Fault tolerant

Phys 523

quantum computationLecture 7

Robert Raussendorf

Course outline — Section 3

• The repetition code again

• Stabilizer states and stabilizer codes

• The quantum error correction condition for stabilizer codes

• The CSS construction

• Examples of stabilizer codes

• Evolution of stabilizer states / the Gottesman-Knill theorem

Recap: Correctable errors and recovery

Definition. A set {Ei} of Kraus operators forming a quantum

operation E is a set of correctable errors for a code defined by

the projector ⇧ if there exists a CPTP map R such that, for all

states ⇢ in the code space, ⇢ = ⇧⇢⇧, it holds that

R � E(⇢) / ⇢. (1)

Definition. Given a quantum code specified by a projector ⇧

and an error channel E based on a set {Ei} of correctable errors,

an error recovery operation R is a CPTP map satisfying Eq. (1).

The point is: The general error-correction condition will be set

up in terms of correctable errors.

Recap: The quantum error correction condition

Theorem. Consider a quantum code described by a projector

⇧ onto the code space, and a quantum operation formed by the

Kraus operators {Ei}. Then, {Ei} is a set of correctable errors

if and only if

⇧E†

i Ej⇧ = cij⇧, (2)

for some Hermitian matrix c = [cij].

The repetition code again!

147=4 I 000) t p l l l D EC c et

VT = ( O o o><ooo I t l l l DC l l l l

*CmrchIloealnm#ipemor

Classical repetition code

8=000 ,T = Ill

.

Decode by majority vote :

010 1-3 ooo =J

O l l l→ I 1 I =Tetc

.

↳ Does not generalize to quantum .

Encodedstate cilearneol in

the majority M !

Classical repetition code

Altamaha method (just as good)

compute a swimming . amend15¥ -- (

'

o

'

. 9) (E) mrazamore

di -Panky - checksyndrome matrix P

cyclenerd, error

PCI te) = PCI) t Pce)in=0 !

= Pce) -learn abouterror--

- - ---- - - - - - - - - - THR k . - - r - -

Cenhmu u- examples Likeliest errors : X , y Xz , Xs , I .such - Coo) , Sy Cx,) - Cd) , Sy Chez)- C !) , Sy Cas ) -(9).

Parity check codes

In general " pGTte) = PCI) tPce)→

=p(e)↳ Nothing learned abouttheencodedrlate Mo

↳Cadegeneralizedtoquantum !

How to generalize to quantum

Continue w .example :

( '→

Z, Zz

→ Zz ZzRecalls 14J =Nooo) tf l l l l>

gym-m-

TIZEN> = 147Stamatiad Zz ⇒ I 47 = 14>

stabilizer ci agood "J = LZ, Zz, ZzZz)Z , -2, I 47 = ( Z, Zz)CZzZ,714> = (Z, -227143 = 14>S = {I, Z, Zz , ZzZz,Z,Zo}

It’s all about (anti) commutation

11,22-2=-2,2-271\ X, ZzZg = t ZzZ, X ,#

↳Z,Zz (X, 147) = (Zzz X i) 147

= - X, Z ,Zz 14>

=①X , 147"-

"is identified when measuring

Z, Zz &

similar : Zzz, (X , 147) = t X, 147Error X, team characteristic syndrome ( Ip!)

Stabilizer states

Definition. A stabilizer group S on n qubits is an Abelian sub-

group of Pn with 2n

elements such that �I 62 S. The corre-

sponding stabilizer state is

g| i = | i 8 g 2 S. (3)e

Simplest examples

Ngenerators b. 2"

amplitude

since : 107=2-10>

5=52-7 = {I , Z}

Smnbitfunex#k :Bellstate IBook 1007ft

Xixzl Doo) = I Doo)2-12-21 Boo) = I Boo>↳

5=44×2,2-12-2> ={I, Yik,ZiZziYYz}

<mgYj÷E¥¥. - EE's.mu?awanmm* 128--27amplitudes

Stabilizer codes

Definition. A stabilizer group S on n qubits is an Abelian sub-

group of Pn such that �I 62 S. The corresponding quantum code

space C(S) ⇢ H is

C(S) = {| i 2 H, s.th. g| i = | i 8 g 2 S} . (4)

teen-Age

Error correction condition for stabilizer codes

Denote by N(S) the normalizer of the stabilizer group S, i.e.,

N(S) = {g|gS = Sg}.

Corollary. A set A of Pauli errors is correctable for a given codewith stabilizer S, if for all Ei,Ej 2 A, either E†

i Ej 62 N(S), or

E†

i Ej 2 ei�S.

(anbedumeuiPoegCa)hn

Proof

CaseI : EetEj E NCS)⇒ Fgoesrun that g. EiTEJ=

-Ei go .

Thus,

A-EYE; D= AEYEj go T= - ftg. EitEj T

= - AEitEjT= O .

Hence, QEC andratshid with cij = 0 .

Case VI : EetEj c-e "9S.Then :

TEFE; T = e ice Ttt = e ice ft↳ QEC and nabskidu . cij = e ice.

pg,

Stabilizer generator matrix

I

* Stabiliser elements arePauli

ops .i

go = ifai- bi ¥,

i)"

( Zi)"

x C- lls,

•

SE ZtzA

§fully determinedby Albis . a, b E (Zz)"

If Only need this information forthe generators :

at:÷:i÷÷÷÷#""is:-stabilizer generator matrix .