Good Quantum Error Correction Codes And Their Decoding

David Poulin Center for the Physics of Information California Institute of Technology

Quantum Information Processing and Communication Barcelona, October 2007

David Poulin (Caltech)

Good Quantum Error Correcting Codes

QIPC07

1 / 28

Rudiments Basic block codes Shor, Steane, 5-qubit, ...

After ` concatenations Remaining error ` pe ∝ 2 . Rate R = (k /n)` . As block size n increases Remaining error pe ∝ exp(−cn). Fixed rate R = k /n. David Poulin (Caltech)

Good Quantum Error Correcting Codes

QIPC07

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Rudiments Basic block codes Shor, Steane, 5-qubit, ...

After ` concatenations Remaining error ` pe ∝ 2 . Rate R = (k /n)` . |ψ!

|0!

U

|0!

As block size n increases Remaining error pe ∝ exp(−cn). Fixed rate R = k /n.

David Poulin (Caltech)

Good Quantum Error Correcting Codes

QIPC07

2 / 28

Rudiments Basic block codes Shor, Steane, 5-qubit, ...

|0! |0!

|ψ!

|0!

U

|0!

|0! |0!

|0! |0!

David Poulin (Caltech)

U

U

U

|0! |0!

U

|0! |0!

U

|0! |0!

U

|0! |0!

U

|0! |0!

U

|0! |0!

U

|0! |0!

U

|0! |0!

U

|0! |0!

U

After ` concatenations Remaining error ` pe ∝ 2 . Rate R = (k /n)` . As block size n increases Remaining error pe ∝ exp(−cn). Fixed rate R = k /n.

Good Quantum Error Correcting Codes

QIPC07

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Rudiments Basic block codes Shor, Steane, 5-qubit, ...

{

|ψ!

|0! |0! |0! |0! |0! |0! |0! |0! |0! |0! |0!

After ` concatenations Remaining error ` pe ∝ 2 .

k

Rate R = (k /n)` . U

David Poulin (Caltech)

n

As block size n increases Remaining error pe ∝ exp(−cn). Fixed rate R = k /n. Good Quantum Error Correcting Codes

QIPC07

2 / 28

Outline 1

Classical preliminaries The decoding problem Probabilistic coding and decoding

2

Quantum error correction

3

Quantum turbo codes Theory Practice

4

Quantum sparse codes Definitions Problems Results

David Poulin (Caltech)

Good Quantum Error Correcting Codes

QIPC07

3 / 28

Classical preliminaries

Outline 1

Classical preliminaries The decoding problem Probabilistic coding and decoding

2

Quantum error correction

3

Quantum turbo codes Theory Practice

4

Quantum sparse codes Definitions Problems Results

David Poulin (Caltech)

Good Quantum Error Correcting Codes

QIPC07

4 / 28

Classical preliminaries

The decoding problem

Classical linear codes Fondations: Shannon 1948. Upper bound on channel capacity R = k /n ≤ 1 − h(p). Bound is achievable given unlimited computational power. A linear classical code encoding k bits into n > k bits is specified by a set of n − k binary constraints: Hx = 0.

Optimal rate is typically achieved by random codes. Decoding such codes is an NP-complete problem.

David Poulin (Caltech)

Good Quantum Error Correcting Codes

QIPC07

5 / 28

Classical preliminaries

The decoding problem

Classical linear codes Fondations: Shannon 1948. Upper bound on channel capacity R = k /n ≤ 1 − h(p). Bound is achievable given unlimited computational power. A linear classical code encoding k bits into n > k bits is specified by a set of n − k binary constraints: Hx = 0. X Hx = 0

Channel

Y Hy = s

Optimal rate is typically achieved by random codes. Decoding such codes is an NP-complete problem.

David Poulin (Caltech)

Good Quantum Error Correcting Codes

QIPC07

5 / 28

Classical preliminaries

The decoding problem

Classical linear codes Fondations: Shannon 1948. Upper bound on channel capacity R = k /n ≤ 1 − h(p). Bound is achievable given unlimited computational power. A linear classical code encoding k bits into n > k bits is specified by a set of n − k binary constraints: Hx = 0. X Hx = 0

Channel

Y Hy = s

Optimal rate is typically achieved by random codes. Decoding such codes is an NP-complete problem.

David Poulin (Caltech)

Good Quantum Error Correcting Codes

QIPC07

5 / 28

Classical preliminaries

The decoding problem

Classical linear codes Fondations: Shannon 1948. Upper bound on channel capacity R = k /n ≤ 1 − h(p). Bound is achievable given unlimited computational power. A linear classical code encoding k bits into n > k bits is specified by a set of n − k binary constraints: Hx = 0. X Hx = 0

Channel

Y Hy = s

Optimal rate is typically achieved by random codes. Decoding such codes is an NP-complete problem.

David Poulin (Caltech)

Good Quantum Error Correcting Codes

QIPC07

5 / 28

Classical preliminaries

The decoding problem

Closing the gap

The field of channel coding started with Claude Shannon’s 1948 landmark paper. For the next half century, its central objective was to find practical coding schemes that could approach channel capacity. — Costello & Forney 2006

David Poulin (Caltech)

Good Quantum Error Correcting Codes

QIPC07

6 / 28

Classical preliminaries

The decoding problem

Algebraic coding For a few decade, the focus was on codes with large minimal distance with efficient minimal distance decoders.

David Poulin (Caltech)

Good Quantum Error Correcting Codes

QIPC07

7 / 28

Classical preliminaries

The decoding problem

Algebraic coding For a few decade, the focus was on codes with large minimal distance with efficient minimal distance decoders.

David Poulin (Caltech)

Good Quantum Error Correcting Codes

QIPC07

7 / 28

,iuwtllo 'snorpal pue lfncglp allnb fiean'd11ec dlluanba:; eql ur enrl ,(llelcedsa ueeq seq slqJ 'sarnpef, ;rlllll||{.ls parJJeJ JI 'SUorlelndrueUrqJnS"q'slrnJJrJ eq} -o-rd uorlf,npar rlletuelsds parldde Surneq rerye paqelqo eads lecrSol eql Surureluretu e[q/v\ 'd1rcr1d er€ r{Jlqrr suorssatdxa JreJqaEIe Jo uor}elndruetu eq} ,Tnr*ifitl[-[J paqrrrsa;d o1 Surprocre aql uodn puB (t) o1 Surpuods ur puads o1 3ur11rmsr aq aIIIl] lunorue Jo ;,,rill"'lll;JL[If Furqclyrs aql ,{;rldurrs o} pua} plnoqs ;au8rsap aq] Jo IIHs eql uo pepuad"p seq sanbruqral aser{J '(Z) uolllpuor d-rerprsqnseq} Jo esn esaql. Jo uollecrldde eql Jo sseualllre#a egl Jo qrnHl 'e:qa81e ;,lllrimEg.lL suorssa;dxa q suorssa-rdxa ( d,;rldurrs eql SurcnpeJ JoJ l) lecrSol leuo]lcunJ Ieruoudlod ,;lii,fidtidLIEO eq lvlou .,(eu suollelndrueur creJqa8m sraqlo pue *,r'euln$ 'ue{lv 'ullex '1leq{rng ,{q padolel 'suorleurqtuoo esoql segrcadsd,1a1a1d -ep uaeq el€q spoqleu rlleuelsds pue ,e:qa81e T'r rn*rylpuoJ eJueH 'I - F q"l.{/n JoJ asoq} }sn[ ate 1ecr3o1uodn .,{1ra,eaq peueal seq slrnJJrc SurqJ}ri!\s ,,11ffi: L{ Jo u8rsap ,'uouueqs Jo >lro^\ eql O1{IAAO-ITO( lt

Good Quantum Error Correcting Codes

QIPC07

7 / 28

,*iiiiiiiliruu,illri$ qJIq/!\

'0:

Senle4

(oXt"

lndUr

JO SUOlleUrqtUOC

r

- :JJ

aZ

' 'zX 'rX)eZ :

, t:l i il i

eSOqJ

o'rX'tX)E

uolllpuor drelplsqns aql .{q passard ll["@uortreler e qcns 'luepuadepur d.1a-rl]uaeq u *t dul aq] uosear srr{} roJ pue '-lnJJo Jelau IIIrle niir- ** sanle^ Jo suolleurqtuo3 urel.Jac '1e;aua8 uI '(uX ( ' ' o '6X'tX)rZ: |,2

tZ

(oX ' ' ' ' 'eX 'tX)rZ :

(oX '

slndur eql Jo uollcunJ lectSol B se passa-rd 1nd1no r{Jea pue uarrr8 .,(1a1a1druoJ ueaq

aql roJ suorlecglcads lecr8ol eql ueq] 's1nd senle^. Jo uorl.eulquroJ elqrssrulpe r{Jea JoJ

rlllllt;",i, 1?

David Poulin (Caltech)

'oa\l JoJeAlodu sgapocerp uI sllq JoJeqruno pepp "gl -ord 'sepocEqlcelep-roJreEqlcn4suocJoJpesneq ^(uurdggsuollgleJ

SFIJ,.3'sleurrpell"c ..slBrruou^d1od *r, Jo sess?lculslJec Jo sJeqrueu egl ueafiueq lsFa ol ua\ogs s1dlqsuollsler clrleru pelcadxerm uv 'c"nII egl uo pelsel ueeq ea"q pu" cp"Illoln" u" rRIlT\asn JoJelqs}Ins 'relnduroc flrupclu"d eJBpeqlJcsep spoqleu egJ 'srrrJel ueealeq sreeddu ,,ro earsnlcxe, uopuredo eql qclgfi\ uI ,,,slu$uou.d1od*r, peil"c (suols -serdxe uBeloog Eqr{gdulls JoJpe$rcsep erB senbpqcel snorr"A 's1nd1noofirl Jo es"c eql q petcese eq fuur uopcnper e1du4s.dlrupcprud v .ulalgord pul8lro eW q slndlno puu slndug Jo srequnu egl Jo runs egl ol requmu rn pnbe ere slndul esoga\ ruelqord lndtno e1tuls p ol pecnper eq dBur sllnc4c gcns roJ ruelqord 1nd1no eldgpur egl Jo slrud uIBUec l"ql ulI\ogs q lI 'urqe81" u?eloog q ..slunuouf1od,, ,fe pelueserdar eq {Bur }BrR s}lncqc o1 fpo sellddu uelqord .t{l Jo lueru}"e4 rpcl}"ru -eglsru egtr '1nd1noeuo usrll erolll easg l"ql sllncrlc tu1qc1lmstqdJ -gdurrs Jo urelqord preue8 egl ol lqtnos sI uollnlos V- fttourutng

xt{flTTflhl.g'O

uorlcerec rorrfl ol puB u8rseq rIncrID Surqcrprs ol Brqe8ry ueeloos Io uorlecUddy SUflJ ndWOJ

JINOV,IJTTE-STVOLLJYS NVVI

g'W' I

For a few decade, the focus was on codes with large minimal distance with efficient minimal distance decoders.

Algebraic coding Classical preliminaries

The decoding problem

,iuwtllo 'snorpal pue lfncglp allnb fiean'd11ec dlluanba:; eql ur enrl ,(llelcedsa ueeq seq slqJ 'sarnpef, ;rlllll||{.ls parJJeJ JI 'SUorlelndrueUrqJnS"q'slrnJJrJ eq} -o-rd uorlf,npar rlletuelsds parldde Surneq rerye paqelqo eads lecrSol eql Surureluretu e[q/v\ 'd1rcr1d er€ r{Jlqrr suorssatdxa JreJqaEIe Jo uor}elndruetu eq} ,Tnr*ifitl[-[J paqrrrsa;d o1 Surprocre aql uodn puB (t) o1 Surpuods ur puads o1 3ur11rmsr aq aIIIl] lunorue Jo ;,,rill"'lll;JL[If Furqclyrs aql ,{;rldurrs o} pua} plnoqs ;au8rsap aq] Jo IIHs eql uo pepuad"p seq sanbruqral aser{J '(Z) uolllpuor d-rerprsqnseq} Jo esn esaql. Jo uollecrldde eql Jo sseualllre#a egl Jo qrnHl 'e:qa81e ;,lllrimEg.lL suorssa;dxa q suorssa-rdxa ( d,;rldurrs eql SurcnpeJ JoJ l) lecrSol leuo]lcunJ Ieruoudlod ,;lii,fidtidLIEO eq lvlou .,(eu suollelndrueur creJqa8m sraqlo pue *,r'euln$ 'ue{lv 'ullex '1leq{rng ,{q padolel 'suorleurqtuoo esoql segrcadsd,1a1a1d -ep uaeq el€q spoqleu rlleuelsds pue ,e:qa81e T'r rn*rylpuoJ eJueH 'I - F q"l.{/n JoJ asoq} }sn[ ate 1ecr3o1uodn .,{1ra,eaq peueal seq slrnJJrc SurqJ}ri!\s ,,11ffi: L{ Jo u8rsap ,'uouueqs Jo >lro^\ eql O1{IAAO-ITO( lt

Good Quantum Error Correcting Codes

QIPC07

7 / 28

,*iiiiiiiliruu,illri$ qJIq/!\

'0:

Senle4

(oXt"

lndUr

JO SUOlleUrqtUOC

David Poulin (Caltech) , t:l i il i

eSOqJ

o'rX'tX)E

'oa\l JoJeAlodu sgapocerp uI sllq JoJeqruno pepp "gl

Eqlcn4suoc JoJpesn eq ^(uur trgfTnn sFI tr$ uolllpuor drelplsqns aql .{q passard -ord 'sepocEqlcelep-roJre 'c 'g dggsuollgleJ 2.'uot't $r6ar SFIJ,.3'sleurrpell"c sess?lculslJec allsJeqrueu ..slBrruou^d1od apooJo *r, p-PJoqfueg s'ranoil ega (! uortreler e qcns aU' eq'o'q-qo d.1a-rl]ua ll["@parepTsuoc sepoc-Jo-ssatc eq o+'luepuadepur rq padolaaapsrrrt egl ueafiueq lsFa ol ua\ogs1-ll s1dlqsuollsler clrleru pelcadxerm uv *tii=:::y"* 3o ea1&aero u *t dul (T + rr) aug+5e+d-rol'rE roJ pue '-lnJJo Jelau IIIrle q1 aq];;; uosear ci+",6r",te ;"'p.dsrr{} q+oq arqr eeE'c 'c"nII egl -trrrE u-Ep€onporlut s{rA niirurel.Jac '1e;aua8 uI ;ilJ ilci{* io e"o,irnc'Ert+^cT-4l T.tuTurEII-.trt.u-fq.snlE uo pelsel ueeq ea"q pu" 4use.r cp"Illoln" u" rRIlT\asn JoJelqs}Ins a ** sanle^ Jo suolleurqtuo3 'relnduroc ;oJ e'mpacord v s4+a11"rrsuoc "i.tr prd'aqr+cqr.r6c-torro-cto ".#d au..rc{6i*rilr.+ flrupclu"d eJBpeqlJcsep spoqleu egJ 'srrrJel ueealeq sreeddu ,,ro '(uX ( ' ' o '6X'tX)rZ: ""pic-ii.ifirreie |,2 earsnlcxe, uopuredo eql qclgfi\ uI ,,,slu$uou.d1od*r, peil"c (suols r

@'I

(oX '

' 'zX 'rX)eZ :

-serdxe uBeloog Eqr{gdulls JoJpe$rcsep erB senbpqcel snorr"A 's1nd1noofirl Jo es"c eql q petcese eq fuur uopcnper e1du4s.dlrupcprud v .ulalgord pul8lro

aZ

q?asqcoesehl t aEPT.rqtuBQ eW q slndlno puu slndug Jo srequnu egl Jo runs egl ol requmu rn : tZ (oX ' ' ' ' 'eX galn?T?sul lrtoilr.E1 oqeT esoga\ rf,roqp s+tesru{c"ss€n -pnbe ere slndul ruelqord lndtno e1tuls p ol pecnper eq dBur Itrolouqcag Jo 'tX)rZ slndur eql Jo uollcunJ lectSol B se passa-rd 1nd1no r{Jea pue uarrr8 .,(1a1a1druoJ ueaq

sllnc4c gcns roJ ruelqord 1nd1no eldgpur egl Jo slrud uIBUec l"ql ulI\ogs q lI 'urqe81" u?eloog q ..slunuouf1od,, ,fe pelueserdar eq {Bur }BrR s}lncqc o1 fpo sellddu uelqord .t{l

Jo lueru}"e4 rpcl}"ru gHf cslv g'lgHgs aql roJ suorlecglcads lecr8ol cNrgocfiI eql ueq] 's1nd -eglsru egtr '1nd1noeuo usrll erolll easg l"ql sllncrlc tu1qc1lmstqdJ V SSI|TC senle^. Jo uorl.eulquroJ sttooc 9M,[CgUUOS-EffiUg-srIdI&TIIr{,{O elqrssrulpe r{Jea JoJ -gdurrs Jo urelqord preue8 egl ol lqtnos sI uollnlos V- fttourutng

- :JJ rlllllt;",i, 1?

xt{flTTflhl.g'O

uorlcerec rorrfl ol puB u8rseq rIncrID Surqcrprs ol Brqe8ry ueeloos Io uorlecUddy SUflJ ndWOJ

JINOV,IJTTE-STVOLLJYS NVVI

g'W' I

For a few decade, the focus was on codes with large minimal distance with efficient minimal distance decoders.

Algebraic coding Classical preliminaries

The decoding problem

,iuwtllo 'snorpal pue lfncglp allnb fiean'd11ec dlluanba:; eql ur enrl ,(llelcedsa ueeq seq slqJ 'sarnpef, ;rlllll||{.ls parJJeJ 'SUorlelndrueUrqJnS"q'slrnJJrJ eq} -o-rd uorlf,npar rlletuelsds parldde Surneq rerye paqelqo JI eads lecrSol eql Surureluretu e[q/v\ 'd1rcr1d er€ r{Jlqrr suorssatdxa JreJqaEIe Jo uor}elndruetu eq} ,Tnr*ifitl[-[J paqrrrsa;d o1 Surprocre (t) o1 Surpuods ur puads o1 3ur11rmsr aq aIIIl] Jo lunorue aql uodn puB ;,,rill"'lll;JL[If Furqclyrs aql ,{;rldurrs o} pua} plnoqs ;au8rsap aq] Jo IIHs eql uo pepuad"p seq sanbruqral aser{J '(Z) uolllpuor d-rerprsqnseq} Jo esn esaql. Jo uollecrldde eql Jo sseualllre#a egl Jo qrnHl ;,lllrimEg.lL ( l) suorssa;dxa leuo]lcunJ eql d,;rldurrs 'e:qa81e lecrSol q suorssa-rdxaIeruoudlod SurcnpeJ JoJ ,;lii,fidtidLIEO eq lvlou .,(eu suollelndrueur creJqa8m sraqlo pue *,r'euln$ 'ue{lv 'ullex '1leq{rng ,{q padolel 'suorleurqtuoo esoql segrcadsd,1a1a1d -ep uaeq el€q spoqleu rlleuelsds pue ,e:qa81e T'r rn*rylpuoJ eJueH 'I - F q"l.{/n JoJ asoq} }sn[ ate 1ecr3o1uodn .,{1ra,eaq peueal seq slrnJJrc SurqJ}ri!\s ,,11ffi: L{ Jo u8rsap ,'uouueqs Jo >lro^\ eql O1{IAAO-ITO( lt

Good Quantum Error Correcting Codes

QIPC07

7 / 28

,*iiiiiiiliruu,illri$ qJIq/!\

'0:

Senle4

(oXt"

lndUr

JO SUOlleUrqtUOC

David Poulin (Caltech) , t:l i il i

eSOqJ

o'rX'tX)E

'oa\l JoJeAlodu sgapocerp uI sllq JoJeqruno pepp "gl

Eqlcn4suoc JoJpesn eq ^(uur trgfTnn sFI tr$ uolllpuor drelplsqns aql .{q passard -ord 'sepocEqlcelep-roJre 'c 'g dggsuollgleJ 2.'uot't $r6ar SFIJ,.3'sleurrpell"c sess?lculslJec allsJeqrueu ..slBrruou^d1od apooJo *r, p-PJoqfueg s'ranoil ega (! uortreler e qcns aU' eq egl d.1a-rl]ua ll["@parepTsuoc sepoc-Jo-ssatc eq o+'luepuadepur 'o'q-qo rq padolaaapsrrrt ueafiueq ol ua\ogs1-ll s1dlqsuollsler clrleru pelcadxerm uv 3o ea1&aero u *t dul (T + rr) lsFa *tii=:::y"* aug+5e+d-rol'rE roJ pue '-lnJJo Jelau IIIrle q1 aq];;; uosear ci+",6r",te ;"'p.dsrr{} q+oq arqr eeE'c 'c"nII egl -trrrE u-Ep€onporlut s{rA niirurel.Jac '1e;aua8 uI ;ilJ ilci{* io e"o,irnc'Ert+^cT-4l T.tuTurEII-.trt.u-fq.snlE uo pelsel ueeq ea"q pu" 4use.r cp"Illoln" u" rRIlT\asn JoJelqs}Ins a ** sanle^ Jo suolleurqtuo3 'relnduroc ;oJ e'mpacord v s4+a11"rrsuoc "i.tr prd'aqr+cqr.r6c-torro-cto ".#d au..rc{6i*rilr.+ flrupclu"d eJBpeqlJcsep spoqleu egJ 'srrrJel ueealeq sreeddu ,,ro '(uX ( ' ' o '6X'tX)rZ: ""pic-ii.ifirreie |,2 earsnlcxe, uopuredo eql qclgfi\ uI ,,,slu$uou.d1od*r, peil"c (suols r

@'I

(oX '

' 'zX 'rX)eZ :

-serdxe uBeloog Eqr{gdulls JoJpe$rcsep erB senbpqcel snorr"A 's1nd1noofirl Jo es"c eql q petcese eq fuur uopcnper e1du4s.dlrupcprud v .ulalgord pul8lro

aZ

q?asqcoesehl t aEPT.rqtuBQ eW q slndlno puu slndug Jo srequnu egl Jo runs egl ol requmu rn : tZ (oX ' ' ' ' 'eX galn?T?sul lrtoilr.E1 oqeT esoga\ rf,roqp s+tesru{c"ss€n -pnbe ere slndul ruelqord lndtno e1tuls p ol pecnper eq dBur Itrolouqcag Jo 'tX)rZ slndur eql Jo uollcunJ lectSol B se passa-rd 1nd1no r{Jea pue uarrr8 .,(1a1a1druoJ ueaq

sllnc4c gcns roJ ruelqord 1nd1no eldgpur egl Jo slrud uIBUec l"ql ulI\ogs q lI 'urqe81" u?eloog q ..slunuouf1od,, ,fe pelueserdar eq {Bur }BrR s}lncqc o1 fpo sellddu uelqord .t{l

Jo lueru}"e4 rpcl}"ru gHf cslv g'lgHgs aql roJ suorlecglcads lecr8ol cNrgocfiI eql ueq] 's1nd -eglsru egtr '1nd1noeuo usrll erolll easg l"ql sllncrlc tu1qc1lmstqdJ V SSI|TC senle^. Jo uorl.eulquroJ sttooc 9M,[CgUUOS-EffiUg-srIdI&TIIr{,{O elqrssrulpe r{Jea JoJ -gdurrs Jo urelqord preue8 egl ol lqtnos sI uollnlos V- fttourutng

- :JJ rlllllt;",i, 1?

xt{flTTflhl.g'O

uorlcerec rorrfl ol puB u8rseq rIncrID Surqcrprs ol Brqe8ry ueeloos Io uorlecUddy SUflJ ndWOJ

JINOV,IJTTE-STVOLLJYS NVVI

g'W' I

For a few decade, the focus was on codes with large minimal distance with efficient minimal distance decoders.

Algebraic coding Classical preliminaries

The decoding problem

Classical preliminaries

The decoding problem

Distance isn’t everything Gilbert-Varshamov Bound For n → ∞, the maximum rate R with of a code with minimal distance d satisfies R ≥ 1 − h(d/n).

d

Z

Minimum distance decoder: up to t ' d/2 errors. Maximum bit flip error p ' t/n = d/2n. Minimum distance decoder: R = 1 − h(2p) Shannon capacity: R = 1 − h(p). David Poulin (Caltech)

Good Quantum Error Correcting Codes

QIPC07

8 / 28

Classical preliminaries

The decoding problem

Distance isn’t everything Gilbert-Varshamov Conjecture For n → ∞, the maximum rate R with of a code with minimal distance d satisfies R = 1 − h(d/n).

d

Z

Minimum distance decoder: up to t ' d/2 errors. Maximum bit flip error p ' t/n = d/2n. Minimum distance decoder: R = 1 − h(2p) Shannon capacity: R = 1 − h(p). David Poulin (Caltech)

Good Quantum Error Correcting Codes

QIPC07

8 / 28

Classical preliminaries

The decoding problem

Distance isn’t everything Gilbert-Varshamov Conjecture For n → ∞, the maximum rate R with of a code with minimal distance d satisfies R = 1 − h(d/n).

d

Z

Minimum distance decoder: up to t ' d/2 errors. Maximum bit flip error p ' t/n = d/2n. Minimum distance decoder: R = 1 − h(2p) Shannon capacity: R = 1 − h(p). David Poulin (Caltech)

Good Quantum Error Correcting Codes

QIPC07

8 / 28

Classical preliminaries

The decoding problem

Distance isn’t everything Gilbert-Varshamov Conjecture For n → ∞, the maximum rate R with of a code with minimal distance d satisfies R = 1 − h(d/n).

d

Z

Minimum distance decoder: up to t ' d/2 errors. Maximum bit flip error p ' t/n = d/2n. Minimum distance decoder: R = 1 − h(2p) Shannon capacity: R = 1 − h(p). David Poulin (Caltech)

Good Quantum Error Correcting Codes

QIPC07

8 / 28

Classical preliminaries

The decoding problem

Distance isn’t everything Gilbert-Varshamov Conjecture For n → ∞, the maximum rate R with of a code with minimal distance d satisfies R = 1 − h(d/n).

d

Z

Minimum distance decoder: up to t ' d/2 errors. Maximum bit flip error p ' t/n = d/2n. Minimum distance decoder: R = 1 − h(2p) Shannon capacity: R = 1 − h(p). David Poulin (Caltech)

Good Quantum Error Correcting Codes

QIPC07

8 / 28

Classical preliminaries

The decoding problem

Distance isn’t everything Gilbert-Varshamov Conjecture For n → ∞, the maximum rate R with of a code with minimal distance d satisfies R = 1 − h(d/n).

d

Z

Minimum distance decoder: up to t ' d/2 errors. Maximum bit flip error p ' t/n = d/2n. Minimum distance decoder: R = 1 − h(2p) Shannon capacity: R = 1 − h(p). David Poulin (Caltech)

Good Quantum Error Correcting Codes

QIPC07

8 / 28

Classical preliminaries

Probabilistic coding and decoding

Probabilistic coding Codes that correct almost any error of weight ≤ t.

Random code ensemble (statistical physics). Decoding is an NP-complete problem. Suboptimal decoding: belief propagation. David Poulin (Caltech)

Good Quantum Error Correcting Codes

QIPC07

9 / 28

Classical preliminaries

Probabilistic coding and decoding

Probabilistic coding Codes that correct almost any error of weight ≤ t.

Random code ensemble (statistical physics). Decoding is an NP-complete problem. Suboptimal decoding: belief propagation. David Poulin (Caltech)

Good Quantum Error Correcting Codes

QIPC07

9 / 28

Classical preliminaries

Probabilistic coding and decoding

Probabilistic coding Codes that correct almost any error of weight ≤ t.

Random code ensemble (statistical physics). Decoding is an NP-complete problem. Suboptimal decoding: belief propagation. David Poulin (Caltech)

Good Quantum Error Correcting Codes

QIPC07

9 / 28

Classical preliminaries

Probabilistic coding and decoding

Probabilistic coding Codes that correct almost any error of weight ≤ t.

Random code ensemble (statistical physics). Decoding is an NP-complete problem. Suboptimal decoding: belief propagation. David Poulin (Caltech)

Good Quantum Error Correcting Codes

QIPC07

9 / 28

Classical preliminaries

Probabilistic coding and decoding

Probabilistic coding Codes that correct almost any error of weight ≤ t.

Random code ensemble (statistical physics). Decoding is an NP-complete problem. Suboptimal decoding: belief propagation. David Poulin (Caltech)

Good Quantum Error Correcting Codes

QIPC07

9 / 28

Classical preliminaries

Probabilistic coding and decoding

Probabilistic coding Codes that correct almost any error of weight ≤ t.

Random code ensemble (statistical physics). Decoding is an NP-complete problem. Suboptimal decoding: belief propagation. David Poulin (Caltech)

Good Quantum Error Correcting Codes

QIPC07

9 / 28

Classical preliminaries

Probabilistic coding and decoding

Quantum belief propagation

David Poulin (Caltech)

Good Quantum Error Correcting Codes

QIPC07

10 / 28

Classical preliminaries

Probabilistic coding and decoding

Quantum belief propagation

David Poulin (Caltech)

Good Quantum Error Correcting Codes

QIPC07

10 / 28

Classical preliminaries

Probabilistic coding and decoding

Quantum belief propagation

David Poulin (Caltech)

Good Quantum Error Correcting Codes

QIPC07

10 / 28

Classical preliminaries

Probabilistic coding and decoding

Quantum belief propagation

David Poulin (Caltech)

Good Quantum Error Correcting Codes

QIPC07

10 / 28

Classical preliminaries

Probabilistic coding and decoding

Sub-optimal decoding

The decoder does not always return the most likely error. Two type of errors: Errors that cannot be corrected by the code. Errors that are misidentified by the decoder.

Block error rate pe = 1 − P(all n bits perfect). To be distinguished from (qu)bit error rate.

For good codes, pe decreases with n, with k /n fixed.

David Poulin (Caltech)

Good Quantum Error Correcting Codes

QIPC07

11 / 28

Classical preliminaries

Probabilistic coding and decoding

Sub-optimal decoding

The decoder does not always return the most likely error. Two type of errors: Errors that cannot be corrected by the code. Errors that are misidentified by the decoder.

Block error rate pe = 1 − P(all n bits perfect). To be distinguished from (qu)bit error rate.

For good codes, pe decreases with n, with k /n fixed.

David Poulin (Caltech)

Good Quantum Error Correcting Codes

QIPC07

11 / 28

Classical preliminaries

Probabilistic coding and decoding

Sub-optimal decoding

The decoder does not always return the most likely error. Two type of errors: Errors that cannot be corrected by the code. Errors that are misidentified by the decoder.

Block error rate pe = 1 − P(all n bits perfect). To be distinguished from (qu)bit error rate.

For good codes, pe decreases with n, with k /n fixed.

David Poulin (Caltech)

Good Quantum Error Correcting Codes

QIPC07

11 / 28

Classical preliminaries

Probabilistic coding and decoding

Sub-optimal decoding

The decoder does not always return the most likely error. Two type of errors: Errors that cannot be corrected by the code. Errors that are misidentified by the decoder.

Block error rate pe = 1 − P(all n bits perfect). To be distinguished from (qu)bit error rate.

For good codes, pe decreases with n, with k /n fixed.

David Poulin (Caltech)

Good Quantum Error Correcting Codes

QIPC07

11 / 28

Quantum error correction

Outline 1

Classical preliminaries The decoding problem Probabilistic coding and decoding

2

Quantum error correction

3

Quantum turbo codes Theory Practice

4

Quantum sparse codes Definitions Problems Results

David Poulin (Caltech)

Good Quantum Error Correcting Codes

QIPC07

12 / 28

Quantum error correction

Decoding problem Hx = 0, ∀x ∈ C ⊂ {0, 1}n becomes Sj |ψi = |ψi, ∀|ψi ∈ C ⊂ (C2 )⊗n

|ψ!

Channel

Sj |ψ! = |ψ!

|φ!

Sj |φ! = sj |φ!

When Sj are random element of Pauli group, achieve hashing capacity QH ≤ Q. Such random codes are hard to decode (NP-hard for Pauli channels). We want to devise good probabilistic quantum codes. Good decoding algorithms are crucial. David Poulin (Caltech)

Good Quantum Error Correcting Codes

QIPC07

13 / 28

Quantum error correction

Decoding problem Hx = 0, ∀x ∈ C ⊂ {0, 1}n becomes Sj |ψi = |ψi, ∀|ψi ∈ C ⊂ (C2 )⊗n

|ψ!

Channel

Sj |ψ! = |ψ!

|φ!

Sj |φ! = sj |φ!

When Sj are random element of Pauli group, achieve hashing capacity QH ≤ Q. Such random codes are hard to decode (NP-hard for Pauli channels). We want to devise good probabilistic quantum codes. Good decoding algorithms are crucial. David Poulin (Caltech)

Good Quantum Error Correcting Codes

QIPC07

13 / 28

Quantum error correction

Decoding problem Hx = 0, ∀x ∈ C ⊂ {0, 1}n becomes Sj |ψi = |ψi, ∀|ψi ∈ C ⊂ (C2 )⊗n

|ψ!

Channel

Sj |ψ! = |ψ!

|φ!

Sj |φ! = sj |φ!

When Sj are random element of Pauli group, achieve hashing capacity QH ≤ Q. Such random codes are hard to decode (NP-hard for Pauli channels). We want to devise good probabilistic quantum codes. Good decoding algorithms are crucial. David Poulin (Caltech)

Good Quantum Error Correcting Codes

QIPC07

13 / 28

Quantum error correction

Decoding problem Hx = 0, ∀x ∈ C ⊂ {0, 1}n becomes Sj |ψi = |ψi, ∀|ψi ∈ C ⊂ (C2 )⊗n

|ψ!

Channel

Sj |ψ! = |ψ!

|φ!

Sj |φ! = sj |φ!

When Sj are random element of Pauli group, achieve hashing capacity QH ≤ Q. Such random codes are hard to decode (NP-hard for Pauli channels). We want to devise good probabilistic quantum codes. Good decoding algorithms are crucial. David Poulin (Caltech)

Good Quantum Error Correcting Codes

QIPC07

13 / 28

Quantum error correction

Decoding problem Hx = 0, ∀x ∈ C ⊂ {0, 1}n becomes Sj |ψi = |ψi, ∀|ψi ∈ C ⊂ (C2 )⊗n

|ψ!

Channel

Sj |ψ! = |ψ!

|φ!

Sj |φ! = sj |φ!

When Sj are random element of Pauli group, achieve hashing capacity QH ≤ Q. Such random codes are hard to decode (NP-hard for Pauli channels). We want to devise good probabilistic quantum codes. Good decoding algorithms are crucial. David Poulin (Caltech)

Good Quantum Error Correcting Codes

QIPC07

13 / 28

Quantum error correction

Minimal distance vs belief propagation

5-qubit code

!"

"!

!#

"!

Repeated concatenation of the 5-qubist code.

!$

pe

"!

!%

"!

Rate = ( 15 )` .

!(

"!

40 dB improvement due to decoding.

!'

"!

!&

"!

!

"

# )*+,-./+-.0*+

David Poulin (Caltech)

$

%

Good Quantum Error Correcting Codes

QIPC07

14 / 28

Quantum error correction

Minimal distance vs belief propagation

Repeated concatenation of the 5-qubist code.

5-qubit code

0.40 0.35

Rate = ( 15 )` .

0.30

pe

0.25

Threshold under min distance ≈ 0.1376.

0.20 0.15

Threshold under belief prop. ≈ 0.1885.

0.10 0.05 0

0

1

2

3

4 5 6 Concatenation

David Poulin (Caltech)

7

8

9

10

Threshold for random codes ≈ 0.189.

Good Quantum Error Correcting Codes

QIPC07

15 / 28

Quantum error correction

Minimal distance vs belief propagation

Repeated concatenation of the 5-qubist code.

5-qubit code

0.40 0.35

Rate = ( 15 )` .

0.30

pe

0.25

Threshold under min distance ≈ 0.1376.

0.20 0.15

Threshold under belief prop. ≈ 0.1885.

0.10 0.05 0

0

1

2

3

4 5 6 Concatenation

David Poulin (Caltech)

7

8

9

10

Threshold for random codes ≈ 0.189.

Good Quantum Error Correcting Codes

QIPC07

15 / 28

Quantum turbo codes

Outline 1

Classical preliminaries The decoding problem Probabilistic coding and decoding

2

Quantum error correction

3

Quantum turbo codes Theory Practice

4

Quantum sparse codes Definitions Problems Results

David Poulin (Caltech)

Good Quantum Error Correcting Codes

QIPC07

16 / 28

Quantum turbo codes

Theory

Quantum turbo code (blueprint) |0! |0! |0!

|0!

U

|0! |0!

DATA

U |0! |0!

W DATA

U |0! |0!

DATA

at

ut

W

U |0! |0!

rm Pe

|0!

n

io

DATA

|0!

U |0! |0!

U |0! |0!

U |0! |0!

DATA

U

...

|0! |0!

...

|0!

U |0! |0!

W DATA

Code parameters: length N = 6, rate R = 2/9, memory m = 4. Belief propagation decoding complexity ∝ N4m (loopy). Ollivier, Poulin, Tillich, in preparation. David Poulin (Caltech)

Good Quantum Error Correcting Codes

QIPC07

17 / 28

Quantum turbo codes

Theory

Quantum turbo code (blueprint) |0! |0! |0!

|0!

U

|0! |0!

DATA

U |0! |0!

W DATA

U |0! |0!

DATA

at

ut

W

U |0! |0!

rm Pe

|0!

n

io

DATA

|0!

U |0! |0!

U |0! |0!

U |0! |0!

DATA

U

...

|0! |0!

...

|0!

U |0! |0!

W DATA

Code parameters: length N = 6, rate R = 2/9, memory m = 4. Belief propagation decoding complexity ∝ N4m (loopy). Ollivier, Poulin, Tillich, in preparation. David Poulin (Caltech)

Good Quantum Error Correcting Codes

QIPC07

17 / 28

Quantum turbo codes

Theory

Quantum turbo code (blueprint) |0! |0! |0!

|0!

U

|0! |0!

DATA

U |0! |0!

W DATA

U |0! |0!

DATA

at

ut

W

U |0! |0!

rm Pe

|0!

n

io

DATA

|0!

U |0! |0!

U |0! |0!

U |0! |0!

DATA

U

...

|0! |0!

...

|0!

U |0! |0!

W DATA

Code parameters: length N = 6, rate R = 2/9, memory m = 4. Belief propagation decoding complexity ∝ N4m (loopy). Ollivier, Poulin, Tillich, in preparation. David Poulin (Caltech)

Good Quantum Error Correcting Codes

QIPC07

17 / 28

Quantum turbo codes

Theory

Good codes?

Theorem Let C be a quantum turbo code obtained by the concatenation of two d ∗ −1 non-catastrophic recursive convolutional codes. Then dC ≥ N d ∗ . Theorem There are no non-catastrophic and recursive quantum convolutional codes. Ollivier, Poulin, Tillich, in preparation.

We cannot prove that these are good codes... resort to numerical analysis.

David Poulin (Caltech)

Good Quantum Error Correcting Codes

QIPC07

18 / 28

Quantum turbo codes

Theory

Good codes?

Theorem Let C be a quantum turbo code obtained by the concatenation of two d ∗ −1 non-catastrophic recursive convolutional codes. Then dC ≥ N d ∗ . Theorem There are no non-catastrophic and recursive quantum convolutional codes. Ollivier, Poulin, Tillich, in preparation.

We cannot prove that these are good codes... resort to numerical analysis.

David Poulin (Caltech)

Good Quantum Error Correcting Codes

QIPC07

18 / 28

Quantum turbo codes

Theory

Good codes?

Theorem Let C be a quantum turbo code obtained by the concatenation of two d ∗ −1 non-catastrophic recursive convolutional codes. Then dC ≥ N d ∗ . Theorem There are no non-catastrophic and recursive quantum convolutional codes. Ollivier, Poulin, Tillich, in preparation.

We cannot prove that these are good codes... resort to numerical analysis.

David Poulin (Caltech)

Good Quantum Error Correcting Codes

QIPC07

18 / 28

Quantum turbo codes

Practice

Turbo code performances on depolarization channel 0

10

!1

Rate is fixed at 19 .

Overhead = 9 Memory = 3

Block error probability

10

Error probability decreases as number of encoded qubits increases.

!2

10

!3

10

50 100 250 500 2000 4000

!4

10

Error-free "phase transition" at 0.1.

!5

10

0.08

0.09

0.1

0.11

0.12

0.13

With finite size, 10−4 threshold around  = 0.08.

Channel error probability

Best performance to date at this rate.

David Poulin (Caltech)

Good Quantum Error Correcting Codes

QIPC07

19 / 28

Quantum turbo codes

Practice

Turbo code performances on depolarization channel 0

10

!1

Rate is fixed at 19 .

Overhead = 9 Memory = 3

Block error probability

10

Error probability decreases as number of encoded qubits increases.

!2

10

!3

10

50 100 250 500 2000 4000

!4

10

Error-free "phase transition" at 0.1.

!5

10

0.08

0.09

0.1

0.11

0.12

0.13

With finite size, 10−4 threshold around  = 0.08.

Channel error probability

Best performance to date at this rate.

David Poulin (Caltech)

Good Quantum Error Correcting Codes

QIPC07

19 / 28

Quantum turbo codes

Practice

Turbo code performances on depolarization channel 0

10

!1

Rate is fixed at 19 .

Overhead = 9 Memory = 3

Block error probability

10

Error probability decreases as number of encoded qubits increases.

!2

10

!3

10

50 100 250 500 2000 4000

!4

10

Error-free "phase transition" at 0.1.

!5

10

0.08

0.09

0.1

0.11

0.12

0.13

With finite size, 10−4 threshold around  = 0.08.

Channel error probability

Best performance to date at this rate.

David Poulin (Caltech)

Good Quantum Error Correcting Codes

QIPC07

19 / 28

Quantum turbo codes

Practice

Turbo code performances on depolarization channel 0

10

!1

Rate is fixed at 19 .

Overhead = 9 Memory = 3

Block error probability

10

Error probability decreases as number of encoded qubits increases.

!2

10

!3

10

50 100 250 500 2000 4000

!4

10

Error-free "phase transition" at 0.1.

!5

10

0.08

0.09

0.1

0.11

0.12

0.13

With finite size, 10−4 threshold around  = 0.08.

Channel error probability

Best performance to date at this rate.

David Poulin (Caltech)

Good Quantum Error Correcting Codes

QIPC07

19 / 28

Quantum turbo codes

Practice

Turbo code performances on depolarization channel 0

10

!1

Rate is fixed at 19 .

Overhead = 9 Memory = 3

Block error probability

10

Error probability decreases as number of encoded qubits increases.

!2

10

!3

10

50 100 250 500 2000 4000

!4

10

Error-free "phase transition" at 0.1.

!5

10

0.08

0.09

0.1

0.11

0.12

0.13

With finite size, 10−4 threshold around  = 0.08.

Channel error probability

Best performance to date at this rate.

David Poulin (Caltech)

Good Quantum Error Correcting Codes

QIPC07

19 / 28

. In the case of dual-containing codes, this is the noise level at which each of the two identical constituent codes (see (19)) has an error probab Quantum turbo codescodesPractice As an aid to the eye, lines have been added between the four unicycle U; between a sequence of Bicycle codes B all of block length rates; and between a sequence of of BCH codes with increasing block length. The curve labeled S2 is the Shannon limit if the correlations betw errors are neglected, (45). Points “ ” are codes invented elsewhere. All other point styles denote codes presented for the first time in this pape

Code performances

MacKay, Mitchison, McFadden, IEEE’04.

David Poulinof (Caltech) Good Quantum Error(depolarizing Correcting Codes / 28 ar mmary of performances several codes on the 4-ary symmetric channel channel). The additional points atQIPC07 the right and20 bottom

Quantum sparse codes

Outline 1

Classical preliminaries The decoding problem Probabilistic coding and decoding

2

Quantum error correction

3

Quantum turbo codes Theory Practice

4

Quantum sparse codes Definitions Problems Results

David Poulin (Caltech)

Good Quantum Error Correcting Codes

QIPC07

21 / 28

Quantum sparse codes

Definitions

Sparse quantum code

Stabilizer generators Sj have low weight (act as identity on most qubits). Each qubit is involved in a small number of checks. S1

X

David Poulin (Caltech)

X

S2

Y

Y

Z

S3

X

X

Good Quantum Error Correcting Codes

Z

Y

QIPC07

22 / 28

Quantum sparse codes

Problems

Difficulty with CSS construction

CSS codes: constructed from two dual classical codes Cx and Cz . Idea: find two dual sparse codes! Since sparse codes are good error correction codes, they don’t have sparse duals.

David Poulin (Caltech)

Good Quantum Error Correcting Codes

QIPC07

23 / 28

Quantum sparse codes

Problems

Difficulty with CSS construction

CSS codes: constructed from two dual classical codes Cx and Cz . Idea: find two dual sparse codes! Since sparse codes are good error correction codes, they don’t have sparse duals.

David Poulin (Caltech)

Good Quantum Error Correcting Codes

QIPC07

23 / 28

Quantum sparse codes

Problems

Difficulty with CSS construction

CSS codes: constructed from two dual classical codes Cx and Cz . Idea: find two dual sparse codes! Since sparse codes are good error correction codes, they don’t have sparse duals.

David Poulin (Caltech)

Good Quantum Error Correcting Codes

QIPC07

23 / 28

Quantum sparse codes

Problems

Sparse code constructions

CSS with cyclic classical code. MacKay, Mitchison, and McFadden IEEE’04

Almost surely has low weight errors by construction.

Fixing commutations by adding qubits. Poulin, Chung, and Ollivier, in preparation

Poor control of weight distribution.

Small-depth random Clifford circuits.1 Complete failure so far.

1

Personal favorite David Poulin (Caltech)

Good Quantum Error Correcting Codes

QIPC07

24 / 28

Quantum sparse codes

Problems

Sparse code constructions

CSS with cyclic classical code. MacKay, Mitchison, and McFadden IEEE’04

Almost surely has low weight errors by construction.

Fixing commutations by adding qubits. Poulin, Chung, and Ollivier, in preparation

Poor control of weight distribution.

Small-depth random Clifford circuits.1 Complete failure so far.

1

Personal favorite David Poulin (Caltech)

Good Quantum Error Correcting Codes

QIPC07

24 / 28

Quantum sparse codes

Problems

Sparse code constructions

CSS with cyclic classical code. MacKay, Mitchison, and McFadden IEEE’04

Almost surely has low weight errors by construction.

Fixing commutations by adding qubits. Poulin, Chung, and Ollivier, in preparation

Poor control of weight distribution.

Small-depth random Clifford circuits.1 Complete failure so far.

1

Personal favorite David Poulin (Caltech)

Good Quantum Error Correcting Codes

QIPC07

24 / 28

Quantum sparse codes

Problems

Sparse code constructions

CSS with cyclic classical code. MacKay, Mitchison, and McFadden IEEE’04

Almost surely has low weight errors by construction.

Fixing commutations by adding qubits. Poulin, Chung, and Ollivier, in preparation

Poor control of weight distribution.

Small-depth random Clifford circuits.1 Complete failure so far.

1

Personal favorite David Poulin (Caltech)

Good Quantum Error Correcting Codes

QIPC07

24 / 28

Quantum sparse codes

Problems

Sparse code constructions

CSS with cyclic classical code. MacKay, Mitchison, and McFadden IEEE’04

Almost surely has low weight errors by construction.

Fixing commutations by adding qubits. Poulin, Chung, and Ollivier, in preparation

Poor control of weight distribution.

Small-depth random Clifford circuits.1 Complete failure so far.

1

Personal favorite David Poulin (Caltech)

Good Quantum Error Correcting Codes

QIPC07

24 / 28

Quantum sparse codes

Problems

Sparse code constructions

CSS with cyclic classical code. MacKay, Mitchison, and McFadden IEEE’04

Almost surely has low weight errors by construction.

Fixing commutations by adding qubits. Poulin, Chung, and Ollivier, in preparation

Poor control of weight distribution.

Small-depth random Clifford circuits.1 Complete failure so far.

1

Personal favorite David Poulin (Caltech)

Good Quantum Error Correcting Codes

QIPC07

24 / 28

Quantum sparse codes

Problems

Difficulties with decoding

Tanner graph must contain small loops, which affects belief propagation. S1

X

David Poulin (Caltech)

X

S2

Y

Y

Z

S3

X

X

Good Quantum Error Correcting Codes

Z

Y

QIPC07

25 / 28

Quantum sparse codes

Problems

Difficulties with decoding

Tanner graph must contain small loops, which affects belief propagation. S1

X

X

S2

Y

Y

Z

S3

X

X

Z

Y

X error

David Poulin (Caltech)

Good Quantum Error Correcting Codes

QIPC07

25 / 28

Quantum sparse codes

Problems

Difficulties with decoding

Tanner graph must contain small loops, which affects belief propagation. S1

X

David Poulin (Caltech)

X

S2

Y

Y

Z

S3

X

X

Good Quantum Error Correcting Codes

Z

Y

QIPC07

25 / 28

Quantum sparse codes

Problems

Difficulties with decoding

Tanner graph must contain small loops, which affects belief propagation. S1

X

David Poulin (Caltech)

X

S2

Y

Y

Z

S3

X

X

Good Quantum Error Correcting Codes

Z

Y

QIPC07

25 / 28

Quantum sparse codes

Problems

More difficulties with decoding

Belief propagation is a qubit-wise decoding scheme. Sparse codes are by definition highly degenerate. Example Stabilizer generators S1 = XX and S2 = ZZ . Error X on second qubit: Acceptable corrections: IX , XI, ZY , or YZ .

By symmetry, we always have Pr1BP (σ) = Pr2BP (σ).

David Poulin (Caltech)

Good Quantum Error Correcting Codes

QIPC07

26 / 28

Quantum sparse codes

Problems

More difficulties with decoding

Belief propagation is a qubit-wise decoding scheme. Sparse codes are by definition highly degenerate. Example Stabilizer generators S1 = XX and S2 = ZZ . Error X on second qubit: Acceptable corrections: IX , XI, ZY , or YZ .

By symmetry, we always have Pr1BP (σ) = Pr2BP (σ).

David Poulin (Caltech)

Good Quantum Error Correcting Codes

QIPC07

26 / 28

Quantum sparse codes

Problems

More difficulties with decoding

Belief propagation is a qubit-wise decoding scheme. Sparse codes are by definition highly degenerate. Example Stabilizer generators S1 = XX and S2 = ZZ . Error X on second qubit: Acceptable corrections: IX , XI, ZY , or YZ .

By symmetry, we always have Pr1BP (σ) = Pr2BP (σ).

David Poulin (Caltech)

Good Quantum Error Correcting Codes

QIPC07

26 / 28

Quantum sparse codes

Problems

More difficulties with decoding

Belief propagation is a qubit-wise decoding scheme. Sparse codes are by definition highly degenerate. Example Stabilizer generators S1 = XX and S2 = ZZ . Error X on second qubit: Acceptable corrections: IX , XI, ZY , or YZ .

By symmetry, we always have Pr1BP (σ) = Pr2BP (σ).

David Poulin (Caltech)

Good Quantum Error Correcting Codes

QIPC07

26 / 28

Quantum sparse codes

Problems

More difficulties with decoding

Belief propagation is a qubit-wise decoding scheme. Sparse codes are by definition highly degenerate. Example Stabilizer generators S1 = XX and S2 = ZZ . Error X on second qubit: Acceptable corrections: IX , XI, ZY , or YZ .

By symmetry, we always have Pr1BP (σ) = Pr2BP (σ).

David Poulin (Caltech)

Good Quantum Error Correcting Codes

QIPC07

26 / 28

Quantum sparse codes

Results

Bicycle codes 0

N = 800 and K = 400 (rate 21 ).

!2

Average qubit degree = 15. Check degree = 30.

!4 Basic Random perturbation Collision & freeze

!6 0.01

0.015

0.02

0.025

13 dB improvement due to decoding. 10−4 threshold around  = 0.014.

0.03

All observed errors are detected: caused by the decoder.

David Poulin (Caltech)

Good Quantum Error Correcting Codes

QIPC07

27 / 28

Quantum sparse codes

Results

Bicycle codes 0

N = 800 and K = 400 (rate 21 ).

!2

Average qubit degree = 15. Check degree = 30.

!4 Basic Random perturbation Collision & freeze

!6 0.01

0.015

0.02

0.025

13 dB improvement due to decoding. 10−4 threshold around  = 0.014.

0.03

All observed errors are detected: caused by the decoder.

David Poulin (Caltech)

Good Quantum Error Correcting Codes

QIPC07

27 / 28

Quantum sparse codes

Results

Bicycle codes 0

N = 800 and K = 400 (rate 21 ).

!2

Average qubit degree = 15. Check degree = 30.

!4 Basic Random perturbation Collision & freeze

!6 0.01

0.015

0.02

0.025

13 dB improvement due to decoding. 10−4 threshold around  = 0.014.

0.03

All observed errors are detected: caused by the decoder.

David Poulin (Caltech)

Good Quantum Error Correcting Codes

QIPC07

27 / 28

Quantum sparse codes

Results

Bicycle codes 0

N = 800 and K = 400 (rate 21 ).

!2

Average qubit degree = 15. Check degree = 30.

!4 Basic Random perturbation Collision & freeze

!6 0.01

0.015

0.02

0.025

13 dB improvement due to decoding. 10−4 threshold around  = 0.014.

0.03

All observed errors are detected: caused by the decoder.

David Poulin (Caltech)

Good Quantum Error Correcting Codes

QIPC07

27 / 28

Conclusion

Summary

The best classical coding schemes use Probabilistic code ensembles. Belief propagation decoding (sub-optimal heuristic).

These codes were commercially used long before they were proven to be good. Quantum turbo codes: Complete control of code design: N, R, m. Proof of high minimal distance requires too strong assumptions. Simulations: highest error threshold.

Sparse quantum codes: No systematic method to construct them. Problems with belief propagation decoding. Simulations using heuristic methods: outperform all algebraic codes. Biggest challenge remains decoding.

David Poulin (Caltech)

Good Quantum Error Correcting Codes

QIPC07

28 / 28

Conclusion

Summary

The best classical coding schemes use Probabilistic code ensembles. Belief propagation decoding (sub-optimal heuristic).

These codes were commercially used long before they were proven to be good. Quantum turbo codes: Complete control of code design: N, R, m. Proof of high minimal distance requires too strong assumptions. Simulations: highest error threshold.

Sparse quantum codes: No systematic method to construct them. Problems with belief propagation decoding. Simulations using heuristic methods: outperform all algebraic codes. Biggest challenge remains decoding.

David Poulin (Caltech)

Good Quantum Error Correcting Codes

QIPC07

28 / 28

Conclusion

Summary

The best classical coding schemes use Probabilistic code ensembles. Belief propagation decoding (sub-optimal heuristic).

These codes were commercially used long before they were proven to be good. Quantum turbo codes: Complete control of code design: N, R, m. Proof of high minimal distance requires too strong assumptions. Simulations: highest error threshold.

Sparse quantum codes: No systematic method to construct them. Problems with belief propagation decoding. Simulations using heuristic methods: outperform all algebraic codes. Biggest challenge remains decoding.

David Poulin (Caltech)

Good Quantum Error Correcting Codes

QIPC07

28 / 28

Conclusion

Summary

The best classical coding schemes use Probabilistic code ensembles. Belief propagation decoding (sub-optimal heuristic).

These codes were commercially used long before they were proven to be good. Quantum turbo codes: Complete control of code design: N, R, m. Proof of high minimal distance requires too strong assumptions. Simulations: highest error threshold.

Sparse quantum codes: No systematic method to construct them. Problems with belief propagation decoding. Simulations using heuristic methods: outperform all algebraic codes. Biggest challenge remains decoding.

David Poulin (Caltech)

Good Quantum Error Correcting Codes

QIPC07

28 / 28

Conclusion

Summary

The best classical coding schemes use Probabilistic code ensembles. Belief propagation decoding (sub-optimal heuristic).

These codes were commercially used long before they were proven to be good. Quantum turbo codes: Complete control of code design: N, R, m. Proof of high minimal distance requires too strong assumptions. Simulations: highest error threshold.

Sparse quantum codes: No systematic method to construct them. Problems with belief propagation decoding. Simulations using heuristic methods: outperform all algebraic codes. Biggest challenge remains decoding.

David Poulin (Caltech)

Good Quantum Error Correcting Codes

QIPC07

28 / 28

Conclusion

Summary

The best classical coding schemes use Probabilistic code ensembles. Belief propagation decoding (sub-optimal heuristic).

These codes were commercially used long before they were proven to be good. Quantum turbo codes: Complete control of code design: N, R, m. Proof of high minimal distance requires too strong assumptions. Simulations: highest error threshold.

Sparse quantum codes: No systematic method to construct them. Problems with belief propagation decoding. Simulations using heuristic methods: outperform all algebraic codes. Biggest challenge remains decoding.

David Poulin (Caltech)

Good Quantum Error Correcting Codes

QIPC07

28 / 28