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Quantum Generalized Reed-Solomon codes concatenated with random rate one inner stabilizer codes asymptotically attain the Quantum Gilbert-Varshamov bound

Yingkai Ouyang

Published Apr 7, 2010• 0 citations
OpenAlex

Abstract

AbstractA good quantum code corrects for a linear number of errors. It has been shownthat random codes attains the QGVB with a relative distance H −1q 2 ((1−R)/2) wherethe code is q-ary and R is the rate of the code (number of encoded systems / blocklength). However,randomcodeshavelittlestructure. Inthispaper,westudyafamilyofconcatenatedq-arystabilizercodes. Eachoftheinnercodesisarandomrate1q-arystabilizer code of block length n. The outer code is a quantum MDS code with blocklengthq n andalphabetsizeq n ,anarbitraryrateR≤1,anddistanceN(1−R)/2+1thatmeets the Quantum Singleton bound. Fixing the outer code rate and letting n grow,the concatenated stabilizer code has a distance that almost surely attains the QGVB.This partially generalizes Thommesen’s result, where heshowed thatthedistanceof aconcatenated code with a Reed-Solomon outer code and random inner linear codes ofarbitraryrateattainstheGilbert-Varshamovboundalmostsurely. 1 Introduction A family of q-ary quantum codes of increasing block length is defined to be good if the ratioof its distance to its block length approaches a non-zero constant. Designing good quantumcodes is highly nontrivial, just as it is in the classical case. The quantum Gilbert-Varshamovbound (QGVB) is a lower bound on an achievable relative distance of a quantum code ofa fixed rate. Explicit constructions of good quantum codes for q ≤ 7 have been studied[1, 2, 3], but they all fail to satisfy the QGVB. The QGVB bound is attainable for the familyof all random quantum codes [4], the family of random stabilizer codes [5], and the familyof random nondegenerate stabilizer codes [6], and the family of random degenerate stabilizercodes [6]. We show that concatenated quantum codes, with a quantum outer code havinga known structure and being efficiently decodable, and randomly chosen independent innerquantum codes also attains the QGVB. Thus our family of random quantum codes has morestructure than previously studied examples.In this paper, we generalize a special case of Thommesen’s result [7] to the quantumcase. He showed that a code from the family of binary concatenated codes made with a1

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AbstractA good quantum code corrects for a linear number of errors. It has been shownthat random codes attains the QGVB with a relative distance H −1q 2 ((1−R)/2) wherethe code is q-ary and R is the rate of the code (number of encoded systems / blocklength). However,randomcodeshavelittlestructure. Inthispaper,westudyafamilyofconcatenatedq-arystabilizercodes. Eachoftheinnercodesisarandomrate1q-arystabilizer code of block length n. The outer code is a quantum MDS code with blocklengthq n andalphabetsizeq n ,anarbitraryrateR≤1,anddistanceN(1−R)/2+1thatmeets the Quantum Singleton bound. Fixing the outer code rate and letting n grow,the concatenated stabilizer code has a distance that almost surely attains the QGVB.This partially generalizes Thommesen’s result, where heshowed thatthedistanceof aconcatenated code with a Reed-Solomon outer code and random inner linear codes ofarbitraryrateattainstheGilbert-Varshamovboundalmostsurely. 1 Introduction A family of q-ary quantum codes of increasing block length is defined to be good if the ratioof its distance to its block length approaches a non-zero constant. Designing good quantumcodes is highly nontrivial, just as it is in the classical case. The quantum Gilbert-Varshamovbound (QGVB) is a lower bound on an achievable relative distance of a quantum code ofa fixed rate. Explicit constructions of good quantum codes for q ≤ 7 have been studied[1, 2, 3], but they all fail to satisfy the QGVB. The QGVB bound is attainable for the familyof all random quantum codes [4], the family of random stabilizer codes [5], and the familyof random nondegenerate stabilizer codes [6], and the family of random degenerate stabilizercodes [6]. We show that concatenated quantum codes, with a quantum outer code havinga known structure and being efficiently decodable, and randomly chosen independent innerquantum codes also attains the QGVB. Thus our family of random quantum codes has morestructure than previously studied examples.In this paper, we generalize a special case of Thommesen’s result [7] to the quantumcase. He showed that a code from the family of binary concatenated codes made with a1
We show that for any R,ε>0R,ε>0R,ε>0 and any prime p≥2p\geq 2p≥2, there exists an infinite family of ppp-ary quantum low-density parity-check (QLDPC) codes, rate RRR, checks of weight Oε(1)O_ε(1)Oε​(1), and normalized distance at least δGV(p,R)−εδ_{\mathrm{GV}}(p,R)-εδGV​(p,R)−ε. Here, δGV(p,R)δ_{\mathrm{GV}}(p,R)δGV​(p,R) denotes the quantum Gilbert-Varshamov (GV) bound for ppp-ary stabilizer codes. In fact, we construct an ensemble of such QLDPC codes, such that a random code from this ensemble is close to the GV bound with high probability. Moreover, this ensemble matches the performance of random stabilizer codes on several quantum channels. Specifically, it approaches the quantum capacity of the erasure channel and the hashing bound for memoryless Pauli channels, including the depolarizing channel. A significant challenge in working with QLDPC codes is that they are necessarily \emph{degenerate}, i.e., contain many low-weight stabilizers. A key contribution of our work is a construction of QLDPC codes with quantitative control on their degeneracy. These codes are obtained by combining known constructions of asymptotically good QLDPC codes with the expander-based distance amplification procedure of Alon, Edmonds, and Luby [FOCS'95]. Our random ensemble is constructed by starting with these low-degeneracy QLDPC codes near the quantum Singleton bound and concatenating each coordinate with a random inner code. This can be viewed as a quantum analogue of Thommesen's construction, and as an LDPC version of a result of Ouyang, with an appropriately designed outer code.
A family of quantum codes of increasing block length with positive rate is asymptotically good if the ratio of its distance to its block length approaches a positive constant. The asymptotic quantum Gilbert-Varshamov (GV) bound states that there exist q -ary quantum codes of sufficiently long block length N having fixed rate R with distance at least NH-1((1-R)/2), where Hq2is the q2-ary entropy function. For q<;7 , only random quantum codes are known to asymptotically attain the quantum GV bound. However, random codes have little structure. In this paper, we generalize the classical result of Thommesen to the quantum case, thereby demonstrating the existence of concatenated quantum codes that can asymptotically attain the quantum GV bound. The outer codes are quantum generalized Reed-Solomon codes, and the inner codes are random independently chosen stabilizer codes, where the rates of the inner and outer codes lie in a specified feasible region.
Entanglement-assisted concatenated quantum codes (EACQCs) are constructed by concatenating two entanglement-assisted quantum error-correcting codes (EAQECCs). By selecting the inner and outer component codes carefully, it is able to construct state-of-the-art EACQCs with parameters better than previous quantum codes. In this work, we use almost maximum-distance-separable (MDS) codes and ℏ\hbarℏ-MDS codes as the outer codes to construct EACQCs. Because the range of code length of almost MDS and ℏ\hbarℏ-MDS codes is much more free than that of the commonly used MDS codes. We derive several families of new EACQCs with parameters better than the previously best known EAQECCs and standard quantum error-correcting codes (QECCs) of the same length and net transmissions. Moreover, we demonstrate that EACQCs are with maximal entanglement if both the inner and outer component codes are with maximal entanglement. As a result, we construct three new maximal-entanglement EACQCs which have optimal parameters. In addition, we present several new maximal-entanglement EACQCs whose minimum distance is only one less than the minimum distance of the optimal codes. In particular, we propose two new families of asymptotically good maximal-entanglement EACQCs with explicit constructions by using entanglement-assisted quantum algebraic geometry codes as the outer codes. At last, we prove that EACQCs can attain the quantum Gilbert-Varshamov bound for EAQECCs asymptotically.
Given a classical error-correcting block code, the task of quantum list decoding is to produce from any quantumly corrupted codeword a short list containing all messages whose codewords exhibit high "presence" in the quantumly corrupted codeword.Efficient quantum list decoders have been used to prove a quantum hardcore property of classical codes.However, the code rates of all known families of efficiently quantum list-decodable codes are, unfortunately, too small for other practical applications.To improve those known code rates, we prove that a specific code family of polynomially small code rate over a fixed code alphabet, obtained by concatenating generalized Reed-Solomon codes as outer codes with Hadamard codes as inner codes, has an efficient quantum list-decoding algorithm if its codewords have relatively high codeword presence in a given quantumly corrupted codeword.As an immediate application, we use the quantum list decodability of this code family to solve a certain form of quantum search problems in polynomial time.When the codeword presence becomes smaller, in contrast, we show that the quantum list decodability of generalized Reed-Solomon codes with high confidence is closely related to the efficient solvability of the following two problems: the noisy polynomial interpolation problem and the bounded distance vector problem.Moreover, assuming that NP BQP, we also prove that no efficient quantum list decoder exists for the generalized Reed-Solomon codes.
Guruswami et al. showed that the list-decodability of random linear codes is as good as that of general random codes. In this paper, we further strengthen the result by showing that the list-decodability of random Euclidean self-orthogonal codes is as good as that of general random codes as well, i.e., achieves the classical Gilbert-Varshamov bound. In particular, we show that, for any fixed finite field Fq, error fraction δ ∈ (0,1 - 1/q) satisfying 1 - Hq(δ) ≤ 1/2, and small ε > 0, with high probability a random Euclidean self-orthogonal code over Fqof rate 1 - Hq(δ) - ε is (δ, O(1/ε))-list-decodable. This generalizes the result of linear codes to Euclidean self-orthogonal codes. In addition, we extend the result to list decoding symplectic dual-containing codes by showing that the list-decodability of random symplectic dual-containing codes achieves the quantum Gilbert-Varshamov bound as well. This implies that list-decodability of quantum stabilizer codes can achieve the quantum Gilbert-Varshamov bound. The counting argument on self-orthogonal codes is an important ingredient to prove our result.