Asymmetric Quantum LDPC Codes
Recently, quantum error-correcting codes were proposed that capitalize on the fact that many physical error models lead to a significant asymmetry between the probabilities for bit flip and phase flip errors. An example for a channel which exhibits s…
Authors: Pradeep Kiran Sarvepalli, Martin Roetteler, Andreas Klappenecker
Asymmetric Quantum LDPC Codes Pradeep Kiran Sarvepalli Departmen t of C omputer Science T exas A&M Univ ersity College Station , TX 778 43 Email: p radeep@cs.tamu .edu Martin R ¨ otteler NEC L aborato ries America 4, Indep endenc e W ay , Suite 200 Princeton, NJ 08 540 Email: mr oetteler@nec-lab s.com Andreas Klappenec ker Departmen t of Com puter Scien ce T exas A&M University College Station, TX 7784 3 Email: k lappi@cs.tamu.ed u Abstract — Recently , quantum error -corr ecting codes were pro- posed that capitalize on the fact that many p hysical error models lead to a significant asym metry between the probabilities f or bit flip and ph ase fli p errors. A n example for a chan nel which exhibits such asymmetry is the combined amplitude damping and dephasing ch annel, wher e th e p robabilities of bit flips and p hase flips can be related to relaxation and deph asing time, respecti vely . W e give systematic constructions of asymmetric quantum stabi - lizer codes that exploit th is asymmetry . Our approach is based on a CSS constru ction that combines BCH and fin ite geometry LDPC codes. I . I N T R O D U C T I O N In ma ny q uantum m echanical systems the m echanisms for the occurre nce of bit flip and p hase flip errors are quite different. In a r ecent p aper Ioffe and M ´ ezard [ 10] postulated that qu antum error-correction should take into accoun t this asymmetry . The main argument given in [10] is that most of the known qu antum co mputing devices have relaxation times ( T 1 ) that are a round 1 − 2 orders of magnitude larger tha n the co rrespond ing deph asing times ( T 2 ) . In general, r elaxation leads to both bit flip and ph ase flip er rors, wh ereas deph asing only lead s to phase flip errors. This large asymmetr y between T 1 and T 2 suggests that bit flip e rrors occur less fre quently than phase flip errors and a well designed qu antum code would explo it this a symmetry of errors to provide better perfor mance. In fact, th is o bservation and its co nsequence s for quantum erro r correctio n, e specially q uantum fault tolera nce, have pr ompted in vestigations from various other researcher s [1], [8], [2 0]. Our goal will be as in [10] to con struct asymmetric quantum codes for quantum memories and at present we do not consider the issue of fault tolerance. W e first quan titativ ely justify how noise pr ocesses, cha racterized in ter ms of T 1 and T 2 , lead to an asymmetry in the b it flip and phase flip errors. As a concrete illustration of this we consider the amplitude damping and dephasing channel. For this channel we can co mpute the probab ilities of bit flip and phase flips in closed form . In particular, by giving explicit expre ssions for th e ratio of these probab ilities in terms of th e ra tio T 1 /T 2 , we show how the channel asy mmetry ar ises. After p roviding the necessary backgr ound , we give two systematic co nstructions o f asym metric quantum cod es based on BCH an d LDPC co des, as an alter native to th e randomize d construction of [10]. I I . B AC K G RO U N D Recall th at a qu antum ch annel that map s a state ρ to ( 1 − p x − p y − p z ) ρ + p x XρX + p y YρY + p z ZρZ, (1) with 1 = 1 0 0 1 , X = 0 1 1 0 , Y = 0 − i i 0 , Z = 1 0 0 − 1 is called a P auli channel . For a Pauli ch annel, o ne can resp ectiv ely determine the proba bilities p x , p y , p z that an input qubit in state ρ is subjec ted to a Pauli X , Y , or Z error . A co mbined a mplitude damping a nd d ephasin g channel E with relaxation time T 1 and deph asing time T 2 that acts on a qubit with den sity matrix ρ = ( ρ ij ) i,j ∈ { 0,1 } for a time t yields the d ensity m atrix E ( ρ ) = 1 − ρ 11 e − t/T 1 ρ 01 e − t/T 2 ρ 10 e − t/T 2 ρ 11 e − t/T 1 . This chann el is intere sting as it mo dels co mmon decoher ence processes fairly well. W e would like to d etermine the proba- bility p x , p y , and p z such that a n X , Y , or Z erro r occurs in a comb ined am plitude damp ing an d deph asing ch annel. Howe ver , it turns o ut that this q uestion is not well-p osed, since E is not a Pauli c hannel, that is, it cannot be written in the form (1). However , we can ob tain a Pauli channel E T by a technique called twirling [7], [ 5]. In our case, the twirlin g consists of conjugating the channel E by Pauli matrices and av eraging over the results. The r esulting channel E T is called the Pauli-twirl of E and is explicitly g iv en by E T ( ρ ) = 1 4 X A ∈ { 1 ,X,Y ,Z } A † E ( AρA † ) A. Theor em 1: Giv en a co mbined amp litude dampin g and dephasing channel E as above, th e associated Pauli-twirled channel is of th e fo rm E T ( ρ ) = ( 1 − p x − p y − p z ) ρ + p x XρX + p y YρY + p z ZρZ, where p x = p y = ( 1 − e − t/T 1 ) /4 an d p z = 1/2 − p x − 1 2 e − t/T 2 . In p articular, p z p x = 1 + 2 1 − e t/T 1 ( 1 − T 1 /T 2 ) e t/T 1 − 1 . If t ≪ T 1 , then we can approx imate th is ratio as 2T 1 /T 2 − 1 . Pr o of: The Kr aus o perator d ecompo sition [18] o f E is E ( ρ ) = 2 X k = 0 A k ρA † k , (2) where A 0 = h 1 0 0 √ 1 − λ − γ i ; A 1 = h 0 0 0 √ λ i ; A 2 = h 0 √ γ 0 0 i , and √ 1 − γ − λ = e − t/T 2 , 1 − γ = e − t/T 1 . W e can rewrite the Kraus o perator s A i as A 0 = 1 + √ 1 − λ − γ 2 1 + 1 − √ 1 − λ − γ 2 Z, A 1 = √ λ 2 1 − √ λ 2 Z, A 2 = √ γ 2 X − √ γ 2i Y . Re writing E ( ρ ) in ter ms of Pauli matrices yields E ( ρ ) = 2 − γ + 2 √ 1 − λ − γ 4 ρ + γ 4 XρX + γ 4 YρY + 2 − γ − 2 √ 1 − λ − γ 4 ZρZ − γ 4 1 ρZ − γ 4 Zρ 1 + γ 4i XρY − γ 4i YρX. (3) It fo llows that the Pauli-twirl channe l E T is o f th e claim ed form, see [5, Lem ma 2 ]. Com puting th e ratio p z /p x we get p z p x = 2 − γ − 2 √ 1 − λ − γ γ = 1 + e − t/T 1 − 2e − t/T 2 1 − e − t/T 1 , = 1 + 2 e − t/T 1 − e − t/T 2 1 − e − t/T 1 = 1 + 2 1 − e t/T 1 − t/T 2 e t/T 1 − 1 = 1 + 2 1 − e t/T 1 ( 1 − T 1 /T 2 ) e t/T 1 − 1 . If t ≪ T 1 , then we can approxim ate the ratio as 2T 1 /T 2 − 1 , as cla imed. Thus, an asym metry in the T 1 and T 2 times d oes translate to an asymmetry in the o ccurren ce of bit flip and p hase flip errors. Note that p x = p y indicating tha t the Y er rors ar e as unlikely as the X erro rs. W e shall refer to the ratio p z /p x as the channe l asymmetry and d enote this par ameter by A . Asymmetric q uantum cod es use the fact that the p hase flip errors are much mor e likely than the b it flip erro rs o r the combined bit-phase flip errors. Therefore the code has different error correcting capability fo r handling different type of e rrors. W e r equire the co de to corr ect many phase flip errors but it is not req uired to han dle the sam e nu mber of bit flip err ors. If we assume a CSS co de [4], then we can m eaningf ully speak of X - distance an d Z -d istance. A CSS stabilizer code that ca n detect all X e rrors u p to weigh t d x − 1 is said to have an X -d istance of d x . Similarly if it can de tect all Z erro rs upto weigh t d z − 1 , then it is said to have a Z -distance of d z . W e shall den ote such a code by [[ n, k, d x /d z ]] q to indicate it is an asymmetric code, see also [1 9] who was the first to use a notatio n that allowed to distinguish between X - a nd Z -distances. W e could also view this code as an [[ n, k, min { d x , d z } ]] q stabilizer code. Fur ther extension of these metr ics to an ad ditiv e no n-CSS cod e is a n interesting problem , but we will not g o into the de tails h ere. Recall th at in the CSS constructio n a pair of codes are used, one for cor recting the bit flip errors and the o ther f or correcting the phase flip err ors. Our choice of the se c odes will b e such th at the code for correctin g the phase flip error s has a larger distance than th e co de fo r co rrecting the bit flip errors. W e restate the CSS constructio n in a form convenient for asymmetric stabilizer c odes. Lemma 2 ( CSS Con struction [4]): Let C x , C z be linear codes over F n q with the p arameters [ n, k x ] q , and [ n, k z ] q respectively . Let C ⊥ x ⊆ C z . Th en th ere exists an [[ n, k x + k z − n, d x /d z ]] q asymmetric quantum code, where d x = wt ( C x \ C ⊥ z ) an d d z = wt ( C z \ C ⊥ x ) . If in the above co nstruction d x = wt ( C x ) and d z = wt ( C z ) , then we say that the code is pur e. In the theor em above and else where in this paper F q denotes a finite field with q elements. W e also den ote a q -ary narrow- sense primitive BCH co de of length n = q m − 1 and design distance δ as B C H ( δ ) . I I I . A S Y M M E T R I C Q UA N T U M C O D E S F RO M L D P C C O D E S In [ 10], Io ffe and M ´ ezard used a comb ination of BCH and LDPC co des to con struct asymme tric codes. The intu- ition b eing that the stro nger L DPC code should be used for correcting the phase flip erro rs and th e BCH code can be used for the inf requen t b it flips. This essentially red uces to finding a go od LDPC cod e such that the dual o f the LDPC code is con tained in the BCH code . They solve th is prob lem by ran domly cho osing codewords in the BCH co de which are of low weig ht (so that they can be used for th e par ity check matrix of the LDPC cod e). Howev er , this method leav es op en how good the resultin g LDPC co de is. For instance, the d egree profiles of the resulting code ar e n ot regular and there is little control over the fin al degree p rofiles o f th e cod e. Fu rthermo re, it is not app arent what ensemble or degree profiles one will use to an alyze th e cod e. W e p ropose an alternate schem e that uses LDPC cod es to con struct asymmetric stabilize r cod es. W e propo se two families of quantum cod es based on LDPC codes. In the first case we use LDPC codes for b oth the X and Z channe l while in the second constru ction we will use a co mbination of BCH and L DPC cod es. But first, we will need the f ollowing facts about gen eralized Reed-Muller codes and finite ge ometry LDPC co des. A. F inite Geo metry LDP C Cod es ([14], [2 1]) Let us denote b y E G ( m, p s ) th e Euc lidean finite g eometry over F p s consisting of p ms points. For our p urpo ses it suffices to use th e fact that th is geom etry is e quiv alent to the vector space F m p s . A µ -dimensional subspa ce of F m p s or its coset is called a µ -flat . Assume that 0 ≤ µ 1 < µ 2 ≤ m . Then we denote by N EG ( µ 2 , µ 1 , s, p ) the n umber of µ 1 -flats in a µ 2 - flat an d by A EG ( m, µ 2 , µ 1 , s, p ) , the num ber of µ 2 -flats tha t contain a given µ 1 -flat. These a re g i ven by (see [21]) N EG ( µ 2 , µ 1 , s, p ) = q ( µ 2 − µ 1 ) µ 1 Y i = 1 q µ 2 − i + 1 − 1 q µ 1 − i + 1 − 1 , (4) A EG ( m, µ 2 , µ 1 , s, p ) = µ 2 Y i = µ 1 + 1 q m − i + 1 − 1 q µ 2 − i + 1 − 1 , (5) where q = p s . In dex all the µ 1 -flats fro m i = 1 to n = N EG ( m, µ 1 , s, p ) as F i . Let F be a µ 2 -flat in E G ( m, p s ) . The n we can associate an inciden ce vector to F with re spect to the µ 1 flats as follows. i F = i j | i j = 1 if F j is co ntained in F i j = 0 otherwise. . Index th e µ 2 -flats from j = 1 to J = N EG ( m, µ 2 , s, p ) . Construct the J × n m atrix H ( 1 ) EG ( m, µ 2 , µ 1 , s, p ) whose rows are the incidence vectors of all the µ 2 -flats with respect to the µ 1 -flats. This matrix is also referred to as the incidence matrix. Then the type-I Eu clidean geometry code f rom µ 2 -flats and µ 1 -flats is define d to be the n ull space, i. e., Euclidean dual code) of the F p -linear span of H ( 1 ) EG ( m, µ 2 , µ 1 , s, p ) . This is denoted as C ( 1 ) EG ( m, µ 2 , µ 1 , s, p ) . L et H ( 2 ) EG ( m, µ 2 , µ 1 , s, p ) = H ( 1 ) EG ( m, µ 2 , µ 1 , s, p ) t . T hen the type-I I Euclidean geometry code C ( 2 ) EG ( m, µ 2 , µ 1 , s, p ) is defined to be th e null space of H ( 2 ) EG ( m, µ 2 , µ 1 , s, p ) . Let us n ow consider th e µ 2 -flats and µ 1 -flats that do no t contain the origin of EG ( m, p s ) . Now form the incidenc e matrix of the µ 2 -flats with respect to the µ 1 -flats not con taining the o rigin. The null spa ce of this incidence matrix giv es us a quasi-cyclic c ode in general, which we denote b y C ( 1 ) EG ,c ( m, µ 2 , µ 1 , s, p ) , see [2 1]. B. Gene ralized Reed-Mu ller codes ( [12]) Let α be a prim itiv e element in F q m . The cyclic g eneralized Reed-Muller c ode of length q m − 1 an d order ν is defined as the cyclic code with th e gener ator polyn omial wh ose roo ts α j satisfy 0 < j ≤ m ( q − 1 ) − ν − 1 . Th e gener alized Reed-Muller code is the singly extended code of length q m . It is deno ted as GRM q ( ν, m ) . The dual of a GRM code is also a GRM code [2], [3 ], [ 12]. I t is kn own that GRM q ( ν, m ) ⊥ = GRM q ( ν ⊥ , m ) , (6) where ν ⊥ = m ( q − 1 ) − 1 − ν . Let C be a linear code over F n q s . Then we define C | F q , the subfield subcod e of C over F n q as th e codewords of C which are entirely in F n q , (see [9, pages 116-1 20]). Formally this ca n be expr essed as C | F q = { c ∈ C | c ∈ F n q } . (7) Let C ⊆ F n q l . The th e trace code of C over F q is d efined as tr q l /q ( C ) = { tr q l /q ( c ) | c ∈ C } . (8) There are interesting relation s between the trace code and the subfield subcode. O ne o f which is the following result which we will n eed late r . Lemma 3 : Let C ⊆ F n q l . Then C | F q , the subfield subco de of C is contained in tr q l /q ( C ) , the trace code of C . In other words C | F q ⊆ tr q l /q ( C ) . Pr o of: Let c ∈ C | F q ⊆ F n q and α ∈ F q l . Then tr q l /q ( αc ) = c tr q l /q ( α ) as c ∈ F n q . Since trace is a surjecti ve form, there exists some α ∈ F q l , such that tr q l /q ( α ) = 1 . This implies that c ∈ tr q l /q ( C ) . Since c is an arbitr ary element in C | F q it f ollows th at C | F q ⊆ tr q l /q ( C ) . Let q = p s , then the Euclidean geo metry code of ord er r over EG ( m, p s ) is defined as th e dual of the subfield subco de of GRM q (( q − 1 )( m − r − 1 ) , m ) , [3, p age 44 8]. The type- I LDPC c ode C ( 1 ) EG ( m, µ, 0, s , p ) code is an Euclidean g eometry code of or der µ − 1 over EG ( m, p s ) , see [21]. Hence its d ual is the subfield subcode o f GRM q (( q − 1 )( m − µ ) , m ) co de. In o ther words, C ( 1 ) EG ( m, µ, 0, s , p ) ⊥ = GRM q (( q − 1 )( m − µ ) , m ) | F p . (9) Further, Delsarte’ s theor em [6] tells us that C ( 1 ) EG ( m, µ, 0, s , p ) = GRM q (( q − 1 )( m − µ ) , m ) | ⊥ F p , = tr q/p GRM q (( q − 1 )( m − µ ) , m ) ⊥ = tr q/p ( GRM q ( µ ( q − 1 ) − 1, m )) . Hence, C ( 1 ) EG ( m, µ, 0, s , p ) code can also be re lated to GRM q ( µ ( q − 1 ) − 1, m ) as C ( 1 ) EG ( m, µ, 0, s , p ) = tr q/p ( GRM q ( µ ( q − 1 ) − 1 ) , m ) . ( 10) C. New families o f asymmetric qu antum co des W ith the previous prepa ration we are now ready to construct asymmetric q uantum cod es from finite geo metry LDPC co des. Theor em 4 (Asymmetric EG LDPC Codes): Let p be a prime, with q = p s and s ≥ 1, m ≥ 2 . Let 1 < µ z < m and m − µ z + 1 ≤ µ x < m . Then there exists an [[ p ms , k x + k z − p ms , d x /d z ]] p asymmetric EG LDPC cod e, wh ere k x = dim C ( 1 ) EG ( m, µ x , 0, s, p ); k z = dim C ( 1 ) EG ( m, µ z , 0, s, p ) . For th e distanc es d x ≥ A EG ( m, µ x , µ x − 1, s, p ) + 1 and d z ≥ A EG ( m, µ z , µ z − 1, s, p ) + 1 hold. Pr o of: Let C z = C ( 1 ) EG ( m, µ z , 0, s, p ) . Th en f rom eq ua- tion ( 10) we have C z = tr q/p ( GRM q ( µ z ( q − 1 ) − 1, m ) . By Lemma 3 we kn ow that C z ⊇ GRM q ( µ z ( q − 1 ) − 1, m ) | F p , C z ⊇ GRM q (( q − 1 )( m − ( m − µ z + 1 )) , m ) | F p , where the last inc lusion f ollows fro m the n esting prop erty of the gener alized Reed-Muller codes. For any ord er µ x such that m − µ z + 1 ≤ µ x < m , let C x = C ( 1 ) EG ( m, µ x , 0, s, p ) . Th en C x is an LDPC co de who se dual C ⊥ x = GRM q (( q − 1 )( m − µ x ) , m ) | F p is co ntained in C z . Th us we can use Lemma 2 to form an a symmetric co de with th e par ameters [[ p ms , k x + k z − p ms , d x /d z ]] p The distance of C z and C x are at lower bounded as d x ≥ A EG ( m, µ x , µ x − 1, s, p ) + 1 an d d z ≥ A EG ( m, µ z , µ z − 1, s, p ) + 1 (see [21]). In th e constru ction just prop osed, we sh ould cho ose C z to be a stro nger code com pared to C x . W e have given the construction over a n onbina ry alphab et ev en though th e ca se p = 2 migh t be of par ticular interest. Our n ext construction makes use of the cyclic finite geom - etry codes. Our goal will be to find a small BCH code whose dual is con tained in a cyclic Euclidean ge ometry LDPC cod e. For solv ing th is proble m we need to know th e cyclic structu re of C ( 1 ) EG ,c ( m, µ, 0, s , p ) . Let α be a primitive element in F p ms . Then the roots o f g enerator po lynomial of C ( 1 ) EG ,c ( m, µ, 0, s , p ) are g iv en by [1 1, Theo rem 6], see also [13], [15]. Now , Z = { α h | 0 < max 0 ≤ l
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