Fault tolerant quantum computation - University of British ...
Transcript of Fault tolerant quantum computation - University of British ...
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 .