SCHOOLOF - Carleton

CARLETON UNIVERSITY

SCHOOL OF MATHEMATICS AND STATISTICS

HONOURS PROJECT

TITLE: Quantum Error-Correcting Codes

AUTHOR: Alexandre Conlon

SUPERVISOR: Jason Crann

DATE: December 19, 2018

1

QUANTUM ERROR-CORRECTING CODES

ALEXANDRE CONLON

Abstract. This paper is an introduction to quantum error-correcting codes. It begins bylaying the mathematical framework with which it then develops fundamental concepts inquantum information theory. Following this, it proves important results to the study ofquantum error correction. The discussion then centres on a paper by Calderbank et al.[1], which presents a geometric formalism to produce quantum error-correcting codes via agroup theoretic framework. Finally, the paper demonstrates that quantum error correctingcodes can be constructed from classical error-correcting codes and inherit a portion of theirerror-correcting properties.

1. Introduction

Quantum physics is renowned for its property of wave-particle duality, famously demon-strated by the double slit experiment where it shows that observing (interacting with) aparticle which exhibits a wave-like behaviour will induce a particle-like behaviour in theparticle immediately after the observation (interaction).

Quantum computers work by exploiting the wave-like behaviour of particles. However,this wave-like behaviour is subject to involuntary observations (interactions) by its environ-ment during computations which may cause errors. In fact, this is a major hurdle in theconstruction of a reasonably sized quantum computer. So far, we have only managed to con-struct quantum computers which are small enough to e�ciently model classical computers.This motivates the study quantum error correction so that we may one day build a largerquantum computer which can out perform our current classical ones.

We begin by covering the mathematical concepts we require in order to introduce basicideas in quantum theory, such as quantum states and entanglement. We assume the readeris familiar with elementary linear algebra and group theory, however we assume no priorknowledge of quantum theory. We then present concepts of quantum information theory,such as the qubit and quantum channels. Next, we focus our attention to quantum codes anddiscuss the conditions in which errors are correctable on quantum codes. Following this, wedescribe a geometric formalism for producing quantum codes, and we show that quantumerror-correcting codes can be built from classical error-correcting codes. Finally, we presenttwo examples of quantum error-correcting codes, the first of which is constructed from aclassical error-correcting code, and the second is the five qubit code.

2. Mathematical Preliminaries

Definition 2.1 (Trace). The trace of an n⇥n matrix A is the sum of its diagonal entries{Aii}n

i=1, and is denoted

tr(A) :=nX

i=1

Aii.

2

QUANTUM ERROR-CORRECTING CODES 3

Definition 2.2 (Sesquilinear Inner-Product). For a finite-dimentional complex vector spaceH, the function h·, ·i : H ⇥H ! C is a sesquilinear inner-product over H if

i) h + �, 0 + �0i = h , 0i+ h ,�0i+ h�, 0i+ h�,�0iii) h↵ , ��i = ↵�h ,�i

For all ,�, 0,�0 2 H and for all ↵, � 2 C, where ↵ is the complex conjugate of ↵.

Any sesquilinear inner product on H induces a norm via k k =p

h , i.

Definition 2.3 (Hilbert space). A Hilbert space is a finite-dimensional vector space Hequipped with a sesquilinear inner-product.

For Hilbert spaces H, K, let h·, ·iH : H ⇥ H ! C and h·, ·iK : K ⇥ K ! C be theinner products over H and K respectively, and let L(H,K) denote the space of linear mapsA : H ! K. The adjoint of an element A 2 L(H,K) is the unique operator A* 2 L(K,H)satisfying

h�, A* iH = hA�, iK, for every � 2 H, 2 K.

When H = K, an operator A 2 L(H) := L(H,H) is self-adjoint if A = A*, and normal ifAA* = A*A.

Definition 2.4 (Positive operator). Let H be a Hilbert space. Then A : H ! H is a positiveoperator if h , A i � 0, for all 2 H.

Definition 2.5 (Conjugate linear). Let H and K be Hilbert spaces. Then the map A :H ! K is conjugate linear if

A(� + �) = �A( ) + A(�)

for all � 2 C and ,� 2 H.

Throughout this paper, we are interested in the structure preserving maps between Hilbertspaces called unitary maps (Hilbert-space-isomorphisms).

Definition 2.6 (Unitary map). Let H and K be Hilbert spaces. Then any bijection U 2L(H,K), which preserves the inner product is a unitary map.

We observe that, U 2 L(H,K) is unitary if and only if for all ,� 2 H :

hU , U�iK = hU*U ,�iK = h ,�iH.

Definition 2.7 (Orthogonal Complement). Let M ✓ H be a subspace of H. Then the set

M? := { 2 H | for all � 2 M : h�, i = 0}is the orthogonal complement of M in H. Then M? is a subspace such that

H = M �M?.

Definition 2.8 (Orthogonal Projection). Let H be a Hilbert space. A projection is anoperator P 2 L(H) such that P = P 2 = P *.

If P 2 L(H) is a projection, then (PH)? = (1 � P )H, so the projection (1 � P ) is theorthogonal complement of P . As above, it then follows that

H = PH � (1� P )H.

4 ALEXANDRE CONLON

Moreover, if { i}ki=1 is an orthonormal basis of M = PH then

P� =kX

i=1

h i,�i i, � 2 H.

Definition 2.9 (Spectrum). The set of eigenvalues {�1,�2, ...,�n} of an operator A 2 L(H)is the spectrum of A, denoted �(A).

Given a normal matrix A and a function f : �(A) ! C, it is possible to make sense off(A). Let A =

P

��P� be the spectral decomposition of the normal matrix A. Then define

f(A) :=X

�

f(�)P�.

We observe that when f is a polynomial,

f(z) = a0 + a1z + · · ·+ anzn,

thenf(A) = a0 + a1A+ · · ·+ anA

n.

We can use this idea to define the square root of a positive operator. Suppose we have apositive operator E with spectral decomposition E =

P

��P�, then we may define

pE =

X

�

p�P�,

as each eigenvalue � � 0.

Example 2.10. Suppose we have an operator A where

A = diag(�1, ...,�n) =

0

@

�1 0. . .

0 �n

1

A ,

with �i � 0, then

pA = diag(

p�1, ...,

p�n) =

0

B

@

p�1 0

. . .

0p�n

1

C

A

.

Remark 2.11. An operator A is positive if and only if there exists B such that A = B*B.Then A* =

�

B*B�*

= B*B = A, thus positive implies self-adjoint.

In this case, one may take B =pA, since

B*B =pA*

pA =

pApA = A.

Proposition 2.12 (Polar Decomposition). Let A 2 L(H), then there exists a unitary U 2L(H) such that

A = UpA*A.


3. Tensor Products

3.1. Tensor Product of Hilbert Spaces. For Hilbert spaces H1 and H2, consider the set,T , of all finite linear combinations of elements in H1 ⇥H2, defined by

T :=n

JX

j=1

aj( j,�j) | a1, ..., aJ 2 C, j 2 H1,�j 2 H2

o

,

and the equivalence relation ⇠ on T such that

1)( ,�1 + �2) ⇠ ( ,�1) + ( ,�2),

2)( 1 + 1,�2) ⇠ ( 1,�) + ( 2,�),

3)�( ,�) ⇠ (� ,�) ⇠ ( ,��), for all � 2 C.Then tensor products between Hilbert spaces are defined as follows.

Definition 3.1 (Tensor Product Space). The tensor product of Hilbert spaces H1 and H2

is defined asH1 ⌦H2 := T/ ⇠

whereD

JX

j=1

cj( j,�j) ,KX

k=1

dk( 0k,�

0k)E

H1⌦H2

:=JX

j=1

KX

k=1

cjdkh j, 0kiH1

h�j, 0kiH2

with respect to which the closure is taken. So, we get the tensor product space⇣

H1 ⌦H2 , h·, ·i⌘

.

3.2. Tensor Product of Vectors. In the subsection above, we defined a tensor productspace as the space of equivalence classes corresponding to ⇠. Perhaps unsurprisingly, wedefine the tensor product between 2 H1 and � 2 H2 by their corresponding equivalenceclass.

Definition 3.2 (Tensor Product of Vectors). The tensor product between vectors is definedby

⌦ � := [( ,�)]⇠ 2 H1 ⌦H2.

Such a tensor is a simple tensor. The inner product on simple tensors in H1 ⌦H2 is simplydefined by

h ⌦ �, 0 ⌦ �0iH1⌦H2:= h , 0iH1

· h�,�0iH2

Any element of H1 ⌦H2 is therefore of the formPn

i=1ci i ⌦ �i.

To gain a feel for the mechanics of the tensor product we introduce what is known as theKronecker product. It o↵ers an explicit representation of the tensor product.

Definition 3.3. Let be an element of the n-dimensional Hilbert space Cn and � bean element of an m-dimensional Hilbert space Cm. Then the Kronecker product, which

6 ALEXANDRE CONLON

represents the tensor product, is defined by

⌦ � =

0

@

1

... n

1

A⌦

0

@

�1

...�m

1

A =

0

B

B

B

B

B

B

B

B

B

B

@

1 ·

0

@

�1

...�m

1

A

...

n ·

0

@

�1

...�m

1

A

1

C

C

C

C

C

C

C

C

C

C

A

=

0

B

B

B

B

B

B

B

B

B

B

B

B

@

1 · �1

... 1 · �m

...

... n · �1

... n · �m

1

C

C

C

C

C

C

C

C

C

C

C

C

A

.

This explicit view given by the Kronecker product can help to determine useful propertiesof the tensor product.

Theorem 3.4. Let H1, H2 be n- and m-dimensional Hilbert spaces, respectively. Let {ei}ni=1

and {fi}mi=1 be orthonormal basis for H1 and H2, respectively. Then

(1) {ei ⌦ fj}n,mi,j=1 is an orthonormal basis for H1 ⌦H2;

(2) dim(H1 ⌦H2) = dim(H1) · dim(H2);

(3) H1 ⌦H2 :=n nP

i=1

mP

j=1

aijei ⌦ fj | aij 2 Co

.

Remark 3.5. Some practical rules for tensor products, which follow directly from ⇠ :

(1) ( + a�)⌦ ( 0 + b�0) = ⌦ 0 + b ⌦ �0 + a�⌦ 0 + ab�⌦ �0

where ,� 2 H1, 0,�0 2 H2 and a, b 2 C;

(2) for all 2 H1 ⌦H2, there is an aij 2 C such that :=P

i,jaijei ⌦ fj

where {ei}ni=1 and {fi}m

i=1 are orthonormal basies for H1 and H2, respectively.

3.3. Tensor Product of Operators. So far we have seen tensor products between Hilbertspaces, H1⌦H2, and tensor products between vectors, ⌦�. We now look at tensor productsbetween linear maps A and B, which we denote A⌦ B.

Definition 3.6 (Tensor Product of Linear Operators). Let A : H1 ! H1 and B : H2 ! H2

be linear operators. Then the tensor product A⌦ B is the unique linear operator

A⌦ B : H1 ⌦H2 ! H1 ⌦H2

satisfying(A⌦ B)( ⌦ �) := A ⌦ B�, for all 2 H1,� 2 H2.

Theorem 3.7. If A,B are self-adjoint operators, then A⌦ B is a self-adjoint operator.

4. Dirac Notation

We begin by providing a result which motivates and justifies the use of Dirac notation.Consider 2 H. Define a linear map

T :H ! C� 7! h ,�i.


This is linear since for a 2 C, and ⇠, ⌘ 2 H we have

T (a⇠ + ⌘) = h , a⇠ + ⌘i= ah , ⇠i+ h , ⌘i= aT (⇠) + T (⌘).

Moreover,

k T k1 := sup�2H

|T (�)|k � k = sup

�2H

|h ,�i|k � k sup

�2H

k k · k � kk � k = k k,

where the last inequality holds by Cauchy-Schwartz, and�

�

�

�

T

✓

k k

◆

�

�

�

�

=

k k |T ( )| =

k k |h , i| = k k

implieskT k1 � k k.

Thus, kT k1 = k k.

Theorem 4.1. Every T 2 L(H,C) can be uniquely written as

T = T .

Proof. If T = 0L(H,C), then take = 0H, and we are done. If T 6= 0L(H,C), then ker(T ) is aproper linear subspace of H, and

H = ker(T )� ker(T )?,

with ker(T )? 6= {0}. Then we may pick some ⇠ 2 ker(T )? such that ⇠ 6= 0H, and withoutloss of generality we may assume k⇠k = 1. Take = T (⇠)⇠ 2 ker(T )? ✓ H, and let ⌘ 2 H.Then,

T (⌘)� T (⌘) = h , ⌘i � T (⌘)k⇠k2

= hT (⇠)⇠, ⌘i � T (⌘)h⇠, ⇠i= h⇠, T (⇠)⌘i � h⇠, T (⌘)⇠i= h⇠, T (⇠)⌘ � T (⌘)⇠i.

Then we observe that

T⇣

T (⇠)⌘ � T (⌘)⇠⌘

= T (⇠)T (⌘)� T (⌘)T (⇠) = 0

implying thatT (⇠)⌘ � T (⌘)⇠ 2 ker(T ),

and since ⇠ 2 ker(T )?, then

T (⌘)� T (⌘) = h⇠, T (⇠)⌘ � T (⌘)⇠i = 0.

Hence, for T 2 L(H,C), there exists 2 H such that T = T . Now we show that isunique.

8 ALEXANDRE CONLON

Let 1, 2 be two such constructed vectors so that

T 1= T = T 2

.

Then, for all ⌘ 2 H

0 = T 1� T 2

= h 1, ⌘i � h 2, ⌘i= h 1 � 2, ⌘i

and thus, 1 = 2.

Hence, for every T 2 L(H,C) there exists a unique 2 H such that T = T . ⇤Remark 4.2.

T� (�) = h� ,�i = �h ,�i = �T (�), for all � 2 C.Together with the proof above, this shows that H 3 7! T 2 L(H,C) is a conjugate linearunitary map.

4.1. Dirac bra-ket Notation. Dirac’s “bra-ket” notation takes full advantage of Theorem4.1. Since for any T 2 L(H,C) there exists a unique 2 H such that

T (⌘) := h , ⌘ior

T (·) := h , ·i : H ! C,Dirac uses the shorthand notation:

T = h |where the h·| is referred to as a bra. Moreover, Dirac notation expresses 2 H as | i, wherethe |·i is known as a ket.

Then,T (�) = h ||�i = h |�i := h ,�iH.

So, we may think of | i as the column vector 2 H, and h | as the conjugate row vector| i*. Dirac notation provides a useful shorthand for the standard basis of C2:

|0i :=✓

10

◆

, |1i :=✓

01

◆

where the number found in the ”ket” signifies which position the non-zero entry is in. Thenfor any | i 2 C2, we may write

| i = h0| ip

h | i|0i+ h1| i

p

h | i|1i.

This idea extends to Cn for any n 2 N; the ith standard basis vector for Cn can be writtenas |ii where i indicates the position of the non-zero entry. Moreover, for the purposes ofquantum information theory, it is useful to write the position i as its base-2 representationwith respect to the dimensionality of the tensor product space.

Example 4.3. Suppose we have the tensor product space (C2)⌦3 which has dimension 23 = 8,then we represent the standard basis vector |5i as |101i. A nice feature of this notation isthat |101i = |1i⌦ |0i⌦ |1i. This will come in handy for applying operations to multi-partitequantum systems, and developing quantum error-correcting codes from classical codes.


For | i, |�i 2 H, | ih�| 2 L(H) is defined by�

| ih�|��

|⇠i�

:= h�|⇠i| i and is a rank-1projector onto C| i. In particular, if | i = |�i and k| ik = 1, then | ih | is the rank-1orthogonal projection onto C| i.

Example 4.4. Consider the Hilbert space H = C2n. Let |eii denote the ith standard basisvector for C2n. Then we can define idH 2 L(H) by

idH :=2nX

i=1

|eiihei|.

5. Quantum States and Operations

We begin by stating Postulate 1 of quantum mechanics to motivate the subsequent dis-cussion.

With every quantum system, there is an associated complex Hilbert space,known as the state space of the system. The states of the system are allpositive linear maps ⇢ : H ! H for which

tr(⇢) = 1

and can be completely described by its state vector, which is a unit vector inthe system’s state space.

Suppose we had a machine which could produce specified pure quantum vector states.Suppose further that this machine is error-prone and with certain probabilities produces arange of di↵erent pure states around the one specified. The set of pure vector states | ii onemay observe with respective probability pi, is denoted by {pi, | ii}, and is referred to as anensemble of pure states.

A useful object in the study of the ensemble {pi, | ii}n�1i=0 is its density operator, defined

by

⇢ :=n�1X

i=0

pi| ii(| ii)⇤ =n�1X

i=0

pi| iih i|

Theorem 5.1 (Characterization of Density Operators). An operator ⇢ 2 L(H) is the densityoperator associated to some ensemble {pi, | ii} if and only if ⇢ satisfies both of the followingconditions:

1. (Trace Condition): tr(⇢) = 1;2. (Positivity Condition): ⇢ is a positive operator.

10 ALEXANDRE CONLON

Proof. Suppose ⇢ =P

i

pi| iih i| is the density operator associated to some ensemble {pi, | ii}.

Then,

tr(⇢) = tr⇣

n�1X

i=0

pi| iih i|⌘

=n�1X

i=0

pi · tr(| iih i|)

=n�1X

i=0

pih i| ii

=n�1X

i=0

pi

= 1.

So, condition (1.) is satisfied. Suppose |�i is some vector in the state space. Then,

h�, ⇢�i =X

i

pih� | iih i | �i

=X

i

pi|h� | ii|2

� 0.

Hence, condition 2. is satisfied.Conversely, suppose ⇢ is an operator satisfying conditions (1.) and (2.). Since ⇢ is positive,

it must have spectral decomposition

⇢ =X

j

�j|jihj|

where the vectors |ji are orthogonal, and �j are real, non-negative eigenvalues of ⇢.Then, from condition (1.), we get that

P

j�j = 1. Therefore, a system in state |ji with

probability �j will have density operator ⇢. ⇤This theorem gives us an intrinsic characterization of density operators and motivates the

following definition.

Definition 5.2 (Density Operator, Pure/Mixed State). An operator ⇢ 2 L(H) is a densityoperator if it is positive and tr(⇢) = 1. A quantum system with vetor state | i which isexactly known is a pure state, and ⇢ = | ih |. On the other hand, if we have the systemensemble {pi, | ii}, where probabilities pi < 1, for all i, then

⇢ =X

i

pi| iih i|

and the system is in a mixed state.

5.1. Evolution. Postulate 2 of quantum theory states:

The evolution of a closed quantum system is described by a unitary transfor-mation. That is, the vector state | i of the system at time t1 is related tothe vector state | 0i of the system at time t2 by a unitary operator U which


depends only on the times t1 and t2,

| 0i = U | i.We observe that the unitary evolution of quantum systems arrises naturally from the

condition that both | i and | 0i be vector states for H and H 0, respectively:

h | i = 1 = h 0| 0i= h 0| · | 0i= | 0i⇤ · | 0i= (U | i)⇤U | i= h |U ⇤U | i.

Hence,U ⇤U = 1

implying that U is unitary. Furthermore, if the system has corresponding ensemble {pi, | ii},we can describe the state of the system following an evolution U by

⇢0 =n�1X

i=0

piU | ii(U | ii)⇤ =n�1X

i=0

piU | iih i|U ⇤ = U⇢U ⇤.

5.2. Measurement. Postulate 3 of quantum mechanics states that quantum measurementsare described by a collection {Mm}n

m=1 of operators called a measurement system. Thesemeasurement operators act on the state space of the quantum system being measured. Theindex m refers to the potential measurement outcomes in an experiment. If the vector stateof some system is immediately before the measurement Mm then the probability that moccurs is given by

p(m) = h |M ⇤mMm| i = hMm ,Mm i = kMm k2

and the vector state of the system after the measurement is

Mm

kMm kthen we have

h | i = 1 =nX

m=1

p(m) =nX

m=1

h ,M ⇤mMm i = h ,

nX

m=1

M ⇤mMm i

which motivates the completeness equation, measurement operators must satisfynX

m=1

M ⇤mMm = I.

Definition 5.3 (Positive Operator-Valued Measure; POVM). A POVM is a finite set ofpositive operators {Ei}n

i=1 such thatnX

i=1

Ei = I.

Proposition 5.4. If {Mi}ni=1 is a measurement system, then {Ei}n

i=1 is a POVM, whereEi = M *

i Mi. Conversely, if {Ei}ni=1 is a POVM, then {

pEi}n

i=1 is a measurement system.

12 ALEXANDRE CONLON

Definition 5.5 (Projector Valued Measure; PVM). A PVM is a POVM {Ei}ni=1 such that

each Ei is an orthogonal projection, i.e.

Ei = E*i = E2

i .

Definition 5.6 (Observable). An observable is a self-adjoint operator A : H ! H.

By the spectral theorem, any observable A can be written as A =Pn

i=1�i|eiihei|, where

�i is an eigenvalue of A and {|eii}ni=1 is an orthonormal basis of eigenvectors of A. Then

{|eiihei|} is the PVM associated to the observable A.

Example 5.7. Consider the measurement operators M0 = |0ih0| and M1 = |1ih1| acting on| i = 0|0i+ 1|1i, where 0, 1 2 C. Then the probability of observing the vector state |0iis given by p(0) = kM0| ik2 with

M0| i = |0ih0|✓

0

1

◆

=

✓

1 00 0

◆✓

0

1

◆

=

✓

0

0

◆

= 0|0i.

This givesp(0) = k 0|0ik2 = | 0|2,

and the state of the system after the measurement is

M0| ikM0| ik

=M0| ik 0|0ik

= 0|0i| 0|

= 0

| 0||0i.

Similarly, p(1) = | 1|2 and the state after the measurement is

1

| 1||1i.

Note that the measurement statistics for the two states ( 1/| 1|)|1i and |1i are the same,that is,

kMj

1

| 1||1ik2 = kMj|1ik2, for all j.

In this case, the two states are equal up to global phase factor 1/| 1|.5.3. Distinguishability of states.

Definition 5.8 (Distinguishable). A set of states {| ii}mi=1 is distinguishable if there exists

a measurement system {Mi}ni=1, n � m, such that kMi| jik2 = �i,j for i, j 2 {1, ...,m}.

Theorem 5.9. A set of states {| ii}mi=1 is distinguishable if and only if, for all i, j 2

{1, ...,m},| ii ? | ji.

Proof. ()) Suppose we have a measurement system {Mi}ni=1 such that kMi| jik = �ij, for

i, j 2 {1, ..., n}. Consider | 1i and | 2i. Then | 2i can be expressed as | 2i = ↵| 1i+ �|⌘i,where |⌘i ? | 1i, k|⌘ik = 1. Since 1 = k k2 = |↵|2 + |�|2, we have

1 = kM2| 2ik2 = kM2

�

↵| 1i+ �|⌘i�

k2

= |�|2kM2|⌘ik2

|�|2k|⌘ik2

1.


Then 1 |�|2 1, which implies |�|2 = 1 and ↵ = 0. Therefore, | 2i and |⌘i are collinear.Hence, | 2i ? | 1i.

(() Let Mi be the orthogonal projection onto the one-dimensional subspace spanned by| ii. Then, Mi = M *

i = M *i Mi, for i = 1, ..., n, and

Pn

i=1M *

i Mi is the orthogonal projectiononto span{ 1, ..., n}. Let M0 be the orthonormal projection onto span{ 1, ..., n}?. ThenPn

j=0M *

j Mj =Pn

j=0Mj = I. Furthermore, Mi|�ji = �ij| ji for all i, j 2 {1, ..., n}, so that

kMi|�jik2 = �ij for all i, j 2 {1, ..., n}. Hence, {Mi}ni=1 is a measurement system. ⇤

5.4. The Qubit. In the classical setting information is stored as bits, where a single bitcan take on one of two values: 0 or 1. We refer to the value of the bit as the state of thebit. When one considers the state of an n-bit system, one is concerned with the cartesianproduct of the state of each bit, i.e. the state of the classical system can be viewed as ann-dimensional vector {ai}n

i=1 2 Zn2 where ai 2 Z2.

In the quantum setting however, information is stored as quantum bits, or qubits. Asingle qubit | i is described by a unit vector in the Hilbert space H = C2. Then H is a2-dimensional state space.

We have just described the simplest state space possible for quantum information, i.e.a single qubit. For multi-qubit systems we must make use of tensor products. A 2-qubitsystem is described by the tensor product between two individual single qubit systems. Forthe two qubits | 1i 2 C2 and | 2i 2 C2 we get the vector state of the multi-qubit systemdescribed by

| 1i ⌦ | 2i 2 C2 ⌦ C2

If | 1i = a1|0i+b1|1i and | 2i = a2|0i+b2|1i, then by the rules for tensor products outlinedearlier we get:

| i = | 1i ⌦ | 2i = a1a2|0i ⌦ |0i + a1b2|0i ⌦ |1i + b1a2|1i ⌦ |0i + b1b2|1i ⌦ |1i= a1a2|00i + a1b2|01i + b1a2|10i + b1b2|11i.

We observe that,

h | i =p

|a1a2|2 + |a1b2|2 + |b1a2|2 + |b1b2|2

=p

(|a1|2 + |b1|2)(|a2|2 + |b2|2)= h 1 | 1i · h 2 | 2i= 1.

Hence, | i is a vector state for the system C2 ⌦ C2. This extends to systems of n-qubits.

Example 5.10. Let H1 = H2 = C2. Then

|01i � |10i :=✓

10

◆

⌦✓

01

◆�

�✓

01

◆

⌦✓

10

◆�

=

0

B

B

@

0100

1

C

C

A

�

0

B

B

@

0010

1

C

C

A

=

0

B

B

@

01�10

1

C

C

A

2 C2 ⌦ C2.

14 ALEXANDRE CONLON

Remark 5.11. This element cannot be written as a simple tensor. We observe that for ,� 2 C2, if

⌦ � =

0

B

B

@

1 · �1

1 · �2

2 · �1

2 · �2

1

C

C

A

=

0

B

B

@

01�10

1

C

C

A

,

then 1 · �2 = 1 implies 1 6= 0,

so 1 · �1 = 0 implies �1 = 0.

But 2 · �1 = �1 implies �1 6= 0,

which is a contradiction. Thus, for all ,� 2 C2

⌦ � 6=

0

B

B

@

01�10

1

C

C

A

.

In other words,H1 ⌦H2 * { ⌦ � | 2 H1, � 2 H2}.

Definition 5.12 (Separable). An element 2 H1 ⌦H2 is separable if there exists � 2 H1,and 2 H2 such that

= �⌦ .

If is not of this form, then it is non-separable.

Definition 5.13 (Entanglement). A pure state ⇢ 2 L(H1 ⌦ H2) is called non-entangle ifthere exist pure states ⇢� 2 L(H1) and ⇢ 2 L(H2) such that

⇢ = ⇢� ⌦ ⇢ .

Remark 5.14. Let | i 2 H1 ⌦H2 be separable, that is, there exists | i 2 H1 and |�i 2 H2

such that | i = | i ⌦ |�i. Then the pure state ⇢ 2 L(H1 ⌦H2) corresponding to | i is⇢ = | ih |

=�

| i ⌦ |�i��

h |⌦ h�|�

= | ih |⌦ |�ih�|= ⇢ ⌦ ⇢�.

Thus, separable elements in H1⌦H2 have corresponding non-entangled pure states in L(H1⌦H2). Moreover, non-separable elements in H1⌦H2 have corresponding entangled pure statesin L(H1 ⌦H2).

5.5. Environments and Quantum Channels.

Definition 5.15 (Partial Trace). Consider L(H1 ⌦ H2). We define the partial trace overL(H2) by

trL(H2):= IL(H1) ⌦ tr : L(H1 ⌦H2) ! L(H1),

where for a simple element ⇢⌦ � 2 L(H1 ⌦H2)

⇢⌦ � 7! tr(�)⇢.


Definition 5.16 (Reduced Density Operator). Suppose we have a quantum system H1⌦H2

with corresponding density operator ⇢ 2 L(H1⌦H2). We define the reduced density operator,⇢H1

, as the partial trace of ⇢ over L(H2), that is

⇢H1:= trL(H2)(⇢).

The dynamics of a closed quantum system can be described by a unitary transformation.Conceptually, one can think of the unitary transformation as a box into which the inputstate enters and from which the output exits.

A natural way to describe the dynamics of an open quantum system is to regard it asarising from an interaction between our system of interest, called the principal system, andan environment in which the principal system lives, which together form a closed system. Inother words, suppose we have a principal system in state ⇢, and in an environment in state⇢env, then ⇢⌦ ⇢env describes the state of the closed system. In general, the final state of theprincipal system, E(⇢), may not be related by a unitary transformation to the initial state⇢. We assume that the system-environment input state is a non-entangled state, ⇢ ⌦ ⇢env.Thus, we perform a partial trace over the environment to obtain the reduced state of theprincipal system alone

E(⇢) = trenv

⇥

U(⇢⌦ ⇢env)U*⇤

,

which motivates the following definition.

Definition 5.17 (Quantum Channel). Let H1, H2 be state spaces. A quantum channel is alinear mapping E : L(H1) ! L(H2) satisfying

1. E is trace preserving ;

tr�

E(⇢)) = tr(⇢), for all ⇢ 2 L(H1).

2. E is completely positive;

id⌦ E : L(Ck)⌦ L(H1) ! L(Ck)⌦ L(H2)

is a positive map, for all k.

Mathematically a quantum channel is a completely-positive trace-preserving map, other-wise referred to as a CPTP map [5].

Theorem 5.18. The operator E : L(H1) ! L(H2) is a quantum operator if and only if

E(⇢) =nX

i=1

Ei⇢E⇤i

for some collection {Ei}ni=1 ✓ L(H1, H2) satisfying

Pn

i=1E⇤

i Ei = I.

For the proof see [4].

Proposition 5.19. Let E(⇢) =Pn

i=1Ei⇢E*

i , and let U be an n⇥n unitary matrix with complexentries uij where i refers to the row position and j refers to the column position. Set Fi =Pn

j=1uijEj. Then,

E(⇢) =nX

i=1

Fi⇢F*i .

16 ALEXANDRE CONLON

Proof.nX

i=1

Fi⇢F*i =

nX

i=1

⇣

nX

j=1

uijEj

⌘

⇢⇣

nX

j=1

uijEj

⌘⇤

=nX

i=1

nX

j,k=1

uijuikEj⇢E*k

=nX

j,k=1

⇣

nX

i=1

uijuik

⌘

Ej⇢E*k

=nX

j,k=1

Ej⇢E*k

= E(⇢)

sincenP

i=1

uijuik equals 0 if j 6= k and equals 1 if j = k. ⇤

6. Quantum Error-Correcting Conditions, and Pauli Operators

We begin by drawing analogies from classical coding theory to build an intuition for thetheory of quantum-error correction. A classical binary linear code is a subspace Ccl of alarger binary vector space Zn

2 . Suppose we send a codeword c through a noisy channel andget y = c + e on the receiving end, where e 2 Zn

2 . Then the error may be detected bydetermining whether or not y is in Ccl by performing a syndrome measurement on y, wherethe possible error-syndromes correspond to distinct cosets of Ccl. Recovery of the originalcodeword is possible once we have identified the coset of Ccl in which y lies.

Analogously, a quantum code, C, is a subspace of a larger complex Hilbert space. Supposewe send a codeword in state ⇢ through a noisy channel and get � = E(⇢) on the receiving end,where E is a quantum operator. Then the quantum-error may be detected by determiningwhether or not � lies in C by a performing a syndrome measurement on �, where thepossible error-syndromes correspond to distinct orthogonal subspaces to C in the largercomplex Hilbert space. Recovery of the original codeword is possible once we have identifiedthe subspace in which � lies.

Figure 1. The box on the left represents the quantum (classical) code withina code space represented by the left circle, the boxes on the right representthe possible subspaces (cosets) in which a transmitted codeword may land.(Figure taken from [3])

Definition 6.1. Let C be a subspace of the complex Hilbert space H, and let E : L(H) !L(H) be a quantum channel, where E = {Ei}n

i=1. If there exists an operation R : E(L(C)) !


L(C) such that, for any density operator ⇢ 2 L(C), we have

R�

E(⇢)�

= ⇢,

then R corrects E on quantum code C, and {Ei}ni=1 is a set of correctable errors on C.

Theorem 6.2 (Quantum error-correcting conditions). Let P be the projector onto somequantum code C, and suppose E is a quantum channel with elements {Ei}n

i=1. Then thereexists an error-correction operation R correcting E on C if and only if

PE*iEjP = ↵ijP

for some complex self-adjoint matrix ↵.

The elements of the noise operation E = {Ei}ni=1 are known as errors, and if R corrects E

on C then {Ei}ni=1 is a correctable set of errors.

Proof. Suppose {Ei}ni=1 is a set of operators which satisfy the error-correcting conditions.

Then each pair of elements satisfies

PE*iEjP = ↵ijP.

By assumption, the matrix ↵ is self-adjoint, and thus can be diagonalized

d = u⇤↵u

where u is unitary and d is diagonal. Now define the operators,

Fk :=nX

i=1

uikEi

then as was shown earlier {Fk} is also a set of quantum operators for E . By substitution weget

PF *kFtP = P

⇣

nX

i=1

u⇤ikE

⇤i

⌘⇣

nX

j=1

ujtEj

⌘

P

=nX

i=1

nX

j=1

u⇤ikujtPE⇤

i EjP

=nX

i=1

nX

j=1

u⇤ikujt↵ijP

=nX

i=1

nX

j=1

u⇤ik↵ijujtP

= dktP.

We observe that if k 6= t, then dkt = 0.· Detection:Now we define the syndrome measurement using PF *

kFtP = dktP and polar decomposition.From the polar decomposition of FkP we have

FkP = Uk

p

PF *kFkP = Uk

p

dkkP =p

dkkUkP

18 ALEXANDRE CONLON

where Uk is some unitary matrix. So Fk can be seen as a rotation of the coding subspaceinto the subspace defined by the projector

Pk := UkPU *k =

1pdkk

FkPU *k

where if dkk = 0 then Pk = 0. Then we note that, for j 6= k, we have

PjPk = P *j Pk =

1p

djjdkk

UjPF *j FkPU *

k = 0.

· Recovery:Given some output state � we can recover the original input state ⇢ 2 L(C) by applying

a unitary operator U *k . Thus the error-correcting operation R, i.e. combined detection and

recovery, corresponds to

R(�) =nX

k=1

U *kPk�PkUk +Q�Q,

where Q := I �Pn

k=1Pk. Since

nX

k=1

�

U *kPk

�*�

PkUk

�

+Q*Q =nX

k=1

PkUkU*kPk +Q2

=nX

k=1

Pk + I �nX

k=1

Pk

= I,

we know that R is a CPTP map. Moreover, observe that

QPj =⇣

I �nX

k=1

Pk

⌘

Pj = Pj � P 2j = 0,

since PkPj = 0, for all k 6= j.


For states ⇢ which have undergone some error resulting in E(⇢), correction is possible with

R(E(⇢)) =nX

k=1

U *kPk

⇣

nX

t=1

Ft⇢F*t

⌘

PkUk +Q⇣

nX

t=1

Ft⇢F*t

⌘

Q

=nX

k,t=1

U *kPkFt⇢F

*t PkUk +

nX

t=1

QFtP⇢PF *t Q

=nX

k,t=1

U *k

�

P *k

�

FtP⇢PF *t

�

Pk

�

Uk +nX

t=1

dttQPtUt⇢U*t PtQ

=nX

k,t=1

U *k

⇣

Uk

1pdkk

PF *k

⌘

FtP⇢PF *t

⇣

FkP1pdkk

U *k

⌘

Uk

=nX

k,t=1

1pdkk

dktP⇢ dtkP1pdkk

=nX

k=1

1pdkk

dkkP⇢Pdkk

1pdkk

=nX

k=1

dkkP⇢P

=nX

k=1

dkk⇢

= ⇢,

where in the fourth equality we use that QPt = 0. For the converse see Theorem 10.2 of[3]. ⇤

Theorem 6.3. Let C be a quantum code with error-correcting operation R corresponding tothe quantum channel E with elements {Ei}n

i=1. Let F be the quantum channel with elements{Ft}m

t=1 where Ft =Pn

i=1�tiEi for some matrix � with complex entries. Then R also corrects

for error process F .

Proof. The set {Ei}ni=1 must satisfy the quantum error-correcting condition PE*

iEjP = ↵ijP .From the proof of the previous theorem, without loss of generality we may take ↵ to be adiagonal matrix, ↵ij = dij. Moreover, we can choose Uk so that EkP =

pdkkUkP . Then let

Pk :=1

pdkkEkPU *

k . Observe that, for states ⇢ 2 L(C), we have

U *kPkEi

p⇢ = U *

kP*kEiP

p⇢

=1pdkk

U *kUkPE*

kEiPp⇢

=dkiP

p⇢

pdkk

, equals 0 for i 6= k

= �kidkkP

p⇢

pdkk

= �kip

dkk

p⇢

20 ALEXANDRE CONLON

Similarly,p⇢E*

iPkUk = �kipdkk

p⇢.

By substituting Ft =Pn

i=1�tiEi, we get

U *kPkFt

p⇢ = U *

kPk

⇣

nX

i=1

�tiEi

⌘p⇢

=nX

i=1

�tiU*kPkEi

p⇢

=nX

i=1

�ti�kip

dkk

p⇢

= �tk

p

dkk

p⇢.

Similarly,p⇢F *

t PkUk = �tk

pdkk

p⇢. Then,

R(F(⇢)) =nX

k=1

U *kPk

⇣

mX

t=1

Ft⇢F*t

⌘

PkUk

=nX

k=1

mX

t=1

U *kPkFt

p⇢p⇢F *

t PkUk

=nX

k=1

mX

t=1

(�tk

p

dkk

p⇢)(�tk

p

dkk

p⇢)

=nX

k=1

mX

t=1

|�tk|2dkk⇢

= ↵⇢

= ⇢

where ↵ = 1 since R,F are CPTP maps. ⇤

6.1. The Pauli Matrices. An important set of unitary operators to the field of quantuminformation theory is the set of 2⇥ 2 Pauli matrices. The three Pauli matrices are denotedX, Z, Y , and are defined as follows:

X =

✓

0 11 0

◆

, Y =

✓

0 �ii 0

◆

, Z =

✓

1 00 �1

◆

.

It is easy to see that X*X = Y *Y = Z*Z = I. Hence the Pauli matrices are unitary. Toget an idea of how the Pauli matrices operate, let | i 2 C2. In terms of the standard basis,| i = a|0i+ b|1i for some a, b 2 C, so we get

X| i = X�

a|0i+ b|1i�

=

✓

0 11 0

◆✓

ab

◆

=

✓

ba

◆

= b|0i+ a|1i.

Thus, X interchanges the coe�cients in the standard basis description of | i. In the com-putational interpretation, X acts as the bit flip operation. Similarly, for Y and Z we get

Y | i = �ib|0i+ a|1i,Z| i = a|0i � b|1i.


Then Z is the phase flip operation, as it changes the phase on the |1i coe�cient.Next we find the eigenvalues and eigenvectors of elements X, Z, and Y with respect to

the standard basis.The eigenvalues of X satisfy

det(X � �I) = det

✓

�� 11 ��

◆

= �2 � 1 = (�+ 1)(�� 1)

and hence�X+ = 1, and �X� = �1.

A normalized eigenvector | i = a|0i+ b|1i corresponding to �X+ satisfies

a|0i+ b|1i = �X+

�

a|0i+ b|1i�

= X

✓

ab

◆

= b|0i+ a|1i

implying that a = b. We obtain

X+ =1p2

✓

11

◆

=1p2

�

|0i+ |1i�

.

Similarly,

X� =1p2

✓

1�1

◆

=1p2

�

|0i � |1i�

.

In the literature, these are often denoted as |+i and |�i, respectively. The eigenvalues andeigenvectors of the other Pauli matrices are found in the same way, which gives us

�Y + = 1, Y + =1p2

✓

1i

◆

, �Z+ = 1, Z+ =

✓

10

◆

= |0i,

�Y � = �1, Y � =1p2

✓

1�i

◆

, �Z� = �1, Z� =

✓

01

◆

= |1i.

As a consequence, all Pauli matrices have eigenvalues of ±1.

6.2. The Real Pauli Group. Now we define a group G constructed from Pauli matricesthat will be central to our study of quantum error-correcting codes. Let G = hX, Zi be thegroup generated by X and Z via matrix multiplication. Then, it is easy to check that

X2 = Z2 = I 2 G,

We also have

XZ =

✓

0 11 0

◆✓

1 00 �1

◆

=

✓

0 �11 0

◆

= �✓

1 00 �1

◆✓

0 11 0

◆

= �ZX 2 G

and(XZ)2 = (XZ)(XZ) = �(ZX)(XZ) = �Z(XX)Z = �I 2 G.

Hence, the group G isG = {±I, ±X, ±Z, ±XZ}.

Remark 6.4. Suppose w is in the centre of G, defined by

Z(G) := {g 2 G | gg0 = g0g for all g0 2 G}.Then wX = Xw and wZ = Zw, which implies w = ±I. Thus, Z(G) = {±I}.

22 ALEXANDRE CONLON

We observe that G is a subgroup of the orthogonal group O(2).

7. The Error Group E

The remainder of the discussion is centered on a paper by Calderbank et al. [1], whichpresents a geometric formalism to produce quantum error-correcting codes via a group the-oretic framework.

Let E denote the set of elements e 2 L((C2)⌦n) of the form

e = w1 ⌦ · · ·⌦ wn

where wj is an element of the real Pauli group G = {±I, ±X, ±Z, ±XZ}. Since,aU1 ⌦ bU2 = abU1 ⌦ U2

we may rewrite elements e 2 E as e = ±w1 ⌦ · · · ⌦ wn where the wj now lie in the set{I, X, Z, XZ}.

We define multiplication of elements e1, e2 2 E with e1 = ⌦ni=1ui and e2 = ⌦n

i=1vi, by

e1e2 =� n

⌦i=1

ui

�� n

⌦i=1

vi

�

=n

⌦i=1

uivi

Then, uivi 2 G, for all i = 1, ..., n, so e1e2 2 E for all e1, e2 2 E. Therefore E is closed undermultiplication. The element IE = I⌦n is clearly the identity element of multiplication in E.Let e = ⌦n

i=1wi in E. Then, since w�1i 2 G for all i = 1, ...n, we have that e�1 = ⌦n

i=1w�1i is

in E, for all e 2 E. Thus, E is a group under multiplication.

7.1. Properties of the group E. To construct an element of the group E, we must choosefrom the 4 = 22 elements {I,X, Z,XY } for each of the n tensor entries, which gives us(22)n = 22n elements of the group E. Then, since each element of E has a factor of ±1, E isa group of 2(22n) = 22n+1 orthogonal 2n ⇥ 2n matrices.

The quotient group E/Z(E), plays an important role in the construction of quantum errorcorrecting codes, so we determine Z(E).

Suppose e = ⌦ni=1wi 2 Z(E). Then, let eXj

= ⌦ni=1ui, eZj

= ⌦ni=1vi 2 E, with uj = X,

vj = Z, and ui, vi = I, 8 i 6= j. Since e 2 Z(E), we have that eeXj= eXj

e and eeZj= eZj

e.In particular, we have that wjX = Xwj and wjZ = Zwj. Which implies wj 2 Z(G). Sincej is a arbitrary in {1, ..., n}, this holds for any j = 1, ..., n. So wi = ±I, for all i = 1, ..., n.Thus, Z(E) = {±I}.

With this, we define the quotient group

E := E/Z(E) = {eZ(E) | e 2 E}.

Proposition 7.1. E is abelian.


Proof.

e1e2 = (e1Z(E))(e2Z(E))

= e1(Z(E)e2)Z(E)

= e1(e2Z(E))Z(E)

= (e1e2)Z(E)

= (�e2e1)Z(E)

= (�e2Z(E))(e1Z(E))

= e2(�I)Z(E)e1

= e2Z(E)e1

= e2e1.

Hence, E is abelian.⇤

We can express e 2 E as a 2n-dimensional vector v = (v1, ..., v2n) 2 Z2n2 in the following

way.

If e contains an I on the jth qubit, then vj = 0 = vn+j. If e contains an X on the jth

qubit, then vj = 1 and vn+j = 0. If e contains an Z on the jth qubit, then vj = 0 andvn+j = 1. If e contains both an X and a Z on the jth qubit, then vj = 1 = vn+j. We denotethe vector v 2 Z2n

2 corresponding to e 2 E as v = (a|b), where (a|b) is the concatenation ofvector a 2 Zn

2 corresponding to the position of X operations in e 2 E, and vector b 2 Zn2

corresponding to the position of Z operations in e 2 E. Then, we notice that multiplicationin E corresponds to vector addition modulo 2 in E, that is, for e1, e2 2 E with e1 = (a1|b1)and e2 = (a2|b2) in E, the image of e1e2 in E, is given by

e1e2 = (a1 � a2|b1 � b2)

Thus, we definee1e2 := e1e2.

Definition 7.2 (Weight). The weight of an element e = (a|b) 2 E is defined by

w(e) =nX

i=1

(ai + bi).

The distance between two elements e1, e2 2 E is defined to be the weight of their di↵erence.Suppose we have e1, e2 2 E, then

w(e1e2) = w�

(a1 � a2|b1 � b2)�

=X

i

�

(a1i+ a2i

(mod 2)) + (b1i+ b2i

(mod 2))�

X

i

(a1i+ b1i

(mod 2)) +X

i

(a2i+ b2i

(mod 2))

= w(e1) + w(e2).

Thus, w(e1e2) w(e1) + w(e2).

24 ALEXANDRE CONLON

8. Quantum Error-Correcting Codes

To motivate the following discussion, we describe classical binary linear codes from anunusual perspective. A classical code C is a subspace of Zn

2 . But, Zn2 is also the group of

possible errors on C, i.e. C is a subgroup of the error group. Then an error e is in C whentranslation by e takes codewords to codewords, which means e is an undetectable error. A setof errors is a detectable set if the sum of any two errors from the set lies outside of C. Eventhough the sum can be equal to the trivial error 0, which lies in C and thus undetectable, ithas no e↵ect.

In the quantum setting it is possible for errors to be non-trivial and yet have no e↵ect onthe encoded state. Then we construct quantum codes from two subgroups of the quantumerror group E; the subgroup of undetectable errors, S 0, and a subgroup of S 0, denoted S,composed of errors which have no e↵ect. Then S is equivalent to the 0 error in the classicalsetting.

For this construction we will require that every element of S 0 commutes with S, whichimplies S is abelian. So we need a criterion for when two elements of E commute.

Let A be the 2n⇥ 2n matrix defined by

A =

✓

0 In⇥n

0 0

◆

,

and define the binary quadratic form Q : E ! Z2 by Q(e) 7! eAeT . Then for an elemente = (a|b) 2 E we have

eAeT = (a|b)A(a|b)T

= (a|b)(b|0)T

= a · b+ 0

=nX

i=1

aibi (mod 2).

Hence for e = w1 ⌦ · · · ⌦ wn in E with image e 2 E, Q(e) is the parity of the number ofcomponents wj which equal XZ.

Definition 8.1 (Totally singular). A subspace S ✓ E is said to be totally singular if 8 s 2 Swe have Q(s) = 0.

We now define a symplectic inner product h· , ·iE : E ⇥ E ! Z2, which serves as ourcommutativity criterion. Let ⇤ := A+ AT ,

⇤ =

✓

0 In⇥n

In⇥n 0

◆

,

and let e1 = (a1|b1), e2 = (a2|b2) in E. Then define the the map h·, ·iE : E ⇥ E ! Z2 by

he1, e2iE = e1⇤e2T = a1b2 + b1a2 (mod 2).

Then e1, e2 2 E commute if and only if he1, e2iE = 0. Thus, we get

e1e2 = (�1)he1, e2iEe2e1,

for all e1, e2 2 E.

Definition 8.2 (Totally isotropic). A subspace S ✓ E is totally isotropic if for all s1, s2 2 Swe have hs1, s2iE = 0.


Proposition 8.3. If S ✓ E is a totally singular subspace, then S is totally isotropic.

Proof. Let s, s0 2 S. ThenQ(s) = Q(s0) = 0. Suppose that hs, s0iE 6= 0. Then, s00 = s�s0 2 Sso s00 = (a� a0 | b� b0), and we get

Q(s00) = (a� a0 | b� b0)A(a� a0 | b� b0)T

= (a� a0 | b� b0)(b� b0 | 0)T

=nX

i=1

(ai + a0i)(bi + b0i)

=nX

i=1

aibi +nX

i=1

(aib0i + a0

ibi) +nX

i=1

a0ib

0i

= Q(s) + (a | b)(b0 | a0)T +Q(s0)

= hs, s0iE 6= 0,

implying that s00 /2 S.Therefore, for all s, s0 2 S we have hs, s0iE = 0. Hence, S is totally isotropic. ⇤Next we show that, for all e 2 E, e is diagonalizable. Let uX, uZ, and uY be the unitary

matrices which diagonalize X, Z, and XZ, respectively, that is,

dX = u*XXuX,

dY = u*YXZuY ,

dZ = u*ZZuZ.

Then for e = ⌦ni=1wi in E, define Ue := ⌦n

i=1uwi, and so

U *e eUe =

� n

⌦i=1

u*wi

�� n

⌦i=1

wi

�� n

⌦i=1

uwi

�

=n

⌦i=1

u*wiwiuwi

=n

⌦i=1

dwi

= de.

Thus we can diagonalize elements of E.

Definition 8.4 (Simultaneously diagonalizable). Let {ei}ki=1 be a set of self-adjoint operators

on some Hilbert space H. Then the set {ei}ki=1 is simultaneously diagonalizable if there exists

a unitary operator U such thatU *eiU = di ,

for all i = 1, ..., k, where di is some real diagonal matrix. In other words, the ei’s share ajoint orthonormal eigenbasis.

Theorem 8.5. Let {ei}ki=1 be a set of self-adjoint operators on some Hilbert space H. Then

the set {ei}ki=1 is simultaneously diagonalizable if and only if eiej = ejei, for all i, j = 1, ..., k.

For the proof see [6].Then, since S is a commutative set of self-adjoint operators, it is simultaneously diago-

nalizable and thus each element s 2 S shares a joint orthonormal eigenbasis of (C2)⌦n.By the following theorem, we take one of these eigenspaces to be our quantum error-

correcting code.

26 ALEXANDRE CONLON

Theorem 8.6. Suppose that S is a k-dimensional totally singular subspace of E. Let S?

be the subspace orthogonal to S with respect to the symplectic inner product h· , ·iE (that is,S ✓ S?). Further suppose that for any two elements e1, e2 in a quantum channel E ✓ E,either e1e2 2 S or e1e2 /2 S?. Then for any eigenspace C corresponding to S there exists anerror-correcting operation R which corrects for the quantum channel E on C.

Proof. Consider the element s 2 S with eigenvalue �s, where s is the associated representa-tion. Then for any |ci 2 C we have s|ci = �s|ci, and for any e 2 E we have

se|ci = (�1)hs , eiEes|ci = (�1)hs , eiE�se|ci.Then, independent of |ci, for any e /2 S? we get hs , eiE = 1. So, the action of e permutesthe eigenspaces generated by the elements of s.

We proceed by splitting the proof into two cases, e1e2 2 S or e1e2 /2 S?. We begin bydefining a projector P onto eigenspace C. Let the orthonormal basis for C be given by theset {|cii}k

i=1 and define P by

P :=kX

i=1

|ciihci|.

Case 1: Suppose e1e2 2 S. Then

Pe1e2P = Pe1e2

✓ kX

i=1

|ciihci|◆

= P

✓ kX

i=1

e1e2|ciihci|◆

= P

✓ kX

i=1

�e1e2|ciihci|

◆

= �e1e2P 2

= �e1e2P,

where �e1e2is a diagonal entry in the real diagonal matrix (thus, self-adjoint) given by

the diagonalization of e1e2 2 S. Hence, any two elements e1, e2 2 E , with e1e2 2 S, arecorrectable.

Case 2: Suppose e1e2 /2 S?. Then for some s 2 S we have se1e2 = �e1e2s, so

se1e2|cii = �e1e2s|cii = ��se1e2|ciiimplying that

e1e2|cii /2 C.

In particular, e1e2|cii is in an eigenspace, C 0, which is orthogonal to C. Then we have

Pe1e2P = Pe1e2

⇣

kX

i=1

|ciihci|⌘

=kX

i,j=1

|cjihcj|e1e2|ciihci|

= 0,

since hcj|e1e2|cii = 0, for all i, j. ⇤


We observe that if we have e 2 S?, e 6= 0, then se = es, for all s 2 S. In particular,se|cii = es|cii = �ie|cii so, e|cii 2 C. This shows that S? permutes elements of C. Then S?

is analogous to classical errors which map codewords to codewords, that is, S? is the set ofnon-trivial undetectable errors.

Corollary 8.7. Suppose S is a k-dimensional linear subspace of E where S ✓ S? withrespect to the symplectic inner product. Suppose further that there are no vectors of weightless than d in S? \ S. Then there exists a quantum-error-correcting code which can correct(d� 1)/2 errors.

Proof. Let E ✓ E be the set of errors e 2 E with w(e) (d� 1)/2. Then for any e1, e2 2 Ewe get

w(e1e2) w(e1) + w(e2) d� 1 < d

and thus,e1e2 /2 S? \ S.

This shows that either e1e2 2 S, or e1e2 /2 S?. Thus, by the previous theorem, thereis a quantum-error-correcting code which will correct any error e 2 E . Hence, there is aquantum-error-correcting code which will correct (d� 1)/2 errors. ⇤

The remainder of this discussion will be centred around two examples of quantum-error-correcting codes. In the first example we construct a quantum-error-correcting code fromand existing classical binary error-correcting code, in the second example we discuss the fivequbit code.

Example 8.8. Suppose we have a classical linear binary-error-correcting code C ⇢ Zn2 which

maps k bit messages into n bit codewords, and has minimum distance d. Further, supposeC? ⇢ C, where C? is orthogonal to C with respect to the standard dot-product, · , modulo2. Then C corrects t = b(d � 1)/2c errors in the classical setting. We can construct aquantum-error-correcting code from the linear code C as follows.

Define S to consist of all vectors (↵1|↵2) 2 E with ↵1,↵2 2 C?. Then for �1, �2 2 C, wehave that (�1|�2) is in E and that for any (↵1|↵2) 2 S

⌦

(↵1|↵2), (�1|�2)↵

E= ↵1 · �2 + ↵2 · �1 = 0.

Thus the set consisting of vectors (�1|�2) 2 E with �1, �2 2 C is contained in S?.�

(�1|�2) 2 E | �1, �2 2 C

✓ S?.

Now let (�1|�2) 2 S?. Then, for all (↵1|↵2) 2 S, we have⌦

(↵1|↵2), (�1|�2)↵

E= 0.

In particular, we may assume ↵2 = 0 2 C?. Which implies that for all ↵1 2 C?

⌦

(↵1|0), (�1|�2)↵

E= ↵1�2 = 0,

and thus,�2 2 (C?)? = C.

Similarly, �1 2 C. Thus,S? ✓

�

(�1|�2) 2 E | �1, �2 2 C

.

Therefore, we have S? =�

(�1|�2) 2 E | �1, �2 2 C

. Then, since there is no vector in S?

with weight less than d, there exists a quantum-code capable of correcting up to t errors.

28 ALEXANDRE CONLON

Example 8.9. Let C be a quantum code which maps 1 qubit into 5 quits and is composedof the two codewords

|c0i = |00000i+ |11000i+ |01100i+ |00110i+ |00011i+ |10001i� |10100i � |01010i � |00101i � |10010i � |01001i� |11110i � |01111i � |10111i � |11011i � |11101i,

|c1i = |11111i+ |00111i+ |10011i+ |11001i+ |11100i+ |01110i� |01011i � |10101i � |11010i � |01101i � |10110i� |00001i � |10000i � |01000i � |00100i � |00010i.

Then the associated error group E is contained in L((C2)⌦5), and E = Z102 . Consider e =

(11000|00101) 2 E, corresponding to e = ⌦5k=1wk 2 E. Then Q(e) = (11000) · (00101) = 0,

and it easy to see that e|c0i = |c0i and e|c1i = |c1i, that is, |c0i, |c1i are in the +1-eigenspaceof e. Notice that |c0i, |c1i are fixed under cyclic permutations.

Then let ⇡ : S5 ! U((C2)⌦5) ✓ L((C2)⌦5) be defined by

⇡(�)| ki⌦5k=1 = ⇡(�)

�

| 1i ⌦ | 2i ⌦ | 3i ⌦ | 4i ⌦ | 5i�

= | ��1(1)i ⌦ | ��1(2)i ⌦ | ��1(3)i ⌦ | ��1(4)i ⌦ | ��1(5)i,where S5 is the symmetric group on 5 symbols and U((C2)⌦5) is the group of unitary operatorson (C2)⌦5. Then, restricting ⇡ to the subgroup of cyclic permutations C5 ⇢ S5, we get

⇡(�)|cii = |cii,for all � 2 C5, i = 0, 1. In particular,

|cii = ⇡(�)e⇡�1(�)|cii

= ⇡(�)e⇡�1(�)16X

j=1

↵j| ki⌦5k=1

ji

= ⇡(�)e16X

j=1

↵j| �(k)i⌦5k=1

ji

= ⇡(�)16X

j=1

↵j

�

wk| �(k)iji�⌦5

k=1

=16X

j=1

↵j

�

w��1(k)| kiji�⌦5

k=1

=� 5

⌦k=1

w��1(k)

�

|cii,

for all � 2 C5, i = 0, 1. Thus, |c0i, |c1i are fixed under all cyclic permutations of e. Usingthese permutations as generators, we can construct a subspace S ⇢ E which by Proposition8.3 is totally isotropic. Then S corresponds to an abelian group S ✓ E which fixes |c0i, |c1i.


Then S is the 4-dimensional totally singular subspace generated by

(11000|00101)(01100|10010)(00110|01001)(00011|10100)

The fifth permutation is left out of the generators since only four permutations of e arelinearly independent. Then S? is generated by S and the additional vectors (11111|00000),(00000|11111). It is straightforward to check that the minimum weight in S is d = 3. Thus,by Corollary 8.7 there exists a quantum-error-correcting operation which can correct up to1 error.

This geometric formalism presented by Calderbank et al. [1] has also been used to describeimportant error-correcting codes, such as Shor’s nine qubit code, and codes arising from thestabilizer formalism (see Section 10.5 of [3]). For further reading see [1], [2], [3], and [4].

References

[1] A.R. Calderbank, E.M. Rains, P.W. Shor, and N.J.A. Sloane; Quantum Error Correction and OrthogonalGeometry. Physical Review Letters. Vol. 78, No. 3, (1996), 405-407.

[2] A.R. Calderbank, E.M. Rains, P.W. Shor, and N.J.A. Sloane; Quantum Error Correction Via CodesOver GF(4). IEEE Transactions on Information Theory, Vol. 44, No. 4, (July 1998), 1369-1387.

[3] Nielsen, M. and Chuang, I.; Quantum Computation and Quantum Information. Tenth Edition, Cam-bridge University Press, (2010).

[4] Paulsen; W 2016, Section 4: Theory of CP maps, Corollary 4.13, lecture notes, Entanglement andnon-locality, University of Waterloo.

[5] Choi, Man Duen; Completely positive linear maps on complex matrices. Linear Algebra and Appl. 10(1975), 285-290. 15A60 (46L05)

[6] Ho↵man, K. and Kunze, R.A., Linear Algebra, Section 9.5, Prentice-Hall, (1971).

E-mail address: [email protected]

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