

which is satised for n In fact the case n saturates the inequality

A nondegenerate quantum co de if it exists is

perfect The entire dimensional Hilb ert space of the ve qubits is needed

to accommo date all p ossible onequbit errors acting on all co dewords there

is no wasted space

The nocloning b ound

We could still wonder though if there is a degenerate n co de that can

correct one error In fact it is easy to see that no such co de can exist We

already know that a co de that corrects t errors at arbitrary lo cations can

also b e used to correct t errors at known lo cations Supp ose that wehave

a quantum co de Then we could enco de a single qubit in the four

qubit blo ck and split the blo ckinto two subblo cks each containing two

qubits Figure

CHAPTER QUANTUM ERROR CORRECTION

If we app end ji to each of those two subblo cks then the original blo ck

has spawned two ospring eachwithtwo lo cated errors If wewere able to

correct the two lo cated errors in each of the ospring wewould obtain two

identical copies of the parentblockwewould have cloned an unknown

quantum state which is imp ossible Therefore no quantum co de

can exist We conclude that n is the minimal blo ck size of a quantum

co de that corrects one error whether the co de is degenerate or not

The same reasoning shows that an n k d co de can exist only for

nd

The quantum Singleton b ound

We will now see that this result eq can b e strengthened to

n k d

Eq resembles the Singleton b ound on classical co de parameters

n k d

linear and so has b een called the quantum Singleton b ound For a classical

co de the Singleton b ound is a near triviality the co de can have distance d

only if any d columns of the paritycheck matrix are linearly indep endent

Since the columns have length n k atmostn k columns can b e linearly

indep endent therefore d cannot exceed n k The Singleton b ound also

applies to nonlinear co des

An elegant pro of of the quantum Singleton b ound can b e found that

exploits the subadditivity of the Von Neumann entropy discussed in x

We b egin byintro ducing a k qubit ancilla and constructing a pure state

k

that maximally entangles the ancilla with the co dewords of the QECC

X

p

ji jxi jxi

AQ A Q

k

k

where fjxi g denotes an orthonormal basis for the dimensional Hilb ert

A

k

space of the ancilla and fjx i g denotes an orthonormal basis for the

Q

dimensional co de subspace If we trace over the lengthn co de blo ck Q the

whichhasentropy density matrix of the ancilla is

A

k

S Ak S Q

SOME CONSTRAINTS ON CODE PARAMETERS

Now if the co de has distance dthend lo cated errors can b e corrected

or as wehave seen no observable acting on d ofthen qubits can reveal

any information ab out the enco ded state Equivalently the observable can

reveal nothing ab out the state of the ancilla in the entangled state ji

Now since we already knowthat nd if k let us imagine

dividing the co de blo ck Q into three disjoint parts a set of d qubits Q

d

another disjoint set of d qubits Q and the remaining qubits Q

d

nd

If we trace out Q and Q the density matrix we obtain must contain no

correlations b etween Q and the ancilla A This means that the entropyof

system AQ is additive

S Q Q S AQ S A S Q

Similarly

S Q Q S AQ S A S Q

Furthermore in general Von Neumann entropy is subadditive so that

S Q Q S Q S Q

S Q Q S Q S Q

ve wend Combining these inequalities with the equalities ab o

S A S Q S Q S Q

S A S Q S Q S Q

Both of these inequalities can b e simultaneously satised only if

S A S Q

Now Q has dimension n d and its entropy is b ounded ab oveby

its dimension so that

S A k n d

which is the quantum Singleton b ound

The co de saturates this b ound but for most values of n and

k the b ound is not tight Rains has obtained the stronger result that an

n k t co de with k must satisfy

n

t

CHAPTER QUANTUM ERROR CORRECTION

where x o or x is the greatest integer greater than or equal to x

Thus the minimal length of a k co de that can correct t

errors is n resp ectively Co des with all of these parameters

have actually b een constructed except for the co de

Stabilizer Co des

General formulation

We will b e able to construct a nondegenerate quantum co de but

to do so we will need a more p owerful pro cedure for constructing quantum

co des than the CSS pro cedure

Recall that to establish a criterion for when error recovery is p ossible we

found it quite useful to expand an error sup erop erator in terms of the nqubit

Pauli op erators But up until nowwehave not exploited the group structure

of these op erators a pro duct of Pauli op erators is a Pauli op erator In fact

we will see that group theory is a p owerful to ol for constructing QECCs

For a single qubit we will nd it more convenientnowtocho ose all of

the Pauli op erators to b e represented by real matrices so I will now use a

Y denotes the antihermitian matrix notation in which

Y ZX i y

satisfying Y I Then the op erators

fI X Y Z gfI X Y Z g

are the elements of a group of order The nfold tensor pro ducts of single

qubit Pauli op erators also form a group

n

G fI X Y Z g

n

n n

of order jG j since there are p ossible tensor pro ducts and another

n

factorofforthe sign we will refer to G as the nqubit Pauli group

n

In fact we will use the term Pauli group b oth to refer to the abstract

It is not the quaternionic group but the other nonab elian group of order the

symmetry group of the square The element Y of order can b e regarded as the

rotation of the plane while X and Z are reections ab out two orthogonal axes

STABILIZER CODES

n

group G and to its dimension faithful unitary representation by tensor

n

pro ducts of matrices its only irreducible representation of dimension

n

greater than Note that G has the two element center Z fI gIf

n

we quotient out its center we obtain the group G G Z this group can

n n

n

also b e regarded as a binary vector space of dimension a prop erty that

we will exploit b elow

n

The dimensional representation of the Pauli group G evidently has

n

these prop erties

y

i Each M G is unitary M M

n

n

Furthermore M I ii For each element M G M I I

n

if the number of Y s in the tensor pro duct is even and M I if

the number of Y s is o dd

y

iii If M I thenM is hermitian M M if M I then M is

y

antihermitian M M

iv Anytwo elements M N G either commute or anticommute MN

n

NM

We will use the Pauli group to characterize a QECC in the following way

Let S denote an ab elian subgroup of the nqubit Pauli group G Thus all

n

n

elements of S acting on H can b e simultaneously diagonalized Then the

stabilizer code H H n asso ciated with S is the simultaneous eigenspace

S

with eigenvalue of all elements of S Thatis

j iH i M j i j i for all M S

S

The group S is called the stabilizer of the co de since it preserves all of the

co dewords

The group S can b e characterized by its generators These are elements

fM g that are independent no one can b e expressed as a pro duct of others

i

and suchthateach elementof S can b e expressed as a pro duct of elements

of fM gIfS has n k generators we can show that the co de space H has

i S

k

dimension there are k enco ded qubits

Toverify this rst note that each M S must satisfy M I if

M I then M cannot have the eigenvalue Furthermore for each

M I in G that squares to one the eigenvalues and have equal n

CHAPTER QUANTUM ERROR CORRECTION

degeneracy This is b ecause for each M I there is an N G that

n

anticommutes with M

NM MN

therefore M j i j i if and only if M N j iN j i and the action

of the unitary N establishes a corresp ondence b etween the eigen

n n

mutually states of M and the eigenstates Hence there are

orthogonal states that satisfy

M j i j i

where M is one of the generators of S

Now let M b e another elementof G that commutes with M such that

n

M I M We can nd an N G that commutes with M but

n

anticommutes with M therefore N preserves the eigenspace of M

but within this space it interchanges the and eigenstates of M It

follows that the space satisfying

M j i M j i j i

n

has dimension

Continuing in this waywenotethatifM is indep endentof fM M M g

j j

then there is an N that commutes with M M butanticommutes

j

with M well discuss in more detail b elowhowsuchanN can b e found

j

Therefore restricted to the space with M M M M

j j

has as many eigenvectors as eigenvectors So adding another genera

tor always cuts the dimension of the simultaneous eigenspace in half With

nk

n k

n k generators the dimension of the remaining space is

The stabilizer language is useful b ecause it provides a simple wa yto

characterize the errors that the co de can detect and correct Wemay think

of the n k stabilizer generators M M asthecheck operators of

nk

the co de the collective observables that we measure to diagnose the errors

If the enco ded information is undamaged then wewillnd M for each

i

of the generators but if M for some i then the data is orthogonal to

i

the co de subspace and an error has b een detected

Recall that the error sup erop erator can b e expanded in terms of elements

E of the Pauli group A particular E either commutesoranticommutes

a a

with a particular stabilizer generator M IfE and M commute then

a

ME j i E M j i E j i

a a a

STABILIZER CODES

for j i H so the error preserves the value M ButifE and M

S a

anticommute then

ME j i E M j i E j i

a a a

so that the error ips the value of M and the error can b e detected by

measuring M

For stabilizer generators M and errors E wemay write

i a

s

ia

M E E M

i a a i

s

ia

The s s i n k constitute a syndrome for the error E as

ia a

will b e the result of measuring M if the error E o ccurs In the case

i a

of a nondegenerate co de the s s will b e distinct for all E E so that

ia a

measuring the n k stabilizer generators will diagnose the error completely

More generally let us nd a condition to b e satised by the stabilizer

that is sucient to ensure that error recovery is p ossible Recall that it is

sucient that for each E E E and normalized j i in the co de subspace

a b

wehave

y

h jE E j i C

b ab

a

where C is indep endentof j iWe can see that this condition is satised

ab

provided that for each E E E one of the following holds

a b

y

E E S

b

a

y

There is an M S that anticommuteswith E E

b

a

y

Pro of In case h jE E j i h j i for j i H In case

b S

a

y y

supp ose M S and ME E E E M Then

b b

a a

y y

h jE E j i h jE E M j i

b b

a a

y y

h jME E j i h jE E j i

b b

a a

y

E j i and therefore h jE

b a

CHAPTER QUANTUM ERROR CORRECTION

Thus a stabilizer code that corrects fE g is a space H xed by an ab elian

S

subgroup S of the Pauli group where either or is satised by each

y

E E with E E The co de is nondegenerate if condition is not

b ab

a

y

E satised for any E

b

a

Evidently we could also just as well cho ose the co de subspace to b e any

nk

one of the simultaneous eigenspaces of n k indep endentcommuting

elements of G But in fact all of these co des are equivalent Wemay regard

n

two stabilizer co des as equivalent if they dier only according to how the

qubits are lab eled and how the basis for each singlequbit Hilb ert space is

chosen that is the stabilizer of one co de is transformed to the stabilizer

of the other bya permutation of the qubits together with a tensor pro d

uct of singlequbit transformations If we partition the stabilizer generators

into two sets fM M g and fM M g then there exists an

j j nk

N G that commutes with eachmemb er of the rst set and anticommutes

n

with eachmemb er of the second set Applying N to j iH preserves the

s

eigenvalues of the rst set while ipping the eigenvalues of the second set

Since N is just a tensor pro duct of singlequbit unitary transformations

there is no loss of generality up to equivalence in cho osing all of the eigen

values to b e one Furthermore since minus signs dont really matter when

the stabilizer is sp ecied wemay just as well saythattwo co des are equiva

lent if up to phases the stabilizers dier byapermutation of the n qubits

and p ermutations on each individual qubits of the op erators X Y Z

y

Recovery may fail if there is an E E that commutes with the stabilizer

b

a

but do es not lie in the stabilizer This is an op erator that preserves the

co de subspace H but may act nontrivially in that space thus it can mo dify

S

E j i have the same syndrome we enco ded information Since E j i and

b a

might mistakenly interpret an E error as an E error the eect of the error

a b

y

E gets applied to the data together with the attempt at recovery is that E

a

b

which can cause damage

A stabilizer co de with distance d has the prop erty that each E G of

n

weight less than d either lies in the stabilizer or anticommutes with some

element of the stabilizer The co de is nondegenerate if the stabilizer contains

no elements of weight less than d A distance d t co de can correct

t errors and a distance s co de can detect s errors or correct s errors at

known lo cations

STABILIZER CODES

Symplectic Notation

Prop erties of stabilizer co des are often b est explained and expressed using the

nk

language of linear algebra The stabilizer S of the co de an order ab elian

subgroup of the Pauli group with all elements squaring to the identitycan

n

equivalently b e regarded as a dimension n k closed linear subspace of F

self orthogonal with resp ect to a certain symplectic inner pro duct

n

The group G G Z is isomorphic to the binary vector space F We

n n

establish this by observing that since Y ZXanyelement M of the Pauli

group up to the sign can b e expressed as a pro duct of Z s and X s we

may write

M Z X

M M

where Z is a tensor pro duct of Z s and X is a tensor pro duct of X s

M M

More explicitlya Pauli op erator may b e written as

n n

O O

i i

Z X Z X j

i i

where and are binary strings of length nThenY acts at the lo cations

n

where and collide Multiplication in G maps to addition in F

n

j j j

the phase arises b ecause counts the numb er of times a Z is interchanged

with a X as the pro duct is rearranged into the standard form of eq

It follows from eq that the commutation prop erties of the Pauli

op erators can b e expressed in the form

j j j j

Thus twoPauli op erators commute if and only if the corresp onding vectors

are orthogonal with resp ect to the symplectic inner pro duct

We also note that the square of a Pauli op erator is

j I

CHAPTER QUANTUM ERROR CORRECTION

since counts the number of Y s in the op erator it squares to the identity

if and only if

Note that a closed subspace where eachelement has this prop ertyisauto

matically selforthogonal since

in the group language that is a subgroup of G with each element squaring

n

to I is automatically ab elian

Using the linear algebra language some of the statements made earlier

ab out the Pauli group can b e easily veried by counting linear constraints

Elements are indep endent if the corresp onding vectors are linearly indep en

n

dentover F sowemay think of the n k generators of the stabilizer

k We will use the nota as a basis for a linear subspace of dimension n

tion S to denote b oth the linear space and the corresp onding ab elian group

Then S denotes the dimensionn k space of vectors that are orthogonal

to eachvector in S with resp ect to the symplectic inner pro duct Note

that S contains S since all vectors in S are mutually orthogonal In the

group language corresp onding to S is the normalizer or centralizer group

N S S ofS in G the subgroup of G containing all elements that

n n

commute with each elementof S SinceS is ab elian it is contained in its

own normalizer which also contains other elements to b e further discussed

b elow The stabilizer of a distance d co de has the prop ertythateachj

P

whose weight is less than d either lies in the stabilizer subspace

i i

i

S or lies outside the orthogonal space S

A code can be characterized by its stabilizer a stabilizer by its generators

and the n k generators can b e represented byann k n matrix

H H jH

Z X

Here eachrowisaPauli op erator expressed in the j notation The syn

drome of an error E j is determined by its commutation prop erties

a a a

with the generators M j that is

i

i i

j s j

a a ia a a

i i i i

STABILIZER CODES

In the case of a nondegenerate co de each error has a distinct syndrome If

the co de is degenerate there may b e several errors with the same syndrome

y

but wemay apply any one of the E corresp onding to the observed syndrome

a

in order to recover

Some examples of stabilizer co des

a The ninequbit co de This co de has eight stabilizer genera

tors that can b e expressed as

Z Z Z Z Z Z Z Z Z Z Z Z

X X X X X X X X X X X X

In the notation of eq these b ecome

C B

C B

C B

C B

C B

C B

C B

C B

C B

C B

C B

C B

C B

A

b The sevenqubit co deThis co de has six stabilizer genera

tors which can b e expressed as

H

ham

H

H

ham

where H is the paritycheck matrix of the classical

ham

Hamming co de The three check op erators

M Z Z Z Z

M Z Z Z Z

M Z Z Z Z

CHAPTER QUANTUM ERROR CORRECTION

detect the bit ips and the three check op erators

M X X X X

M X X X X

M X X X X

detect the phase errors The space with M M M is

spanned by the co dewords that satisfy the Hamming paritycheck Re

calling that a Hadamard change of basis interchanges Z and X we

see that the space with M M M is spanned by co dewords

that satisfy the Hamming paritycheck in the Hadamardrotated ba

sis Indeed we constructed the sevenqubit co de by demanding that

the Hamming paritycheck b e satised in b oth bases The generators

commute b ecause the Hamming co de contains its dual co de ie each

H satises the Hamming paritycheck rowof

ham

c CSS co des Recall whenever an n k d classical co de C contains its

dual co de C we can p erform the CSS construction to obtain an

n k n d quantum co de The stabilizer of this co de can b e written

as

H

H

H

where H is the n k n paritycheck matrix of C As for the seven

qubit co de the stabilizers commute b ecause C contains C and the

co de subspace is spanned by states that satisfy the H paritycheckin

b oth the F basis and the P basis Equivalentlycodewords ob ey the H

paritycheck and are invariant under

jv i jv w i

where w C

d More general CSS co des Consider more generally a stabilizer

whose generators can eachbechosen to b e either a pro duct of Z s

j or a pro duct of X s j Then the generators have the form

H

Z

H

H X

STABILIZER CODES

Now what condition must H and H satisfy if the Z generators and

X Z

X generators are to commute Since Z s must collide with X s an

even numberoftimeswehave

T T

H H H H

X Z

Z X

But this is just the requirement that the dual C of the co de whose

X

paritycheckis H b e contained in the co de C whose paritycheckis

X Z

H In other words this QECC ts into the CSS framework with

Z

C C C C

Z

X

So wemaycharacterize CSS co des as those and only those for which

the stabilizer has generators of the form eq

However there is a caveat The co de dened by eq will b e non

d mind d degenerate if errors are restricted to weight less than

Z X

where d is the distance of C andd the distance of C But the

Z Z X X

true distance of the QECC could exceed dFor example the qubit

co de is in this generalized sense a CSS co de But in that case the

classical co de C is distance reecting that eg Z Z is contained

X

in the stabilizer Nevertheless the distance of the CSS co de is d

since no weight Pauli op erator lies in S n S

Enco ded qubits

Wehave seen that the troublesome errors are those in S n S those that

commute with the stabilizer but lie outside of it These Pauli op erators are

also of interest for another reason they can b e regarded as the logical

op erations that act on the enco ded data that is protected by the co de

App ealing to the linear algebra viewp oint we can see that the nor

malizer S of the stabilizer contains n k indep endent generators in the

ndimensional space of the j s the subspace containing the vectors that

are orthogonal to eachof n k linearly indep endentvectors has dimension

n n k n k Of the n k vectors that span this space n k

can b e chosen to b e the generators of the stabilizer itself The remaining

k generators preserve the co de subspace b ecause they commute with the

stabilizer but act nontrivially on the k enco ded qubits

In fact these k op erations can b e chosen to b e the singlequbit op erators

Z X i k where Z X are the Pauli op erators Z and X acting

i i i i

CHAPTER QUANTUM ERROR CORRECTION

on the enco ded qubit lab eled by i First note that we can extend the n k

stabilizer generators to a maximal set of n commuting op erators The k

op erators that we add to the set may b e denoted Z Z We can then

k

regard the simultaneous eigenstates of Z Z in the co de subspace H

k S

as the logical basis states jz z iwithz corresp onding to Z

k j j

andz corresp onding to Z

j j

The remaining k generators of the normalizer maybechosen to b e mutu

ally commuting and to commute with the stabilizer but then they will not

commute with anyoftheZ s By invoking a GramSchmidt orthonormaliza

i

tion pro cedure we can cho ose these generators denoted X to diagonalize

i

the symplectic form so that

ij

Z X Z X

i j i j

Thus each X ips the eigenvalue of the corresp onding Z and it can so b e

j j

regarded as the Pauli op erator X acting on enco ded qubit i

a The qubit Co deAswehave discussed previously the logical op er

ators can b e chosen to b e

Z X X X

X Z Z Z

These anticommute with one another an X and a Z collide at p osition

commute with the stabilizer generators and are indep endent of the

generators no element of the stabilizer contains three X s or three

Z s

b The qubit co deWehave seen that

X X X X

Z Z Z Z

then X adds an o dd Hamming co deword and Z ips the phase of an

o dd Hamming co deword These op erations implement a bit ip and

phase ip resp ectively in the basis fji ji g dened in eq

F F

THE QUBIT CODE

The Qubit Co de

All of the QECCs that wehave considered so far are of the CSS typ e each

stabilizer generator is either a pro duct of Z s or a pro duct of X s But not

all stabilizer co des have this prop erty An example of a nonCSS stabilizer

co de is the p erfect nondegenerate co de

Its four stabilizer generators can b e expressed

M XZZXI

M IXZZX

M XIXZZ

M ZXIXZ

M are obtained from M by p erforming a cyclic p ermutation of the

qubits The fth op erator obtained by a cyclic p ermutation of the qubits

M ZZXI X M M M M is not indep endent of the other four

Since a cyclic p ermutation of a generator is another generator the co de itself

is cyclic a cyclic p ermutation of a co deword is a co deword

Clearly each M contains no Y s and so squares to I For each pair

i

sothatthe of generators there are two collisions b etween an X and a Z

generators commute One can quickly check that eachPauli op erator of

weight or weight anticommutes with at least one generator so that the

distance of the co de is

Consider for example whether there are error op erators with supp ort

on the rst two qubits that commute with all four generators The weight

op erator to commute with the IX in M and the XI in M must b e

XXButXX anticommutes with the XZ in M and the ZX in M

In the symplectic notation the stabilizer may b e represented as

B C

B C

B H C

A

This matrix has a nice interpretation as each of its columns can b e regarded

as the syndrome of a singlequbit error For example the singlequbit bit ip

op erator X commutes with M if M has an I or X in p osition j and

j i i

anticommutes if M has a Z in p osition j Thus the table i

CHAPTER QUANTUM ERROR CORRECTION

X X X X X

M

M

M

M

lists the outcome of measuring M in the event of a bit ip For example

if the rst bit ips the measurement outcomes M M M M

diagnose the error Similarly the righthalfof H can b e regarded as the

syndrome table for the phase errors

Z Z Z Z Z

M

M

M

M

Since Y anticommutes with b oth X and Z we obtain the syndrome for the

error Y by summing the ith columns of the X and Z tables

i

Y Y Y Y Y

M

M

M

M

We nd by insp ection that the columns of the X Y and Z syndrome

tables are all distinct and so weverify again that our co de is a nondegenerate

co de that corrects one error Indeed the co de is p erfect each of the

nontrivial binary strings of length app ears as a column in one of the tables

Because of the cyclic prop erty of the co de we can easily characterize all

nontrivial elements of its stabilizer Aside from M XZZXI and

the four op erators obtained from it by cyclic p ermutations of the qubit the

stabilizer also contains

M M YXXYI

plus its cyclic p ermutations and

M M ZYYZI

THE QUBIT CODE

and its cyclic p ermutations Evidently all elements of the stabilizer are

weight Pauli op erators

For our logical op erators wemaycho ose

Z ZZZZZ

X XXXXX

these commute with M square to I and anticommute with one an

other Being weight they are not themselves contained in the stabilizer

Therefore if we dont mind destroying the enco ded state we can determine

the value of Z for the enco ded qubit by measuring Z of each qubit and eval

uating the parity of the outcomes In fact since the co de is distance three

there are elements of S n S of weightthree alternate expressions for Z and

X can b e obtained bymultiplying by elements of the stabilizer For example

wecancho ose

Z ZZZZZ ZY Y ZI IXXI Z

or one of its cyclic p ermutations and

X XXXXX YXXYI ZII ZX

or one of its cyclic p ermutations So it is p ossible to ascertain the value of

X or Z by measuring X or Z of only three of the ve qubits in the blo ck

and evaluating the parity of the outcomes

If we wish we can construct an orthonormal basis for the co de subspace

as follows Starting from anystatej iwe can obtain

X

j i M j i

M S

This unnormalized state ob eys M j i j i for each M S since

multiplication by an element of the stabilizer merely p ermutes the terms in

the sum To obtain the Z enco ded state jiwema y start with the state

ji whichisalsoa Z eigenstate but not in the stabilizer wend

CHAPTER QUANTUM ERROR CORRECTION

up to normalization

X

ji ji

M S

ji M cyclic p erms ji

M M cyclic p erms ji M M cyclic p erms ji

ji i cyclic p erms

ji cyclic p erms

ji cyclic p erms

Wemay then nd ji by applying X to j ithatisby ipping all qubits

ji X j i ji ji cyclic p erms

ji cyclic p erms

ji cyclic p erms

How is the syndrome measured A circuit that can b e executed to mea

sure M XZZXI is

Figure

The Hadamard rotations on the rst and fourth qubits rotate M to the

tensor pro duct of Z s ZZZZI and the CNOTs then imprint the value

of this op erator on the ancilla The nal Hadamard rotations return the

enco ded blo ck to the standard co de subspace Circuits for measuring M

are obtained from the ab oveby cyclically p ermuting the ve qubits in the

co de blo ck

What ab out enco ding Wewant to construct a unitary transformation

U jiaji bji aji bji

enco de

Wehave already seen that ji is a Z eigenstate and that ji is

a Z eigenstate Therefore up to normalization

X

A

jiaji bji aji bji M

M S

THE QUBIT CODE

P

So we need to gure out how to construct a circuit that applies M to

an initial state

Since the generators are indep endent each element of the stabilizer can b e

expressed as a pro duct of generators as a unique wayandwemay therefore

rewrite the sum as

X

M I M I M I M I M

M S

Now to pro ceed further it is convenient to express the stabilizer in an alter

native form Note that wehave the freedom to replace the generator M by

i

M M without changing the stabilizer This replacement is equivalentto

i j

adding the j th rowtotheith row in the matrix H With suchrow op era

tions we can p erform a Gaussian elimination on the matrixH and

X

so obtain the new presentation for the stabilizer

C B

C B

C H B

A

or

M YZIZY

M IXZZX

M ZZXIX

M ZIZY Y

In this form M applies an X ip only to qubits i and in the blo ck

i

p

I M to a state Adopting this form for the stabilizer we can apply

jz z z z i by executing the circuit

Figure

p

The Hadamard prepares ji ji If the rst qubit is ji the other

op erations dont do anything so I is applied But if the rst qubit is ji

then X has b een applied to this qubit and the other gates in the circuit apply

CHAPTER QUANTUM ERROR CORRECTION

ZZI ZY conditioned on the rst qubit b eing ji Hence YZIZY M

p

I M has b een applied Similar circuits can b e constructed that apply

to jz z z z i and so forth Apart from the Hadamard gates eachofthese

circuits applies only Z s and conditional Z s to qubits through these

qubits never ip It was to ensure thus that we p erformed the Gaussian

elimination on H Therefore we can construct our enco ding circuit as

X

Figure

Furthermore each Z gate acting on ji can b e replaced by the identityso

wemay simplify the circuit by eliminating all such gates obtaining

Figure

This pro cedure can b e generalized to construct an enco ding circuit for any

stabilizer co de

Since the enco ding transformation is unitarywe can use its adjointto

deco de And since each gate squares to I the deco ding circuit is just the

enco ding circuit run in reverse

Quantum secret sharing

The co de provides a nice illustration of a p ossible application of

QECCs

Supp ose that some top secret information is to b e entrusted to n parties

Because none is entirely trusted the secret is divided into n shares so that

eachparty with access to his share alone can learn nothing at all ab out the

secret But if enough parties get together and p o ol their shares they can

decipher the secret or some part of it

heme has the prop ertythatm shares In particular an m n threshold sc

are sucient to reconstruct all of the secret information But from m

R Cleve D Gottesman and HK Lo How to Share a Quantum Secret quant ph

QUANTUM SECRET SHARING

shares no information at all can b e extracted This is called a threshold

scheme b ecause as shares m are collected one by one nothing

is learned but the next share crosses the threshold and reveals everything

We should distinguish to o kinds of secrets a classical secret is an a priori

unknown bit string while a quantum secret is an a priori unknown quantum

state Either typ e of secret can b e shared In particular we can distribute

a classical secret among several parties by selecting one from an ensemble

of mutually orthogonal entangled quantum states and dividing the state

among the parties

We can see for example that the co de may b e employed in

a threshold scheme where the shared information is classical One

classical bit is enco ded by preparing one of the two orthogonal states ji or

ji and then the ve qubits are distributed to ve parties Wehaveseenthat

since the co de is nondegenerate if anytwo parties get together then the

density matrix their two qubits is

Hence they learn nothing ab out the quantum state from any measurement

of their two qubits But wehave also seen that the co de can correct two

lo cated errors or two erasures When any three parties get together they

may correct the two errors the two missing qubits and p erfectly reconstruct

the enco ded state j i or ji

It is also clear that by a similar pro cedure a single qubit of quantum infor

mation can b e shared the codeisalsothebasisofa quan

tum threshold scheme we use the m n notation if the shared information

is quantum information and the m n notation if the shared information

is classical How do es this quantumsecretsharing scenario generalize to

more qubits Supp ose we prepare a pure state j i of n qubits can it b e

employed in an m n threshold scheme

We know that m qubits must b e sucient to reconstruct the state hence

n m erasures can b e corrected It follows from our general error correction

criterion that the exp ectation value of anyweightn m observable must

b e indep endent of the state j i

h jE j i indep endentof j i wtE n m

Thus if m parties have all the information the other n m parties have no

information at all That makes sense since quantum information cannot b e cloned

CHAPTER QUANTUM ERROR CORRECTION

On the other hand weknowthatm shares reveal nothing or that

h jE j i indep endentof j i wtE m

It then follows that m erasures can b e corrected or that the other n m

parties have all the information

From these two observations we obtain the two inequalities

n mm nm

m n m nm

It follows that

n m

in an m n pure state quantum threshold scheme where eachparty has

a single qubit In other words the threshold is reached as the number of

qubits in hand crosses over from the minority to the ma jorit yofall n qubits

We see that if each share is a qubit a quantum pure state threshold

scheme is a m km quantum co de with k Butinfactthe

and co des do not exist and it follows from the Rains b ound that the

m co des do not exist In a sense then the co de is the unique

quantum threshold scheme

There are a number of caveats the restriction n m continues to

apply if each share is a q dimensional system rather than a qubit but various

m k

q

co des can b e constructed for q See the exercises for an example

Also wemightallow the shared information to b e a mixed state that

enco des a pure state For example if we discard one qubit of the ve qubit

blo ck wehave a scheme Again once wehave three qubits we can

correct two erasures one arising b ecause the fourth share is in the hands of

another party the other arising b ecause a qubit has b een thrown away

ve assumed that the shared information is quantum infor Finallyweha

mation But if we are only sharing classical information instead then the

conditions for correcting erasures are less stringent For example a Bell pair

may b e regarded as a kind of threshold scheme for two bits of classical

information where the classical information is enco ded bycho osing one of

SOME OTHER STABILIZER CODES

the four mutually orthogonal states j i j i A party in p ossession of one

of the two qubits is unable to access any of this classical information But

this is not a scheme for sharing a quantum secret since linear combinations

if we trace out one of these Bell states do not have the prop ertythat

of the two qubits

Some Other Stabilizer Co des

The
  co de

A k quantum co de has a onedimensional co de subspace that is there is

only one enco ded state The co de cannot b e used to store unknown quantum

information but even so k co des can haveinteresting prop erties Since

they can detect and diagnose errors they might b e useful for a study of the

correlations in decoherence induced byinteractions with the environment

If k thenS and S coincide a Pauli op erator that commutes

with all elements of the stabilizer must lie in the stabilizer In this case

the distance d is dened as the minimum weightofanyPauli op erator in

the stabilizer Thus a distanced co de can detect d errors that is if

anyPauli op erator of weight less than d acts on the co de state the result is

orthogonal to that state

Asso ciated with the codeisa co de whose enco ded state

can b e expressed as

jiji jiji

where j i and ji are the Z eigenstates of the co de You can verify

that this co de has distance d an exercise

The co de is interesting b ecause its co de state is maximally en

tangled Wemaycho ose any three qubits from among the six The density

matrix of those three obtained by tracing over the other three is totally

random I In this sense the state is a natural multiparti

cle analog of the twoqubit Bell states It is far more entangled than the

p

sixqubit cat state ji ji If we measure any one of the six

qubits in the cat state in the fji jig basis weknoweverything ab out the

state wehave prepared of the remaining ve qubits But wemay measure

any observable we please acting on any three qubits in the state and

CHAPTER QUANTUM ERROR CORRECTION

we learn nothing ab out the remaining three qubits which are still describ ed

I by

Our state is all the more interesting in that it turns out but is not

so simple to prove that its generalizations to more qubits do not exist That

is there are no n n binary quantum co des for n Youll see in

the exercises though that there are other nonbinarymaximally entangled

states that can b e constructed

The m m   errordetecting co des

p

The Bell state j i ji ji is a co de with stabilizer gen

erators

ZZ

XX

The co de has distance two b ecause no weightone Pauli op erator commutes

with b oth generators none of X Y Z commute with b oth X and Z Cor

resp ondingly a bit ip X oraphaseipZ or b oth Y acting on either

qubit in j itakes it to an orthogonal state one of the other Bell states

j i j i j i

One way to generalize the Bell states to more qubits is to consider the

n k co de with stabilizer generators

ZZZZ

XXXX

This is a distance d co de for the same reason as b efore The co de

subspace is spanned by states of even parityZZZZ that are invariant

under a simultaneous ip of all four qubits XXX A basis is

ji ji

ji ji

ji ji

ji ji

Evidentlyan X or a Z acting on any qubit takes each of these states to

a state orthogonal to the co de subspace thus any singlequbit error can b e detected

SOME OTHER STABILIZER CODES

A further generalization is the m m co de with stabilizer gen

erators

ZZ Z

XX X

the length is required to b e even so that the generators will commute The

n

co de subspace is spanned by our familiar friends the cat states

p

jxi jxi

where x is an evenweight string of length n m

The    co de

As already noted in our discussion of the co de a stabilizer co de with

generators

H H jH

Z X

can correct one error if the columns of H are distinct a distinct syndrome

for each X and Z error and each sum of a column of H with the

Z

corresp onding column of H is distinct from each column of H and distinct

X

from all other such sums each Y error can b e distinguished from all other

onequbit errors

We can readily construct a matrix H with this prop erty and so

derive the stabilizer of an co de wecho ose

H H

C B

H

A

Here H is the matrix

B C

H

A

whose columns are all the distinct binary strings of length and H is ob

tained from H by p erforming a suitable p ermutation of the columns This

CHAPTER QUANTUM ERROR CORRECTION

p ermutation is chosen so that the eight sums of columns of H with corre

sp onding columns of H are all distinct Wemay see by insp ection that a

suitable choice is

C B

H

A

as the column sums are then

B C

A

The last tworows of H serve to distinguish each X syndrome from each Y

syndrome and the ab ovementioned prop ertyof H ensures syndrome or Z

that all Y syndromes are distinct Therefore wehave constructed a length

co de with k that can correct one error It is actually the simplest

m m

in an innite class of m co des constructed by Gottesman

with m

The quantum co de that wehave just describ ed is a close cousin

of the extended Hamming co de the selfdual classical co de that

is obtained from the dual of the Hamming co de byaddinganextra

parity bit Its paritycheck matrix which is also its generator matrix is

C B

C B

C H B

EH

A

This matrix H has the prop erty that not only are its eight columns dis

EH

tinct but also each sum of two columns is distinct from all columns since

the sum of two columns has not as its fourth bit

Co des Over GF


We constructed the co de by guessing the stabilizer generators and

checking that d Is there a more systematic metho d

CODES OVER GF

In fact there is Our suspicion that the co de might exist was

aroused by the observation that its parameters saturate the quantum sphere

packing inequalityfor t co des

nk

n

for n and k To a co ding theorist this equation might

lo ok familiar

Aside from the binary co des wehave fo cused on up to now classical co des

can also b e constructed from lengthn strings of symb ols that takevalues

not in f g but in the nite eld with q elements GF q Such nite elds

m

exist for any q p where p is prime GF is short for Galois Field in

honor of their discoverer

For suchnonbinary co des wemay mo del error as addition by an element

of the eld a cyclic shift of the q symb ols Then there are q nontrivial

n

is the numb er of its nonzero ele errors The weightofa vector in GF q

ments and the distance b etween twovectors is the weight of their dierence

the numb er of elements that disagree An n k d classical co de consists

q

k n

of q co dewords in GF q where the minimal distance b etweenapairis

d The sphere packing b ound that must b e satised for an n k d co de to

q

exist b ecomes for d

nk

q n q

In fact the p erfect binary Hamming co des that saturate this b ound for q

with parameters

m

n k n m

admit a generalization to any GF q p erfect Hamming co des over GF q

can b e constructed with

m

q

n k n m

q

The quantum co de is descended from the classical Hamming

co de the case q and m

What do the classical GF co des have to do with binary quantum sta

bilizer co des The connection arises b ecause the stabilizer can b e asso ciated

with a set of vectors over GF closed under addition

CHAPTER QUANTUM ERROR CORRECTION

The eld GF has four elements that may b e denoted where

and Thus the additive structure of GF echos the

multiplicative structure of the Pauli op erators X Y Z Indeed the length

n binary string j thatwehave used to denote an elementofthePauli

n

group can equivalently b e regarded as a lengthn vector in GF

j

nk

The stabilizer with elements can b e regarded as a sub co de of GF

nk

ords closed under addition and containing co dew

Note that the co de need not b e a vector space over GF as it is not

required to b e closed under multiplication by a scalar GF In the sp ecial

case where the co de is a vector space it is called a linear co de

Muchisknown ab out co des over GF so this connection op ened the

do or for the classical co ding theorists to construct many QECCs How

n

ever not every sub co de of GF is asso ciated with a quantum co de we

have not yet imp osed the requirement that the stabilizer is ab elian the

j s that span the co de must b e mutually orthogonal in the symplectic

inner pro duct

This orthogonality condition might lo ok strange to a co ding theorist who is

n

more accustomed to dening the inner pro duct of twovectors in GF as

an elementof GF given by

v u v u v u

n n

where conjugation denoted by a bar interchanges and If this hermi

tian inner pro duct of twovectors v and u is

v u a b GF

Calderbank Rains Shor and Sloane Quantum error correction via co des over

GF quantph

CODES OVER GF

then our symplectic inner pro duct is

v u b

Therefore vanishing of the symplectic inner pro duct is a weaker condition

than vanishing of the hermitian inner pro duct In fact though in the sp ecial

case of a linear co de selforthogonality with resp ect to the hermitian inner

pro duct is actually equivalent to selforthogonality with resp ect to the sym

plectic inner pro duct We observethatifv u a b orthogonalityinthe

symplectic inner pro duct requires b Butif u is in a linear co de then so

isu where

v ub a

so that

v ua

We see that if v and u b elong to a linear GF co de and are orthogonal

with resp ect to the symplectic inner pro duct then they are also orthogonal

with resp ect to the hermitian inner pro duct We conclude then that a lin

ear GF co de denes a quantum stabilizer co de if and only if the co de is

selforthogonal in the hermitian inner pro duct Classical co des with these

prop erties havebeenmuch studied

In particular consider again the Hamming co de Its paritycheck

tation can b e expressed as matrix in an unconventional presen

H

which is also the generator matrix of its dual a linear selforthogonal

co de In fact this co de with co dewords is precisely the

stabilizer of the quantum co de By identifying X Z we

recognize the tworows of H as the stabilizer generators M M The dual

of the Hamming co de is a linear co de so linear combinations of the rows are

contained in the co de Adding the rows and multiplying by we obtain

M and multiply by wend whichis M And if we add M to

CHAPTER QUANTUM ERROR CORRECTION

whichisM

The co de is just one example of a quite general construction

n

Consider a sub co de C of GF that is additive closed under addition

and selforthogonal contained in its dual with resp ect to the symplectic

inner pro duct This GF co de can b e identied with the stabilizer of a

nk

binary QECC with length nIftheGF co de contains co dewords

then the QECC has k enco ded qubits The distance d of the QECC is the

minimum weightofa vector in C n C

Another example of a selforthogonal linear GF co de is the dual of the

m Hamming co de with

n

nm m

The Hamming co de has co dewords and its dual has co dewords

We immediately obtain a QECC with parameters

that can correct one error

Go o d Quantum Co des

A family of n k d co des is good if it contains co des whose rate R kn

and error probability p tn where t d b oth approacha

nonzero limit as n We can use the stabilizer formalism to prove

aquantum Gilb ertVarshamov b ound that demonstrates the existence of

go o d quantum co des In fact go o d co des can b e chosen to b e nondegenerate

We will only sketch the argument without carrying out the requisite

counting precisely Let E fE g b e a set of errors to b e corrected and

a

y

denote by E fE E g the pro ducts of pairs of elemen ts of E Thento

b

a

construct a nondegenerate co de that can correct the errors in E wemust

nd a set of stabilizer generators such that some generator anticommutes

with each elementof E

To see if a co de with length n and k qubits can do the job b egin with the

nk

set S of all ab elian subgroups of the Pauli group with n k generators

We will gradually pare away the subgroups that are unsuitable stabilizers for

correcting the errors in E and then see if any are left

SOME CODES THAT CORRECT MULTIPLE ERRORS

nk

Eachnontrivial error E commutes with a fraction of all groups

a

nk

contained in S since it is required to commute with eachofthen k

generators of the group There is a small correction to this fraction that we

may ignore for large n Each time we add another elementto E a fraction

k n

of all stabilizer candidates must b e rejected When E has b een fully

assembled wehave rejected at worst a fraction

k n

jE j

nk

of all the subgroups contained in S where jE j is the numb er of ele

ments of E As long as this fraction is less than one a stabilizer that do es

the job will exist for large n

If wewant to correct t pn errors then E contains op erators of weight

at most t and wemay estimate

n

pn

n H pp log log log jE j

pn

Therefore nondegenerate quantum stabilizer co des that correct pn errors

exist with asymptotic vote R kn given by

log jE j k n or R H p p log

Thus is the asymptotic form of the quantum Gilb ertVarshamov b ound

We conclude that co des with a nonzero rate must exist that protect

against errors that o ccur with any error probability pp The

GV

maxim um error probability allowed by the Rains b ound is p for a

co de that can protect against every error op erator of weight pn

Though go o d quantum co des exist the explicit construction of families

of go o d co des is quite another matter Indeed no such constructions are

known

Some Co des that Correct Multiple Er

rors

Concatenated co des

Up until now all of the QECCs that wehave explicitly constructed have

d or d and so can correct one error at b est Nowwewill

CHAPTER QUANTUM ERROR CORRECTION

describ e some examples of co des that have higher distance

A particularly simple way to construct co des that can correct more errors

is to concatenate co des that can correct one error A concatenated co de is

a co de within a co de Supp ose wehavetwo k QECCs an n d

co de C co de and an n d co de C Imagine constructing a length n

co deword of C and expanding the co deword as a coherent sup erp osition of

pro duct states in whicheach qubit is in one of the states ji or jiNow

replace eachqubitby a lengthn enco ded state using the co de C that is

replace ji by ji and ji by ji of C The result is a co de with length

n n n k and distance no less than d d d We will call C the

outer co de and C the inner co de

In fact wehave already discussed one example of this construction Shors

qubit co de In that case the inner co de is the threequbit rep etition co de

with stabilizer generators

ZZI IZZ

and the outer co de is the threequbit phase co de with stabilizer generators

XXI IX X

the Hadamard rotated rep etition co de We construct the stabilizer of the

concatenated co de as follows Acting on each of the three qubits contained

in the blo ck of the outer co de we include the two generators Z Z Z Z of

the inner co de six generators altogether Then weaddthetwo generators

of the outer co de but with X Z replaced bytheencoded op erations of the

inner co de in this case these are the two generators

X X I I X X

where I II I and X XXX You will recognize these as the eight

stabilizer generators of Shors co de that wehave describ ed earlier In this

case the inner and outer co des b oth have distance eg ZII commutes

with the stabilizer of the inner co de yet the concatenated co de has distance

d d This happ ens b ecause the co de has b een cleverly constructed

so that the weight and enco ded op erations of the inner co de do not

commute with the stabilizer of the outer co de It would have b een dierent

if we had concatenated the rep etition co de with itself rather than with the phase co de

SOME CODES THAT CORRECT MULTIPLE ERRORS

We can obtain a distance co de capable of correcting four errors by

concatenating the co de with itself The length n is the smallest

for any known co de with k and d An n co de with n

would b e consistent with the Rains b ound but it is unknown whether such

a co de really exists

The stabilizer of the concatenated co de has generators Of

these are obtained as the four generators M acting on eachofthe

ve subblo cks of the outer co de and the remaining four are the encoded

op erators M of the outer co de Notice that the stabilizer contains

elements of weight the stabilizer elements acting on each of the ve inner

co des therefore the co de is degenerate This is typical of concatenated

co des

There is no need to stop at twolevels of concatenation from L QECCs

d we can construct a hierarchical with parameters n d n

L L

co de with altogether L levels of co des within co des it has length

n n n n

L

and distance

d d d d

L

In particular by concatenating the co de L times wemay construct

a co de with parameters

L L

Strictly sp eaking this family of co des cannot protect against a number of

errors that scales linearly with the length Rather the ratio of the number t

of errors that can b e corrected to the length n is

L

t

n

which tends to zero for large L But the distance d may b e a deceptive

measure of howwell the co de p erforms it is all right if recovery fails for

some ways of cho osing t pn errors so long as recovery will b e successful

for the typical ways of cho osing pn faulty qubits In fact concatenated co des

can correct pn typical errors for n large and p

Actually the way concatenated co des are usually used do es not fully

exploit their p ower to correct errors To b e concrete consider the

CHAPTER QUANTUM ERROR CORRECTION

co de in the case where each of the ve qubits is indep endently sub jected to

the dep olarizing channel with error probability p that is X Y Z errors each

o ccur with probability p Recovery is sure to succeed if fewer than two

errors o ccur in the blo ck Therefore as in x we can b ound the failure

probabilityby

p p p p

fail

Now consider the p erformance of the concatenated co de To

keep life easywe will p erform recovery in a simple but nonoptimal way

First we p erform recovery on each of the vesubblocks measuring M

to obtain an error syndrome for each subblo ck After correcting the sub

blo cks we then measure the stabilizer generators M of the outer co de

to obtains its syndrome and apply an enco ded X Y orZ to one of the

subblo cks if the syndrome reveals an error

For the outer co de recovery will succeed if at most one of the subblo cks

is damaged and the probability p of damage to a subblo ck is b ounded as

in eq we conclude that the probabilityofabotched recovery for the

co de is b ounded ab oveby

p p p p

Our recovery pro cedure is clearly not the b est p ossible b ecause four errors

can induce failure if there are two eachintwo dierent subblo cks Since the

co de has distance nine there is a b etter pro cedure that would always recover

successfully from four errors so that p would b e of order p rather than

p Still the sub optimal pro cedure has the advantage that it is very easily

generalized and analyzed if there are manylevels of concatenation

Indeed if there are L levels of concatenation we b egin recovery at the

ork our way up Solving the recursion innermost level and w

p C p

starting with p pwe conclude that

L

L

p Cp

C

where here C We see that as long as p we can make the

failure probability as small as wepleaseby adding enough levels to the co de

SOME CODES THAT CORRECT MULTIPLE ERRORS

Wemay write

L

p

L

p p

o

p

o

where p is an estimate of the threshold error probabilitythatcanbe

o

tolerated we will obtain b etter co des and b etter estimates of this threshold

b elow Note that to obtain

L

p

L

wemaycho ose the blo cksize n so that

log

log p

o

n

log p p

o

In principle the concatenated co de at a high level could fail with many

fewer than n errors but these would have to b e distributed in a highly

conspiratorial fashion that is quite unlikely for n large

The concatenated enco ding of an unknown quantum state can b e carried

out level by level For example to enco de aji bji in the blo ck

we could rst prepare the state aji bji in the ve qubit blo ck using the

enco ding circuit describ ed earlier and also prepare four vequbit blo cks in

the state ji The aji j i can b e enco ded at the next level by executing the

enco ded circuit yet again but this time with all gates replaced byencoded

gates acting on vequbit blo cks We will see in the next chapter how these

enco ded gates are constructed

Toric co des

The toric co des are another family of co des that like concatenated co des

oer much b etter p erformance than would b e exp ected on the basis of their

distance Theyll b e describ ed by Professor Kitaev who discovered them

ReedMuller co des

Another way to construct co des that can correct manyerrorsistoinvokethe

CSS construction Recall in particular the sp ecial case of that construction

that applies to a classical co de C that is contained in its dual co de we

CHAPTER QUANTUM ERROR CORRECTION

then saythatC is weakly selfdual In the CSS construction there is a

co deword asso ciated with each coset of C in C Thus we obtain an n k d

quantum co de where n is the length of C d is at least the distance of C

and k dim C dim C Therefore for the construction of CSS co des that

correct many errors we seek weakly selfdual classical co des with a large

minimum distance

One class of weakly selfdual classical co des are the ReedMuller co des

Though these are not esp ecially ecient they are very convenient b ecause

they are easy to enco de recovery is simple and it is not dicult to explain

their mathematical structure

To prepare for the construction of ReedMuller co des consider Bo olean

functions on m bits

m

f f g f g

m

There are such functions forming what wemay regard as a binary vector

m

space of dimension Itwillbeusefultohave a basis for this space Recall

x that any Bo olean function has a disjunctive normal form Since the

x and the OR of two bits x and y can b e expressed as NOT of a bit x is

x y x y xy

any of the Bo olean functions can b e expanded as a p olynomial in the m binary

variables x x x x A basis for the vector space of p olynomials

m m

m

consists of the functions

xxx xx x

i i j i j k

where since x xwemaycho ose the factors of each monomial to b e

distinct Each such function f can b e represented by a binary string of length

m

whose value in the p osition lab eled by the binary string x x x x

m m

See eg MacWilliams and Sloane Chapter

SOME CODES THAT CORRECT MULTIPLE ERRORS

is f x x x x For example for m

m m

x

x

x

x x

x x

x x

x x x

A subspace of this vector space is obtained if we restrict the degree of

the p olynomial to r or less This subspace is the ReedMuller or RM co de

m

denoted Rrm Its length is n and its dimension is

m m m

k

r

Some sp ecial cases of interest are

m

Rm is the length rep etition co de

m

Rm m is the dual of the rep etition co de the space on all length

evenweight strings

R is the n k co de spanned byx x x itisinfact

the extended Hamming co de that wehave already discussed

More generally Rm misad extended Hamming co de for

each m If we puncture this co de remove the last bit from all

m

co dewords we obtain the n k n m d p erfect

Hamming co de

m

n and k m It is the dual of the extended Rmhas d

Hamming co de and is known as a rstorder ReedMuller co de It

is of considerable practical interest in its own right b oth b ecause of its large distance and b ecause it is esp ecially easy to deco de

CHAPTER QUANTUM ERROR CORRECTION

We can compute the distance of the co de Rrmbyinvoking induction

on m First wemust determine how Rm r is related to Rm r A

function of x x can b e expressed as

m

f x x g x x x hx x

m m m m

and if f has degree r then g must b e of degree r and h of degree r

m

Regarding f as a vector of length wehave

f g jg hj

m

where g h are vectors of length Consider the distance b etween f and

f g jg h j

For h h and f f this distance is wtf f wt g g

dist Rrm for h h it is at least wth h dist Rr m If

drm denotes the distance of Rrm then we see that

m drm min drmdr

mr

Nowwe can show that drm by induction on m To start with

r

wecheck that drm for r R is the space of all

length strings and R is the length rep etition co de Next supp ose

mr

that d for all m M and r mThenwe infer that

mr mr mr

drm min

for each r m It is also clear that dm m since

m

Rm m is the space of all binary strings of length and that

m m

dm since Rm is the length rep etition co de

mr

This completes the inductive step and proves drm

It follows in particular that Rm m has distance and therefore

that the dual of Rrmis Rm r m First we notice that the binomial

m

m

sum to so that Rm r has the right dimension co ecients

j

to b e Rrm It suces then to show that Rm r is contained in

Rrm But if f Rrmandg Rm r m their pro duct is a

p olynomial of degree at most m and is therefore in Rm m Each

SOME CODES THAT CORRECT MULTIPLE ERRORS

vector in Rm m has even weight so the inner pro duct f g vanishes

hence g is in the dual Rv m This shows that

Rrm Rm r m

It is b ecause of this nice duality prop erty that ReedMuller co des are well

suited for the CSS construction of quantum co des

In particular the ReedMuller co de is weakly selfdual for r m r

or r m and selfdual for r m In the selfdual case the

distance is

p

m mr

n d

and the numb er of enco ded bits is

m

k n

These selfdual co des for m have parameters

The co de is the extended Hamming co de as wehave already noted

Asso ciated with these selfdual co des are the k quantum co des with

parameters

and so forth

One way to obtain a k quantum co de is to puncture the selfdual

m

ReedMuller co de that is to delete one of the n bits from the co de

It turns out not to matter which bit we delete The classical punctured

q

m m

co de has parameters n d n and

n Furthermore the dual of the punctured co de is its even k

sub co de The even sub co de consists of those RM co dewords for whichthe

bit removed by the puncture is zero and it follows from the selfdualityof

the RM co de that these are orthogonal to all the words b oth o dd and even

weight of the punctured co de From these punctured co des we obtain via

the CSS construction k quantum co des with parameters

CHAPTER QUANTUM ERROR CORRECTION

and so forth The Hamming co de is obtained by puncturing the

RM co de and the corresp onding QECC is of course Steanes

co de These QECCs have a distance that increases like the square ro ot of

their length

These k co des are not among the most ecient of the known QECCs

Nevertheless they are of sp ecial interest since their prop erties are esp ecially

conducive to implementing faulttolerant quantum gates on the enco ded data

as we will see in Chapter In particular one useful prop erty of the selfdual

RM co des is that they are doubly even all co dewords haveaweight that

is an integral multiple of four

Of course we can also construct quantum co des with kby applying

the CSS construction to the RM co des For example R with parameters

m

n

mr

d

k

is dual to R with parameters

m

n

mr

d

k

and so the CSS construction yields a QECC with parameters

Many other weakly selfdual co des are known and can likewise b e employed

The GolayCode

From the p ersp ective of pure mathematics the most imp ortant errorcorrecting

co de classical or quantum ever discovered is also one of the rst ever de

scrib ed in a published article the Golay co de Here we will briey describ e

the Golay co de as it to o can b e transformed into a nice QECC via the CSS

construction Perhaps this QECC is not really imp ortant enough to deserve

a section of this chapter still I have included it just for fun

SOME CODES THAT CORRECT MULTIPLE ERRORS

The extended Golay co de is a selfdual classical co de If we

puncture it removeany one of its bits we obtain the Golay

co de which can correct three errors This co de is actually p erfect as it

saturates the spherepacking b ound

In fact p erfect co des that correct more than one error are extremely rare

It can b e shown that the only p erfect co des linear or nonlinear over any

nite eld that can correct more than one error are the co de and

one other binary co de discovered byGolay with parameters

The Golaycodehasavery intricate symmetry The symmetry

is characterized by its automorphism group the group of p ermutations of

the bits that take co dewords to co dewords This is the Mathieu group

M a sp oradic simple group of order that was discovered in the

tury th cen

The co dewords have the weight distribution in an obvious

notation

Note in particular that eachweightisa multiple of the co de is doubly

even What is the signicance of the numb er In fact it is

and it arises for this combination reason with eachweight co deword we

asso ciate the eightelement set o ctad where the co deword has its supp ort

Each element subset of the bits is contained in exactly one o ctad a

reection of the co des large symmetry

What makes the Golaycodeimportant in mathematics Its discovery

in set in motion a sequence of events that led by around to a

complete classication of the nite simple groups This classication is one

of the greatest achievements of th century mathematics

A group is simple if it contains no nontrivial normal subgroup The nite

simple groups may b e regarded as the building blo cks of all nite groups in

MacWilliams and Sloane x

CHAPTER QUANTUM ERROR CORRECTION

the sense that for any nite group G there is a unique decomp osition of the

form

G G G G G

n

where each G is a normal subgroup of G and eachquotient group

j j

G G is simple The nite simple groups can b e classied into various

j j

innite families plus additional sp oradic simple groups that resist clas

sication

The Golay co de led Leech in to discover an extraordinarily close

packing of spheres in dimensions known as the Leech Lattice The lattice

p oints the centers of the spheres are comp onentintegervalued vectors

with these prop erties to determine if x x x x iscontained in

write each comp onent x in binary notation

j

x x x x x

j j j j j

Then x if

i The x s are either all s or all s

j

ii The x sareaneven parity bit string if the x s are and an o dd

j j

parity bit string if the x s are

j

iii The x s are a bit string contained in the Golay co de

j

When these rules are applied a negativenumb er is represented by its binary

complement eg

etc

We can easily check that is a lattice that is it is closed under addition

Bits other than the last three in the binary expansion of the x s are unre

j

stricted

We can now countthenumb er of nearest neighb ors to the origin or

the numb er of spheres that touchanygiven sphere These p oints are all

SOME CODES THAT CORRECT MULTIPLE ERRORS

distance away from the origin

That is there are neighb ors that haveeight comp onents with the

values their supp ort is on one of the weight Golaycodewords

and the number of signs must b e even There are neighb ors that

have one comp onent with value this comp onent can b e chosen in

ways and the remaining comp onents havethevalue If sayis

chosen then the p osition of the together with the p osition of the s

can b e any of the Golay co dewords with value at the p osition of the

There are neighb ors with two comp onents each taking the value

the signs are unrestricted Altogether the co ordination number of the

lattice is

The Leech lattice has an extraordinary automorphism group discovered

by Conway in This is the nite subgroup of the dimensional rotation

group SO that preserves the lattice The order of this nite group known

as or dot oh is

If its two element center is mo dded out the sp oradic simple group is

obtained At the time of its discovery was the largest of the sp oradic

simple groups that had b een constructed

The Leech lattice and its automorphism group even tually by a route

that wont b e explained here led Griess in to the construction of the

most amazing sp oradic simple group of all whose existence had b een inferred

earlier by Fischer and Griess It is a nite subgroup of the rotation group in

dimensions whose order is approximately This b ehemoth

known as F has earned the nickname the monster though Griess prefers

to call it the friendly giant It is the largest of the sp oradic simple groups

and the last to b e discovered

Thus the classication of the nite simple groups owes much to classical

co ding theory and to the Golay co de in particular Perhaps the theory of

CHAPTER QUANTUM ERROR CORRECTION

QECCs can also b equeath to mathematics something of value and broad

interest

Anyway since the extended Golay co de is selfdual the

co de obtained by puncturing it is weakly self dual its dual is its

even sub co de From it a QECC can b e constructed by the CSS

metho d This co de is not the most ecientquantum co de that can correct

three errors there is a co de that saturates the Rains b ound but it

has esp ecially nice prop erties that are conducive to faulttolerantquantum

computation as we will see in Chapter

The Quantum Channel Capacity

As wehaveformulated it up until now our goal in constructing quantum

error correcting co des has b een to maximize the distance d of the co de

given its length n and the number k of enco ded qubits Larger distance

provides b etter protection against errors as a distance d co de can correct

d erasures or d errorsatunknown lo cations Wehave observed

that go o d co des can b e constructed that maintain a nite rate kn for n

pn that scales linearly with n large and correct a numb er of errors

Nowwe will address a related but rather dierent question ab out the

asymptotic p erformance of QECCs Consider a sup erop erator that acts on

density op erators in a Hilb ert space HNow consider acting indep endently

eachcopyofH contained in the nfold tensor pro duct

n

H H H

n

n

Wewould like to select a co de subspace H of H such that quantum

co de

n

information residing in H can b e sub jected to the sup erop erator

co de

n

and yet can still b e deco ded with high delity

The rate of a co de is dened as

n

log H

co de

R

n

log H

this is the numb er of qubits employed to carry one qubit of enco ded infor

mation The quantum channel capacity Q of the sup erop erator is the

THE QUANTUM CHANNEL CAPACITY

maximum asymptotic rate at whichquantum information can b e sentover

the channel with arbitrarily go o d delityThatisQ is the largest number

n

such that for any RQ and any there is a co de H with rate at

co de

n

least Rsuch that for any j iH the state recovered after j i passes

co de

n

through has delity

F h j j i

Thus Q is a quantum version of the capacity dened by Shannon

for a classical noisy channel As wehave already seen in Chapter this

Q is not the only sort of capacity that can b e asso ciated with a quantum

channel It is also of considerable interest to ask ab out C the maximum

classical information can b e transmitted through a quantum rate at which

channel with arbitrarily small probability of error A formal answer to this

question was formulated in x but only for a restricted class of p ossible

enco ding schemes the general answer is still unknown The quantum channel

capacity Q is even less well understo o d than the classical capacity C of

a quantum channel Note that Q is not the same thing as the maximum

asymptotic rate kn that can b e achieved by good n k d QECCs with

p ositive dn In the case of the quantum channel capacitywe need not insist

that the co de correct any p ossible distribution of pn errors as long as the

errors that cannot b e corrected b ecome highly atypical for n large

Here we will mostly limit the discussion to twointeresting examples of

quantum channels acting on a single qubit the quantum erasure channel

for which Q is exactly known and the dep olarizing channel for which Q

is still unknown but useful upp er and lower b ounds can b e derived

What are these channels In the case of the quantum erasure chan

nel a qubit transmitted through the channel either arrives intact or with

probability p b ecomes lost and is never received We can nd a unitary rep

resentation of this channel b yemb edding the qubit in the threedimensional

Hilb ert space of a qubit with orthonormal basis fji ji jigThechannel

acts according to

q

p

pjiji pjiji jiji

E E E

q

p

jiji pjiji pjiji

E E E

where fji ji ji g are mutually orthogonal states of the environment

E E E

The receiver can measure the observable jihj to determined whether the qubit is undamaged or has b een erased

CHAPTER QUANTUM ERROR CORRECTION

The dep olarizing channel with error probability pwas discussed at

length in x We see that for p wemay describ e the fate of

a qubit transmitted through the channel this way with probability q

where q p the qubit arrives undamaged and with probability q it is

destroyed in which case it is describ ed by the random density matrix

Both the erasure channel and the dep olarizing channel destroy a qubit

with a sp ecied probability The crucial dierence b etween the twochannels

is that in the case of the erasure channel the receiver knows which qubits

have b een destroyed in the case of the dep olarizing channel the damaged

qubits carry no identifying marks whichmakes recovery more challenging

Of course for b oth channels the sender has no waytoknow ahead of time

which qubits will b e obliterated

Erasure channel

The quantum channel capacity of the erasure channel can b e precisely de

termined First we will derive an upp er b ound on Q and then we will show

that co des exist that achieve high delity and attain a rate arbitrarily close

to the upp er b ound

As the rst step in the derivation of an upp er b ound on the capacitywe

show that Q for p

Figure

We observe that the erasure channel can b e realized if Alice sends a qubit

to Bob and a third party Charlie decides at random to either steal the

qubit with probability porallow the qubit to pass unscathed to Bob with

probability p

If Alice sends a large number n of qubits then ab out pn reach Bob

Charlie winds up in and pn are intercepted by Charlie Hence for p

p ossession of more qubits than Bob and if Bob can recover the quantum

information enco ded by Alice then certainly Charlie can as well Therefore

Bob and Charlie can clone the unknown enco ded if Qp for p

quantum states sentby Alice which is imp ossible Strictly sp eaking they

can clone with delity F forany We conclude that Qp

for p

THE QUANTUM CHANNEL CAPACITY

To obtain a b ound on Qp in the case p we will app eal to the

following lemma Supp ose that Alice and Bob are connected by b oth a

p erfect noiseless channel and a noisy channel with capacity Q And

supp ose that Alice sends m qubits over the p erfect channel and n qubits

over the noisy channel Then the number r of enco ded qubits that Bob may

recover with arbitrarily high delitymust satisfy

r m Qn

We derive this inequalityby noting that Alice and Bob can simulate the m

qubits sentover the p erfect channel by sending mQ over the noisy channel

and so achievea rate

r r

R Q

mQ n m Qn

over the noisy channel Were r to exceed m Qn this rate R would exceed

the capacity a contradiction Therefore eq is satised

How consider the erasure channel with error probability p and supp ose

Qp Then we can b ound Qp for p p by

p p

Qp Qp

p p

In other words if we plot Qpinthep Q plane and wedraw a straightline

segment from anypointp Q on the plot to the p ointp Q then

the curve Qpmust lie on or b elow the segmentintheinterval p p if

Qpistwice dierentiable then its second derivative cannot b e p ositive To

obtain this b ound imagine that Alice sends n qubits to Bob knowing ahead

of time that n p p sp ecied qubits will arrive safely The remaining

np p qubits are erased with probability p Therefore Alice and Bob are

using b oth a p erfect channel and an erasure channel with erasure probability

p eq holds and the rate R they can attain is b ounded by

p p

Qp R

p p

On the other hand for n large altogether ab out np qubits are erased and

p n arrive safelyThus Alice and Bob have an erasure channel with

erasure probability p except that they have the additional advantage of

CHAPTER QUANTUM ERROR CORRECTION

knowing ahead of time that some of the qubits that Alice sends are invul

nerable to erasure With this information they can b e no worse o than

without it eq then follows The same b ound applies to the dep olar

izing channel as well

Now the result Qp for p can b e combined with eq

We conclude that the curve Qpmust b e on or b elow the straightline

connecting the p oints p Q and p Q or

Qp p p

In fact there are stabilizer co des that actually attain the rate p for

p We can see this by b orrowing an idea from Claude Shannon

and averaging over random stabilizer co des Imagine cho osing in succession

altogether n k stabilizer generators Each is selected from among the

n

Pauli op erators where all have equal a priori probability except that

each generator is required to commute with all generators chosen in previous

rounds

Now Alice uses this stabilizer co de to enco de an arbitrary quantum state

k

in the dimensional co de subspace and sends the n qubits to Bob over an

erasure channel with erasure probability p Will Bob b e able to recover the

state sentby Alice

Bob replaces each erased qubit by a qubit in the state ji and then

pro ceeds to measure all n k stabilizer generators From this syndrome

measurement he hop es to infer the Pauli op erator E acting on the replaced

y

qubits Once E is known we can apply E to recover a p erfect duplicate

of the state sentby Alice For n large the numb er of qubits that Bob m ust

replace is ab out pn and he will recover successfully if there is a unique Pauli

op erator E that can pro duce the syndrome that he nds If more than one

Pauli op erator acting on the replaced qubits has this same syndrome then

recovery may fail

How likely is failure Since there are ab out pn replaced qubits there are

pn

ab out Pauli op erators with supp ort on these qubits Furthermore for any

particular Pauli op erator E a random stabilizer co de generates a random

syndrome each stabilizer generator has probabilityofcommuting with

E and probabilityofanticommuting with E Therefore the probability

nk

that twoPauli op erators have the same syndrome is

There is at least one particular Pauli op erator acting on the replaced

qubits that has the syndrome found by Bob But the probability that an

THE QUANTUM CHANNEL CAPACITY

other Pauli op erator has this same syndrome and hence the probabilityof

a recovery failure is no worse than

nk

npR pn

P

fail

where R kn is the rate Eq b ounds the failure probabilityif

we average over all stabilizer co des with rate R it follows that at least one

particular stabilizer co de must exist whose failure probability also satises

the b ound

For that particular co de P gets arbitrarily small as n foranyrate

fail

R p strictly less than p Therefore R p is asymptotically

attainable combining this result with the inequality eq we obtain

the capacity of the quantum erasure channel

Qp p p

If wewanted assurance that a distinct syndrome could b e assigned to

all ways of damaging pn erased qubits then wewould require an n k d

quantum co de with distance dpn Our Gilb ertVarshamov b ound of x

guarantees the existence of such a co de for

R H p p log

This rate can b e achieved by a co de that recovers from any of the p ossible

ways of erasing up to pn qubits It lies strictly b elow the capacityforp

b ecause to achievehighaverage delity it suces to b e able to correct the

typical erasures rather than all p ossible erasures

Dep olarizing channel

The capacity of the dep olarizing channel is still not precisely known but we

can obtain some interesting upp er and lower b ounds

As for the erasure channel we can nd an upp er b ound on the capacity

byinvoking the nocloning theorem Recall that for the dep olarizing channel

with error probability p each qubit either passes safely with prob

p or is randomized replaced by the maximally mixed state ability

with probability q p An eavesdropp er Charlie then can

simulate the channel byintercepting qubits with probability q and replacing

CHAPTER QUANTUM ERROR CORRECTION

each stolen qubit with a maximally mixed qubit For q Charlie steals

more than half the qubits and is in a b etter p osition than Bob to deco de the

state sentby Alice Therefore to disallow cloning the rate at which quan

tum information is sent from Alice to Bob must b e strictly zero for q

or p

Qp p

In fact we can obtain a stronger b ound by noting that Charlie can cho ose

a b etter eavesdropping strategy he can employ the optimal approximate

cloner that you studied in a homework problem This device applied to

each qubit sentby Alice replaces it bytwo qubits that eachapproximate the

original with delity F or

j ih j q j ih j q

where F q By op erating the cloner b oth Charlie and

Bob can receive Alices state transmitted through the q dep olarizing

channel Therefore the attainable rate must vanish otherwise bycombin

ing the approximate cloner with quantum error correction Bob and Charlie

would b e able to clone Alices unknown state exactlyWe conclude that the

capacityvanishes for qorp

Qp p

Invoking the b ound eq we infer that

Qp p p

This result actually coincides with our b ound on the rate of n k d co des

with k andd pn found in x A b ound on the capacityis not

the same thing as a b ound on the allowable error probabilityforann k d

co de and in the latter case the Rains b ound is tighter Still the similarity

of the two results b ound may not b e a complete surprise as b oth b ounds are

derived from the nocloning theorem

We can obtain a lower b ound on the capacityby estimating the rate that

can b e attained through random stabilizer co ding as we did for the erasure

THE QUANTUM CHANNEL CAPACITY

channel Now when Bob measures the n k randomly chosen commuting

stabilizer generators he hop es to obtain a syndrome that p oints to a unique

one among the typical Pauli error op erators that can arise with nonnegligible

probability when the dep olarizing channel acts on the n qubits sentby Alice

The number N of typical Pauli op erators with total probability can

typ

b e b ounded by

nH pp log

N

typ

for any andn suciently large Bobs attempt at recovery can fail if

another among these typical Pauli op erators has the same syndrome as the

actual error op erator Since a random co de assigns a random n k bit

syndrome to eachPauli op erator the failure probability can b e b ounded as

k n nH pp log

P

fail

Here the second term b ounds the probabilityofanatypical error and the

rst b ounds the probabilityofanambiguous syndrome in the case of a typical

error We see that the failure probabilityaveraged over random stabilizer

co des b ecomes arbitrarily small for large nforany and rate R such

that

k

H p p log R

n

If the failure probabilityaveraged over co des is small there is a particu

lar co de with small failure probabilityandwe conclude that the rate R is

attainable the capacity of the dep olarizing channel is b ounded b elowas

Qp H p p log

Not coincidentally the rate attainable by random co ding agrees with the

asymptotic form of the quantum Hamming upp er b ound on the rate of nonde

generate n k d co des with dpnwe arrive at b oth results by assigning

a distinct syndrome to each of the typical errors Of course the Gilb ert

Varshamovlower b ound on the rate of n k d co des lies b elow Qp as it

is obtained by demanding that the co de can correct al l the errors of weight

pn or less not just the typical ones

This random co ding argument can also b e applied to a somewhat more

general channel in which X Y andZ errors o ccur at dierent rates Well

CHAPTER QUANTUM ERROR CORRECTION

call this a Pauli channel If an X error o ccurs with probability p a Y

X

error with probability p a Z error with probability p and no error with

Y Z

probability p p p p then the number of typical errors on n

I X Y Z

qubits is

n

nH p p p p

I X Y Z

p np np np n

X Y Z I

where

H H p p p p p log p p log p p log p p log p

I X Y Z I I X X Y Y Z Z

is the Shannon entropy of the probability distribution fp p p p gNow

I X Y Z

we nd

p H p p p p Qp p p

Y Z I X Y Z I X

if the rate R satises R H then again it is highly unlikely that a single

syndrome of a random stabilizer co de will p oint to more than one typical

error op erator

Degeneracy and capacity

Our derivation of a lower b ound on the capacity of the dep olarizing channel

closely resembles the argumentin x for a lower b ound on the capacity

of the classical binary symmetricchannel In the classical case there was

a matching upp er b ound If the rate were larger then there would not b e

enough syndromes to attach to all of the typical errors

In the quantum case the derivation of the matching upp er b ound do es

not carry through b ecause a quantum co de can b e degenerate Wemay

not need a distinct syndrome for eachtypical error as some of the p ossible

errors could act trivially on the co de subspace Indeed not only do es the

derivation fail the matching upp er b ound is actually false rates exceeding

H p p log actually can b e attained

Shor and Smolin investigated the rate that can b e achieved by concate

nated co des where the outer co de is a random stabilizer co de and the inner

PW Shor and JA Smolin Quantum ErrorCorrecting Co des Need Not Completely

Reveal the Error Syndrome quantph DP DiVincen PW Shor and JA

Smolin Quantum Channel CapacityofVery Noisy Channels quantph

THE QUANTUM CHANNEL CAPACITY

co de is a degenerate co de with a relatively small blo ck size Their idea is

that the degeneracy of the inner co de will allowenoughtypical errors to act

trivially in the co de space that a higher rate can b e attained than through

random co ding alone

Toinvestigate this scheme imagine that enco ding and deco ding are each

p erformed in two stages In the rst stage using the random outer co de

that she and Bob have agreed on Alice enco des the state that she has selected

in a large nqubit blo ck In the second stage Alice enco des each of these

nqubits in a blo ckofm qubits using the inner co de Similarly when Bob

receives the nm qubits he rst deco des each inner blo ckofm and then

subsequently deco des the blo ckofn

We can evidently describ e this pro cedure in an alternativelanguage

Alice and Bob are using just the outer co de but the qubits are b eing trans

mitted through a comp osite channel

Figure

This mo died channel consists as shown of rst the inner enco der then

propagation through the original noisy channel and nally inner deco ding

and inner recovery The rate that can b e attained through the original chan

nel via concatenated co ding is the same as the rate that can b e attained

through the mo died channel via random co ding

Sp ecically supp ose that the inner co de is an mqubit rep etition co de

with stabilizer

Z Z Z Z Z Z Z Z

m

This is not much of a quantum co de it has distance since it is insensi

tive to phase errors each Z commutes with the stabilizer But in the

j

presentcontext its imp ortant feature is it high degeneracyall Z errors are

i

equivalent

The enco ding and deco ding circuit for the rep etition co de consists of

just m CNOTs so our comp osite channel lo oks like in the case m Figure

CHAPTER QUANTUM ERROR CORRECTION

where denotes the original noisy channel Wehave also suppressed the

nal recovery step of the deco ding eg if the measured qubits b oth read

we should ip the data qubit In fact to simplify the analysis of the

comp osite channel we will disp ense with this step

Since we recall that a CNOT propagates bit ips forward from control

to target and phase ips backward from target to control we see that for

each p ossible measurement outcome of the auxiliary qubits the comp osite

channel is a Pauli channel If we imagine that this measurementofthe m

inner blo ck qubits is p erformed for eachofthen qubits of the outer blo ck

then Pauli channels act indep endently on eachofthen qubits but the chan

nels acting on dierent qubits have dierent parameters error probabilities

i i i i

p p p p for the ith qubit Now the number of typical error op erators

I X Y Z

acting on the n qubits is

P

n

H

i

i

where

i i i i

H p p p p H

i

I X Y Z

is the Shannon entropy of the Pauli channel acting on the ith qubit By the

law of large numb ers wewillhave

n

X

H nhH i

i

i

m

for large n where hH i is the Shannon entropyaveraged over the p os

sible classical outcomes of the measurement of the extra qubits of the inner

co de Therefore the rate that can b e attained by the random outer co de is

hH i

R

m

we divide by m b ecause the concatenated co de has a length m times longer

than the random co de

Shor and Smolin discovered that there are rep etition co des values of m

for which in a suitable range of p hH i is p ositive while H p p log

is negative In this range then the capacity Qp is nonzero showing that

the lower b ound eq is not tight

THE QUANTUM CHANNEL CAPACITY

Anonvanishing asymptotic rate is attainable through random co ding for

H p p log or pp If a random outer co de is

max

concatenated with a qubit inner rep etition co de m turnsouttobethe

optimal choice then hH i forpp the maximum error

max

probability for which a nonzero rate is attainable increases byabout

It is not obvious that the concatenated co de should outp erform the random

co de in this range of error probability though as wehave indicated it might

have b een exp ected b ecause of the phase degeneracy of the rep etition co de

Nor is it obvious that m should b e the b est choice but this can b e

veried by an explicit calculation of hH i

The dep olarizing channel is one of the very simplest of quantum chan

nels Yet even for this case the problem of characterizing and calculating

the capacity is largely unsolved This example illustrates that due to the

p ossibility of degenerate co ding the capacity problem is considerably more

subtle for quantum channels than for classical channels

Wehave seen that if the errors are well describ ed by the dep olarizing

channel quantum information can b e recovered from a quantum memory

with arbitrarily high delity as long as the probability of error p er qubit is

less than This is an improvement relative to the error rate that

we found could b e handled by concatenation of the co de In fact

n k d co des that can recover from any distribution of up to pn errors do

not exist for p according to the Rains b ound Nonzero capacityis

p ossible for error rates b etween and b ecause it is sucientforthe

QECC to b e able to correct the typical errors rather than all p ossible errors

However the claim that recovery is p ossible even if of the qubits

sustain damage is highly misleading in an imp ortant resp ect This result

applies if enco ding deco ding and recovery can b e executed awlesslyBut

these op erations are actually very intricate quantum computations that in

practice will certainly b e susceptible to error We will not fully understand

howwell co ding can protect quantum information from harm until wehave

learned to design an error recovery proto col that is robust even if the execu

tion of the proto col is awed Such faulttolerant proto cols will b e develop ed

in Chapter

In fact a very slight further improvementcanbeachieved by concatenating a random

co de with the qubit generalized Shor co de describ ed in the exercises then a nonzero

another b etter than the maximum tolerable rate is attainable for pp

max

error probability with rep etition co ding

CHAPTER QUANTUM ERROR CORRECTION

Summary

Quantum errorcorrecting co des Quantum error correction can protect

quantum information from b oth decoherence and unitary errors due to

imp erfect implementations of quantum gates In a binary quantum error

k

correcting code QECC the dimensional Hilb ert space H of k enco ded

co de

n

qubits is emb edded in the dimensional Hilb ert space of n qubits Errors

y

acting on the n qubits are reversible provided that h jM M j ih j i is

indep endentofj i for any j iH and anytwo Kraus op erators M

co de

o ccuring in the expansion of the error sup erop erator The recovery sup erop

erator transforms entanglementoftheenvironment with the co de blo ckinto

entanglement of the environment with an ancilla that can then b e discarded

Quantum stabilizer co des Most QECCs that have b een constructed

des A binary stabilizer co de is characterized by its stabilizer are stabilizer co

n

S an ab elian subgroup of the nqubit Pauli group G fI X Y Z g

n

where X Y Z are the singlequbit Pauli op erators The co de subspace is

the simultaneous eigenspace with eigenvalue one of all elements of S ifS has

n k indep endent generators then there are k enco ded qubits A stabilizer

co de can correct each error in a subset E of G if for each E E E

n a b

y

E either lies in the stabilizer S or outside of the normalizer S of the E

b

a

y

E is in S for E E the co de is degenerate otherwise stabilizer If some E

b ab

a

it is nondegenerate Op erators in S n S are logical op erators that act on

enco ded quantum information The stabilizer S can b e asso ciated with an

additivecodeover the nite eld GF that is selforthogonal with resp ect

to a symplectic inner pro duct The weight of a Pauli op erator is the number

of qubits on which its action is nontrivial and the distance d of a stabilizer

co de is the minimum weight of an elementof S n S A co de with length n

k enco ded qubits and distance d is called an n k d quantum co de If the

co de enables recovery from any error sup erop erator with supp ort on Pauli

op erators of weight t or less wesay that the co de can correct t errors A

co de with distance d can correct d in unknown lo cations or derrors

in known lo cations Go o d families of stabilizer co des can b e constructed

in which dn and kn remain b ounded away from zero as n

Examples The co de of minimal length that can correct one error is a

quantum co de asso ciated with a classical GF Hamming co de

Given a classical linear co de C and sub co de C C a CalderbankShor

Steane CSS quantum co de can b e constructed with k dimC dimC

enco ded qubits The distance d of the CSS co de satises d mind d

EXERCISES

the dual of is the distance of C where d is the distance of C and d

C The simplest CSS co de is a quantum co de constructed from the

classical Hamming co de and its even sub co de An n d quantum

co de can b e concatenated with an n d co de to obtain a degenerate

n n d co de with d d d

Quantum channel capacity The quantum channel capacity of a su

p erop erator noisy quantum channel is the maximum rate at which quantum

information can b e transmitted over the channel and deco ded with arbitrar

ily go o d delity The capacity of the binary quantum erasure channel with

erasure probability p is Qp pfor p The capacityofthe

binary dep olarizing channel is no yet known The problem of calculating the

capacity is subtle b ecause the optimal co de may b e degenerate in particular

random co des do not attain an asymptotically optimal rate over a quantum

channel

Exercises

Phase errorcorrecting co de

a Construct stabilizer generators for an n k co de that can

correct a single bit ip that is ensure that recovery is p ossible for

any of the errors in the set E fIII XI I IXI IIXgFind

an orthonormal basis for the twodimensional co de subspace

b Construct stabilizer generators for an n k co de that can

correct a single phase error that is ensure that recovery is p ossible

for any of the errors in the set E fIII ZII IZI IIZgFind

an orthonormal basis for the twodimensional co de subspace

Errordetecting co des

a Construct stabilizer generators for an n k d quantum

co de With this co de we can detect any singlequbit error Find

the enco ded state Do es it lo ok familiar

with the same length nareequivalent bTwo QECCs C and C

if a p ermutation of qubits combined with singlequbit unitary

transformations transforms the co de subspace of C to that of

C Are all stabilizer co des equivalent

CHAPTER QUANTUM ERROR CORRECTION

cDoesa stabilizer co de exist

Maximal entanglement

Consider the quantum co de whose stabilizer generators are

M XZZXI andM obtained by cyclic p ermutations of M

and cho ose the enco ded op eration Z to b e Z ZZZZZFrom the

enco ded states ji with Z ji ji and j i with Z ji ji construct

the n k co de whose enco ded state is

p

jiji jij i

a Construct a set of stabilizer generators for this n k co de

b Find the distance of this co de Recall that for a k co de the

distance is dened as the minimum weightofany element of the

stabilizer

cFind the density matrix that is obtained if three qubits are

selected and the remaining three are traced out

Co dewords and nonlo cality

For the co de with stabilizer generators and logical op erators as

in the preceding problem

a Express Z as a weight Pauli op erator a tensor pro duct of I s

X s and Z s no Y s Note that b ecause the co de is cyclic

all cyclic p ermutations of your expression are equivalentways to

represent Z

b Use the Einstein lo cality assumption lo cal hidden variables to pre

dict a relation b etween the ve cyclically related observables

foundina and the observable ZZZZZ Is this relation among

observables satised for the state ji

ould Einstein say c What w

Generalized Shor co de

For integer m consider the n m k generalization of Shors

ninequbit co de with co de subspace spanned bythetwo states

m

ji j i j i

m

ji j ij i

EXERCISES

a Construct stabilizer generators for this co de and construct the log

ical op erations Z and X such that

Z ji ji X j i j i

Z ji ji X j i j i

b What is the distance of this co de

c Supp ose that m is o dd and supp ose that eachofthen m qubits

is sub jected to the dep olarizing channel with error probability p

Howwell do es this co de protect the enco ded qubit Sp ecically

i estimate the probability to leading nontrivial order in pofa

ii estimate the probability logical bitip error jij iand

to leading nontrivial order in p of a logical phase error jij i

jiji

d Consider the asymptotic b ehavior of your answer to cform large

What condition on p should b e satised for the co de to provide

go o d protection against i bit ips and ii phase errors in the

n limit

Enco ding circuits

For an nkd quantum co de an enco ding transformation is a unitary

U that acts as

nk

U j iji j i

where j i is an arbitrary k qubit state and j i is the corresp onding

enco ded state Design a quantum circuit that implements the enco ding

transformation for

a Shors co de

b Steanes co de

Shortening a quantum co de

a Consider a binary n k d stabilizer co de Show that it is p ossible

k stabilizer generators so that at most twoact to cho ose the n

nontrivially on the last qubit That is the remaining n k

generators apply I to the last qubit

CHAPTER QUANTUM ERROR CORRECTION

b These n k stabilizer generators that apply I to the last qubit will

still commute and are still indep endentifwe drop the last qubit

Hence they are the generators for a co de with length n andk

enco ded qubits Show that if the original co de is nondegenerate

then the distance of the shortened co de is at least d Hint

First show thatifthereisaweightt elementofthen qubit

Pauli group that commutes with the stabilizer of the shortened

co de then there is an element of the nqubit Pauli group of weight

at most t that commutes with the stabilizer of the original

co de

c Apply the co deshortening pro cedure of a and b to the

QECC Do you recognize the co de that results HintItmay

b e helpful to exploit the freedom to p erform a change of basis on

some of the qubits

Co des for qudits

A qudit is a ddimensional quantum system The Pauli op erators

I X Y Z acting on qubits can b e generalized to qudits as follows

ijd ig denote an orthonormal basis for the Hilb ert Let fji j

space of a single qudit Dene the op erators

X jj ijj mo d di

j

Z jj i jj i

where expid Then the d d Pauli op erators E are

rs

r s

E X Z rs d

rs

a Are the E s a basis for the space of op erators acting on a qudit

rs

y

Are they unitary Evaluate trE E

tu

rs

bThePauli op erators ob ey

E E E E

rs tu rstu tu rs

where is a phase Evaluate this phase

rstu

The nfold tensor pro ducts of these qudit Pauli op erators form a group

d n

G of order d and if we mo d out its delement center we obtain n

EXERCISES

n d

of order d To construct a stabilizer co de for qudits the group G

n

d

wecho ose an ab elian subgroup of G with n k generators the co de

n

is the simultaneous eigenstate with eigenvalue one of these generators

k

If d is prime then the co de subspace has dimension d k logical qudits

are enco ded in a blo ckof n qudits

c Explain how the dimension might b e dierentifd is not prime

Hint Consider the case d and n

Syndrome measurement for qudits

Errors on qudits are diagnosed by measuring the stabilizer generators

For this purp ose wemayinvoke the twoqudit gate SUM which gen

eralizes the controlledNOT acting as

SUM jj ijk ijj ijk j mo d di

a Describ e a quantum circuit containing SUM gates that can b e exe

cuted to measure an nqudit observable of the form

O

s

a

Z

a

a

If d is prime then for each rs d there is a singlequdit

unitary op erator U such that

rs

y

U E U Z

rs rs

rs

b Describ e a quantum circuit containing SUM gates and U gates

rs

d

of that can b e executed to measure an arbitrary elementof G

n

the form

O

E

r s

a a

a

Errordetecting co des for qudits

A qudit with d is called a qutrit Consider a qutrit stabilizer

co de with length n and k enco ded qutrit dened bythetwo

stabilizer generators

ZZZ XXX

CHAPTER QUANTUM ERROR CORRECTION

a Do the generators commute

b Find the distance of this co de

c In terms of the orthonormal basis fji ji jig for the qutrit write

out explicitly an orthonormal basis for the threedimensional co de

subspace

d Construct the stabilizer generators for an n m qutrit co de where

m is any p ositiveinteger with k n that can detect one

error

e Construct the stabilizer generators for a qudit co de that detects one

error with parameters n d k d

Errorcorrecting co de for qudits

Consider an n k qudit stabilizer co de with stabilizer generators

X Z Z X I

I X Z Z X

X I X Z Z

Z X I X Z

the second third and fourth generators are obtained from the rst by

a cyclic p ermutation of the qudits

a Find the order of each generator Are the generators really in

dep endent Do they commute Is the fth cyclic p ermutation

ZZ X IX indep endent of the rest

b Find the distance of this co de Is the co de nondegenerate

c Construct the enco ded op erations X and Z each expressed as an

op erator of weight Be sure to check that these op erators ob ey

the rightcommutation relations for anyvalue of d