Asymmetric Quantum Codes on Toric Surfaces

Transcription

1 Institute of Mathematics Aahus Univesity Octobe 1, 2017

2 Abstact Asymmetic quantum eo-coecting codes ae quantum codes defined ove biased quantum channels: qubit-flip and phase-shift eos may have equal o diffeent pobabilities. The code constuction is the Caldebank-Sho-Steane constuction based on two linea codes. We pesent families of toic sufaces, toic codes and associated asymmetic quantum eo-coecting codes.

3 Backgound on quantum eo-coecting codes Woks of P.W. Sho [17] and A.M. Steane [23], [19] initiated the study and constuction of quantum eo-coecting codes. A.R. Caldebank [5], P.W. Sho [18] and A.M. Steane [22] poduced stabilize codes (CSS) fom linea codes containing thei dual codes. Fo details see fo example [2], [4] and [24].

4 Toic codes Intoduction In [9] and [10] we developed methods to constuct linea eo coecting codes fom toic vaieties. In [11] we genealized this to constuct linea codes suitable fo constucting quantum codes by the Caldebank-Sho-Steane method. Ou constuctions extended simila esults obtained by A. Ashikhmin, S. Litsyn and M.A. Tsfasman in [3] fom Goppa codes on algebaic cuves.

5 Asymmetic quantum eo-coecting codes Asymmetic quantum eo-coecting codes ae quantum codes defined ove biased quantum channels: qubit-flip and phase-shift eos may have equal o diffeent pobabilities. The code constuction is the CSS constuction based on two linea codes. The constuction appeaed oiginally in [7], [13] and [25]. We pesent new families of toic sufaces, toic codes and associated asymmetic quantum eo-coecting codes.

6 Fo the geneal theoy of toic vaieties we efe to [6], [8] and [15]. Let F q be the field with q elements and let be an intege dividing q. Let b Z such that 0 b q 2 with a := b + q 2 q 2. Let b in M R be the 2-dimensional integal convex polytope in M R with vetices (0, 0), (a, 0), (b, q 2) and (0, q 2) popely contained in the squae [0, q 2] [0, q 2], see Figue 1. It is the Minkowski sum of the line segment fom (0, 0) to (b, 0) and the polytope 0, see Figue 2.

7 The polytope b q 2 b q 2 a = b + q 2 Figue: The polytope b is the polytope with vetices (0, 0), (a = b + q 2, 0), (b, q 2), (0, q 2).

8 The polytope 0 q 2 q 2 a = q 2 Figue: The polytope 0 is the polytope with vetices (0, 0), (a = q 2, 0), (0, q 2).

9 The efined nomal fan V (ρ 4 ) V (ρ 3 ) V (ρ 2 ) V (ρ 1 ) Figue: The efined nomal fan and the 1-dimensional cones of the polytope 0 in Figue 2

10 The 1-dimensional cones in the efined nomal fan 0 of the polytope 0 ae geneated by unique pimitive elements n(ρ) such that ρ = R 0 n(ρ), specifically ae ) ) ) ) n ρ1 = ( 1 0, n ρ2 = ( 0 1, n ρ3 = ( 1 0, n ρ4 = ( 1, (1) see Figue 3.

11 Thee ae fou 2-dimensional cones in the efined nomal fan 0 : ( ) 0 1 σ 1 with faces ρ 1, ρ 2 and l σ1 = 0 ) 2 σ 2 with faces ρ 2, ρ 3 and l σ1 = ( q 2 0 ( q 2 3 σ 3 with faces ρ 3, ρ 4 and l σ1 = 0 ( ) 0 4 σ 4 with faces ρ 4, ρ 1 and l σ1 =. q 2 )

12 Let M be an intege lattice M Z 2. LetN = Hom Z (M, Z) be the dual lattice. The 2-dimensional algebaic tous T N k k is defined by T N := Hom Z (M, k ). The multiplicative chaacte e(m), m M is the homomophism e(m) : T k defined by e(m)(t) = t(m) fo t T N. Specifically, if {n 1, n 2 } and {m 1, m 2 } ae dual Z-bases of N and M and we denote u j := e(m j ), j = 1, 2, then we have an isomophism T N k k sending t to (u 1 (t), u 2 (t)). Fo m = λ 1 m 1 + λ 2 m 2 we have e(m)(t) = u 1 (t) λ 1 u 2 (t) λ 2. (2) The toic suface X b associated to the nomal fan b of b is X b = σ U σ, whee U σ is the k-valued points of the affine scheme Spec(k[S σ ]), i.e., mophisms u : S σ k with u(0) = 1 and u(m + m ) = u(m)u(m ) m, m S σ, whee S σ is the additive subsemigoup of M.

13 Toic code Intoduction Definition Fo each t T k k, we evaluate the ational functions in H 0 (X b, O X (D b )) H 0 (X b, O X (D b )) k f f (t). Let H 0 (X b, O X (D b )) Fob denote the ational functions in H 0 (X b, O X (D b )) that ae invaiant unde the action of the Fobenius, i.e. functions that ae F q -linea combinations of the functions e(m) in (2). Evaluating in all points in S = F q F q X b, we obtain the code C b (F q ) #S as the image H 0 (X b, O X (D h )) Fob C b (F q ) #S (3)

14 Paametes Intoduction Theoem Let F q be the field with q elements and let be an intege dividing q. Let b Z such that 0 b q 2 with a := b + q 2 q 2. Let b in M R be the 2-dimensional integal convex polytope in M R with vetices (0, 0), (a, 0), (b, q 2) and (0, q 2) contained in the squae [0, q 2] [0, q 2], see Figue 1. Let C b be the coesponding toic code of (3). Then n := length( C b = (q) 1) 2 and k = dim C b = 1 q q + b(q 1) and the minimum distance d(c b ) = (q 1 a)(q 1).

15 Notation Intoduction Let H be the Hilbet space H = C qn = C q C q... C q. Let x, x F q be an othonomal basis fo C q. Fo a, b F q, the unitay opeatos X (a) and Z(b) in C q ae X (a)x = x + a, Z(b)x = ω t(bx) x, (5) whee ω = exp(2πi/p) is a pimitive pth oot of unity and t is the tace opeation fom F q to F p. Fo a = (a 1,..., a n ) F n q and b = (b 1,..., b n ) F n q X (a) = X (a 1 ) X (a n ) Z(b) = Z(b 1 ) Z(b n ) ae the tenso poducts of n eo opeatos. With n E x = {X (a) = X (a i ) a F n q, a i F q }, i=1 n

16 Eo goups Intoduction The eo goups G x and G z ae G x = {ω c E x = ω c X (a) a F n q, c F p }, G z = {ω c E z = ω c Z(b) b F q nc F p }. It is assumed that the goups G x and G z epesent the qubit-flip and phase-shift eos.

17 Asymmetic quantum code Definition (Asymmetic quantum code) A q-ay asymmetic quantum code Q, denoted by [[n, k, d z /d x ]] q, is a q k dimensional subspace of the Hilbet space C qn and can dx 1 contol all bit-flip eos up to 2 and all phase-flip eos up dz 1 to 2. The code Q detects (d x 1) qubit-flip eos as well as detects (d z 1) phase-shift eos.

18 Let C 1 and C 2 be two linea eo-coecting codes ove the finite field F q, and let [n, k 1, d 1 ] q and [n, k 2, d 2 ] q be thei paametes. Fo the dual codes Ci, we have dim Ci = n k i and if C1 C 2 then C2 C 1. Lemma Let C i fo i = 1, 2 be linea eo-coecting codes with paametes [n, k i, d i ] q such that C1 C 2 and C2 C 1. Let d x = min { wt(c 1 \C2 ), wt(c 2\C1 )}, and d z = max { wt(c 1 \C2 ), wt(c 2\C1 )}. Then thee is an asymmetic quantum code with paametes [[n, k 1 + k 2 n, d z /d x ]] q. The quantum code is pue to its minimum distance, meaning that if wt(c 1 ) = wt(c 1 \C2 ), then the code is pue to d x, also if wt(c 2 ) = wt(c 2 \C1 ), then the code is pue to d z.

19 This constuction is well-known, see fo example [2], [4], [17], [23], [20], [21] [1]. The eo goups G x and G z can be mapped to the linea codes C 1 and C 2.

20 Let F q be the field with q elements and let be an intege dividing q. Let b Z such that 0 b ( 1)(q 2). The polytope b with vetices (0, 0), (a = b + q 2 ), (b, q 2), (0, q 2) is contained in [0, q 2] [0, q 2]. Conside the associated toic code C b of (3). Fom wok of Ruano [16, Theoem 6] we conclude that the dual code Cb is the toic code associated to the polytope b with vetices (0, 0), (a = b + q 2 ), (b, q 2), (0, q 2) whee b = ( 1)(q 2) b, such that a = q 2 b.

21 Fo i = 1, 2 let b i Z with 0 b i ( 1)(q 2) and b 1 + b 2 ( 1)(q 2). We have the inclusions of polytopes b b1 and 2 b b2, see Fig. 4, and coesponding 1 inclusions of the associated toic codes of (3): C b 2 = C b 2 C b1, C b 1 = C b 1 C b2. The nested codes gives by the constuction of Lemma 1 and the discussion above ise to an asymmetic quantum code Q b1,b 2.

22 Theoem (Asymmetic quatum codes Q b1,b 2 ) Let F q be the field with q elements and let be an intege dividing q. Fo i = 1, 2 let b i, a i = b i + q 2 Z 0 b i ( 1)(q 2) and b 1 + b 2 ( 1)(q 2). Then thee is ( an asymmetic ) quantum code Q b1,b 2 with paametes [[(q 1) 2, 1 q q + (b 1 + b 2 )(q 1), d z /d x ]] q, whee d z = (q 1 min{b 1, b 2 })(q 1) d x = (q 1 max{b 1, b 2 })(q 1) If b 1 + b 2 ( 1)(q 2) the quantum code is pue to d x and d z.

23 Poof. The paametes and claims follow diectly fom Lemma 1 and Theoem 2.

24 q 2 b 2 b 1 b 1 b 2 q a2 a 1 a a 2 1 Figue: The polytope bi is the polytope with vetices (0, 0), (a i = b i + q 2, 0), (b i, q 2), (0, q 2). The polytopes giving the dual toic codes have vetices (0, 0), (a = b + q 2, 0), Johan(b P. Hansen, q 2), Asymmetic (0, q 2), Quantum whee Codesb on Toic = qsufaces 2 a i.

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