Constructions of \ell-Adic t-Deletion-Correcting Quantum Codes
CConstructios of ‘ -adic t -deletion-correcting quantum codes Ryutaroh Matsumoto ∗ and Manabu Hagiwara † February 8, 2021
Abstract
We propose two systematic constructions of deletion-correcting codes for protecting quan-tum information. The first one works with qudits of any dimention, but only one deletion iscorrected and the constructed codes are asymptotically bad. The second one corrects multipledeletions and can construct asymptotically good codes. The second one also allows conversionof stabilizer-based quantum codes to deletion-correcting codes, and entanglement assistance.
In the context of conventional (classical) error correction, deletion correction, which was introducedby Levenshtein in 1966 [11], has attracted much attention recently (see, for example, [21] and thereferences therein). In the correction of erasures, the receiver is aware of positions of erasures[2, 9, 12]. In contrast to this, the receiver is unaware of positions of deletions, which adds extradifficulty to correction of deletions and code constructions suitable for deletion correction. Partlydue to the combined difficulties of deletion correction and quantum error correction, the studyof quantum deletion correction has begun very recently [10, 15, 16]. Those researches providedconcrete examples of quantum deletion-correcting codes. The first systematic construction of 1-deletion-correcting binary quantum codes was proposed in [10], where ((2 k +2 − , k )) codes wereconstructed for any positive integer k . Very recently, the first systematic construction of t -deletion-correcting binary quantum codes was proposed [20, 19] for any positive integer t . The number ofcodewords was two in [20, 19]. There are the following problems in the existing studies: (1) Thereis no systematic construction for nonbinary quantum codes correcting more than 1 deletions. (2)Existing studies of quantum error correction cannot be reused in an obvious manner.In this paper, we tackle these problem by proposing two systematic constructions of nonbinaryquantum codes. The first one is based on the method of type in the information theory [6].The constructed codes belong to the class of permutation-invariant quantum codes [18, 19]. Itcan construct quantum codes for qudits of arbitrary dimension, but the codes can correct only 1deletion and asymptotically bad. The second construction converts quantum erasure-correctingcodes to deletion-correcting ones. The construction is asymptotically good, and can correct asmany deletions as the number of correctable erasures of the underlying quantum codes. But the ∗ Department of Information and Communications Enginnering, Tokyo Institute of Technology, Tokyo, 152-8550Japan. Email: [email protected]. Department of Mathematical Sciences, Aalborg University, Aalborg,Denmark. † Department of Mathematics and Informatics, Graduate School of Science, Chiba University, 1-33 Yayoi-cho,Inage-ku, Chiba City, Chiba Pref., 263-0022 Japan. a r X i v : . [ qu a n t - ph ] F e b econd construction has severe limitations on the dimension of qudits. For example, the secondconstruction cannot construct binary or ternary quantum codes.This paper is organized as follows: Section 2 introduces necessary notations and concepts.Section 3 proposes the first construction. Section 4 proposes the second construction. Section 5concludes the paper. Let Z ‘ = {
0, 1, . . . , ‘ − } . A type P [6] of length n on the alphabet Z ‘ is a probability distributionon Z ‘ such that each probability in P is of the form m/n , where m is an integer. The alphabet isfixed to Z ‘ when we consider types. For ~x = ( x , . . . , x n ) ∈ Z n‘ , the type P ~x of ~x is the probabilitydistribution P ~x ( a ) = ] { i | x i = a } /n , where ] denotes the number of elements in a set. For a type P of length n , T ( P ) denotes the set of all sequences with type P , that is, T ( P ) = { ~x ∈ Z n‘ | P ~x = P } . For types P and P , we define P ∼ P if there exists a permutation σ on ‘ numbers in a typesuch that σ ( P ) = P . For example, when P = (1 /
3, 1 /
6, 1 / σ ( P ) can be (1 /
6, 1 /
2, 1 / ∼ is an equivalence relation, and we can consider equivalence classes induced by ∼ . We denote anequivalence class represented by P as [ P ]. We define T [ P ] = S Q ∈ [ P ] T ( Q ). Definition 1
For ≤ t ≤ n − , we say a type P of length n − t to be a type of P after t deletion,where P is a type of length n , if • For each a ∈ Z ‘ , ( n − t ) P ( a ) ≤ nP ( a ) , • and P a ∈ Z ‘ { nP ( a ) − ( n − t ) P ( a ) } = t . We see that P ~y is a type of P ~x after t deletion if ~y is obtained by deleting t components in ~x . Definition 2
Let S = { P , . . . , P M − } be a set of types of length n . We call S to be suitable for t -deletion correction if for any Q ∈ [ P i ] and any Q ∈ [ P j ] with Q = Q there does not exist atype R of length n − t such that R is a type of both Q and Q after t deletion. Let H ‘ be the complex linear space of dimention ‘ . By an (( n, M )) ‘ quantum code we meanan M -dimentional complex linear subspace Q of H ⊗ n‘ . An (( n, M )) ‘ code is said to be ‘ -adic. Theinformation rate of Q is defined to be (log ‘ M ) /n . A code construction is said to be asymptoticallygood if it can give a sequence of codes with which lim inf n →∞ (log ‘ M ) /n > With a given S suitable for t -deletion correction, we construct (( n, M )) ‘ quantum code as follows:An M -level quantum state α | i + · · · + α M − | M − i is encoded to a codeword | ϕ i ∈ Q as M − X k =0 α k q ]T ([ P k ]) X ~x ∈ T ([ P k ]) | ~x i . In the next subsection, we will prove this construction can correct t = 1 deletion.2 .2 Proof of -Deletion Correction We assume t = 1 in this subsection (see Remark 3). The proof argument does not work for t > | ϕ i ∈ Q , any permutation of n qudits in | ϕ i does not change | ϕ i . Ourconstructed codes are instances of the permutation-invariant quantum codes [18, 19]. So any t deletion of | ϕ i is the same as deleting the first, the second, . . . , the t -th qudits in | ϕ i . Therefore, t deletion on | ϕ i ∈ Q can be corrected by assuming t erasures in the first, the second, . . . , the t -thqudits.By using Ogawa et al.’s condition [17, Theorem 1], we show that the code can correct oneerasure at the first qudit by computing the partial trace Tr { } [ | ϕ ih ϕ | ] of | ϕ ih ϕ | over the second,the third, . . . , and the n -th qudits.Let | ϕ k i = √ ]T ([ P k ]) P ~x ∈ T ([ P k ]) | ~x i . We first compute Tr { } [ | ϕ k ih ϕ k | ]. Let D be the deletionmap from Z n‘ to Z n − ‘ deleting the first component. For ~x ∈ Z n‘ , x i denotes the i -component.Tr { } [ | ϕ k ih ϕ k | ]= 1 ]T ([ P k ]) X a,b ∈ Z ‘ | a ih b | ] { ( ~x, ~y ) ∈ T ([ P k ]) × T ([ P k ]) | x = a, y = b, D ( ~x ) = D ( ~y ) } . When a = x = b = y and D ( ~x ) = D ( ~y ) we have P ~x = P ~y . Since there does not exist a type R of length n − R is P ~x after 1 deletion and also R is P ~y after 1 deletion, for any k there cannot exist ~x , ~y ∈ T ([ P k ]) such that a = x = b = y and D ( ~x ) = D ( ~y ). On the otherhand, by the symmetry of the construction, for any a ∈ Z ‘ , ] { ( ~x, ~y ) ∈ T ([ P k ]) ⊗ T ([ P k ]) | x = a = y , D ( ~x ) = D ( ~y ) } has the same size. Therefore, we see that ρ k = Tr { } [ | ϕ k ih ϕ k | ] = 1 ‘ X a ∈ Z ‘ | a ih a | . On the other hand, by the construction, for k = k , ~x ∈ T ([ P k ]), ~y ∈ T ([ P k ]), D ( ~x ) is alwaysdifferent from D ( ~y ), which impliesTr { } [ | ϕ ih ϕ | ] = M − X k =0 | α k | ρ k = I ‘ × ‘ /‘. (1)By [17, Theorem 1], this implies that the constructed code can correct one erasure at the firstqudit, which in turn implies one deletion correction by the symmetry of codewords with respectto permutations. Remark 3
When t > , Eq. (1) depends on the encoded quantum information, and one cannotapply [17, Theorem 1]. Since the number of types is polynomial in n [6], the proposed construction is asymptoticallybad. Let ‘ = n = 3. Then P = (1 , ,
0) and P = (1 / , / , /
3) are suitable for 1-deletion correction.This code was first found by Prof. Mikio Nakahara at Kindai University. Since 1-deletion correctingquantum code of length 2 is prohibited by the quantum no-cloning theorem [22], this code has theshortest possible length among all 1-deletion-correcting quantum codes.3 .3.2 Example 2
Let n = 7, ‘ = 3. Then P = (7 / , , P = (5 / , / , / P = (3 / , / , /
7) are suitable for1-deletion correction.
Let n = 8, ‘ = 4. Then P = (8 / , , , P = (6 / , / , / , P = (4 / , / , , P =(4 / , / , / , /
8) are suitable for 1-deletion correction.
The previous construction allows arbitrary ‘ , but the information rate (log ‘ M ) /n goes to zero as n → ∞ . In this section, we construct a t -deletion-correcting code over H ( t +1) ‘ , that is, we assumethat the qudit has ( t + 1) ‘ levels.We introduce an elementary lemma, which is known in the conventional coding theory [16]. Lemma 4
Let ~x = (0 , , . . . , t , , , . . . ) ∈ Z nt +1 . Let ~y be a vector after deletions of at most t components in ~x . Then one can determine all the deleted positions from ~y . Proof:
Let i = min { j | y j > y j +1 } . Then y , . . . , y i correspond to x , . . . , x t +1 . The set difference { x , . . . , x t +1 } \ { y , . . . , y i } reveals the deleted positions among x , . . . , x t +1 . Repeat the aboveprecedure from y j +1 until the rightmost component in ~y and one gets all the deleted positions.Let Q ⊂ H n‘ be a t -erasure-correcting (( n, M )) ‘ quantum code. A codeword | ψ i ∈ Q canbe converted to a codeword in the proposed t -deletion-correcting code as follows: Firstly, observe H ( t +1) ‘ is isomorphic to H ‘ ⊗ H t +1 . Let | ψ i = | · · · t · · ·i ∈ H ⊗ nt +1 . The sender sends | ψ i ⊗ | ψ i as a codeword in H ⊗ n ( t +1) ‘ .The receiver receives ρ ∈ S ( H ⊗ n − t ( t +1) ‘ ), where 0 ≤ t ≤ t , where S ( H ⊗ n − t ( t +1) ‘ ) denotes the set ofdensity matrices on H ⊗ n − t ( t +1) ‘ . The quantum system of received state can be divided to H ⊗ n − t ‘ and H ⊗ n − t . The receiver make a projective measurement on the subsystem H ⊗ n − t defined by {| ~y ih ~y || ~y ∈ Z n − t t +1 } . Then the receiver knows all the deleted positions. After that, the receiver applies theerasure correction procedure of Q , for example, [13] for quantum stabilizer codes [1, 4, 5, 8, 14].When ‘ is a prime power and t is fixed relative to n , lim n →∞ (log ‘ M ) /n can attain 1 [7], by theabove construction the information rate lim n →∞ (log ( t +1) ‘ M ) /n can attain log ( t +1) ‘ ‘ . Remark 5
Let ρ ∈ S ( H ⊗ n‘ ) be a quantum codeword of an entanglement assisted code [3]. By using ρ in place of | ϕ i in the last section, one can construct t -deletion-correcting entanglement assistedcode. This paper proposes two systematic constructions of quantum deletion-correcting codes. The firstone has advantage of supporting arbitrary dimension of qudits. The second one has advantagesof multiple deletion correction and asymptotic goodness. It is a future research agenda to find aconstruction of having all the above stated advantages.4 cknowledgment
The authors would like to thank Prof. Mikio Nakahara at Kindai University for the helpful dis-cussions.
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