The distance of a stabilizer quantum code is a very important feature since it determines the number of errors that can be detected and corrected. We present three new fast algorithms and implementations for computing the symplectic distance of the associated classical code. Our new algorithms are based on the Brouwer–Zimmermann algorithm. Our experimental study shows that these new implementations are much faster than current state-of-the-art licensed implementations on single-core processors, multicore processors, and shared-memory multiprocessors. In the most computationally-demanding cases, the performance gain in the computational time can be larger than one order of magnitude. The experimental study also shows a good scalability on shared-memory parallel architectures.

Quasi-twisted codes are used here as the classical ingredients in the so-called Construction X for quantum error-control codes. The construction utilizes nearly self-orthogonal codes to design quantum stabilizer codes. We expand the choices of the inner product to also cover the symplectic and trace-symplectic inner products, in addition to the original Hermitian one. A refined lower bound on the minimum distance of the resulting quantum codes is established and illustrated. We report numerous record breaking quantum codes from our randomized search for inclusion in the updated online database.

We show that a relatively simple reasoning using von Neumann entropy inequalities yields a robust proof of the quantum Singleton bound for quantum error-correcting codes (QECC). For entanglement-assisted quantum error-correcting codes (EAQECC) and catalytic codes (CQECC), a type of generalized quantum Singleton bound [Brunet al., IEEE Trans. Inf. Theory 60(6):3073–3089 (2014)] was believed to hold for many years until recently one of us found a counterexample [MG, Phys. Rev. A 103, 020601 (2021)]. Here, we rectify this state of affairs by proving the correct generalized quantum Singleton bound, extending the above-mentioned proof method for QECC; we also prove information-theoretically tight bounds on the entanglement-communication tradeoff for EAQECC. All of the bounds relate block length$n$and code length$k$for given minimum distance$d$and we show that they are robust, in the sense that they hold with small perturbations for codes which only correct most of the erasure errors of less than$d$letters. In contrast to the classical case, the bounds take on qualitatively different forms depending on whether the minimum distance is smaller or larger than half the block length. We also provide a propagation rule: any pure QECC yields an EAQECC with the same distance and dimension, but of shorter block length.

Traditional quantum error-correcting codes are designed for the depolarizing channel modeled by generalized Pauli errors occurring with equal probability. Amplitude damping channels model, in general, the decay process of a multilevel atom or energy dissipation of a bosonic system with Markovian bath at zero temperature. We discuss quantum error-correcting codes adapted to amplitude damping channels for higher dimensional systems (qudits). For multi-level atoms, we consider a natural kind of decay process, and for bosonic systems, we consider the qudit amplitude damping channel obtained by truncating the Fock basis of the bosonic modes (e.g., the number of photons) to a certain maximum occupation number. We construct families of single-error-correcting quantum codes that can be used for both cases. Our codes have larger code dimensions than the previously known single-error-correcting codes of the same lengths. In addition, we present families of multi-error correcting codes for these two channels, as well as generalizations of our construction technique to error-correcting codes for the qutrit V and Λ channels.

We consider the characterization as well as the construction of quantum codes that allow to transmit both quantum and classical information, which we refer to as `hybrid codes'. We construct hybrid codes [n, k:m, d]qwith length n and distance d, that simultaneously transmit k qudits and m symbols from a classical alphabet of size q. Many good codes such as [7,1:1, 3]2, [9,2:2,3]2, [10, 3:2, 3]2, [11,4:2, 3]2, [11,1:2,4]2, [13,1:4,4]2, [13,1:1, 5]2, [14,1:2, 5]2, [15,1:3, 5]2, [19, 9:1,4]2, [20, 9:2,4]2, [21, 9:3, 4]2, [22, 9:4,4]2have been found. All these codes have better parameters than hybrid codes obtained from the best known stabilizer quantum codes.

In this study we present a method that adapts the codeword stabilized (CWS) quantum code framework to the problem of finding asymmetric quantum codes. Making use of the the corresponding Pauli error models for amplitude and phase-damping models, we focus on codes that correct one or two amplitude-damping errors. As a result, we are able to exhaustively search for all possible codes up to length 9 by applying local Clifford operations on graph states. With a similar method, we also look at codes for the Pauli error models that detect a single amplitude error and detect multiple phase damping errors. Many new codes with good parameters are found, including non-additive codes and degenerate codes.

We discuss a method to construct quantum code correcting amplitude-damping errors via code concatenation. The inner codes are chosen as asymmetric Calderbank-Shor-Steane (CSS) codes. By concatenating with outer code-correcting symmetric errors, many new codes with good parameters are found, which outperform amplitude damping codes obtained by any previously known construction.

We consider quantum MDS (QMDS) codes for quantum systems of dimension q with lengths up to q2+ 2 and minimum distances up to q + 1. We show how starting from QMDS codes of length q2+ 1 based on cyclic and constacyclic codes, new QMDS codes can be obtained by shortening. We provide numerical evidence for our conjecture that almost all admissible lengths, from a lower bound n0(q, d) on, are achievable by shortening. Some additional codes that fill gaps in the list of achievable lengths are presented as well along with a construction of a family of QMDS codes of length q2+2, where q = 2m, that appears to be new.

We present new constructions of codes for asymmetric channels for both binary and nonbinary alphabets, based on methods of generalized code concatenation. For the binary asymmetric channel, our methods construct nonlinear single-error-correcting codes from ternary outer codes. We show that some of the Varshamov-Tenengol'ts-Constantin-Rao codes, a class of binary nonlinear codes for this channel, have a nice structure when viewed as ternary codes. In many cases, our ternary construction yields even better codes. For the nonbinary asymmetric channel, our methods construct linear codes for many lengths and distances which are superior to the linear codes of the same length capable of correcting the same number of symmetric errors.

Traditional quantum error-correcting codes are designed for the depolarizing channel modeled by generalized Pauli errors occurring with equal probability. Amplitude damping channels, in general, model the decay process of a multilevel atom or energy dissipation of a bosonic system at zero temperature. We discuss quantum error-correcting codes adapted to amplitude damping channels for higher dimensional systems (qudits). For multi-level atoms, we consider a natural kind of decay process, and for bosonic systems, we consider the qudit amplitude damping channel obtained by truncating the Fock basis of the bosonic modes to a certain maximum occupation number. We construct families of single-error-correcting quantum codes that can be used for both cases. Our codes have larger code dimensions than the previously known single-error-correcting codes of the same lengths.

We present a generalized construction of extended length Goppa codes. Using this construction, we obtain 71 new codes in finite field Fqfor q = 4, 7, 8, 9 with better minimum distance than the previously known codes with the same length and dimension.

We consider asymmetric quantum error-correcting codes that detect a single amplitude error. Both optimal additive and non-additive codes are presented.

Fault-tolerant quantum computation is a technique that is necessary to build a scalable quantum computer from noisy physical building blocks. Key for the implementation of fault-tolerant computations is the ability to perform a universal set of quantum gates that act on the code space of an underlying quantum code. To implement such a universal gate set fault-tolerantly is an expensive task in terms of physical operations, and any possible shortcut to save operations is potentially beneficial and might lead to a reduction in overhead for fault-tolerant computations. We show how the automorphism group of a quantum code can be used to implement some operators on the encoded quantum states in a fault-tolerant way by merely permuting the physical qubits. We derive conditions that a code has to satisfy in order to have a large group of operations that can be implemented transversally when combining transversal CNOT with automorphisms. We give several examples for quantum codes with large groups, including codes with parameters [8, 3, 3], [15, 7, 3], [22, 8, 4], and [31, 11, 5].

The concept of generalized concatenated quantum codes (GCQC) provides a systematic way for constructing good quantum codes from short component codes. We introduce a stabilizer formalism for GCQCs, which is achieved by defining quantum coset codes. This formalism offers a new perspective for GCQCs and enables us to derive a lower bound on the code distance of stabilizer GCQCs from component codes parameters, for both non-degenerate and degenerate component codes. Our formalism also shows how to exploit the error-correcting capacity of component codes to design good GCQCs efficiently.

We present an efficient construction of extended length Goppa codes. Using this construction, we obtain 78 new nonbinary codes with better minimum distance than the previously known codes with the same length and dimension. The construction is based on the observation that certain Goppa codes can be seen as BCH codes.

This paper deals with the problem of increasing the minimum distance of a linear code by adding one or more columns to the generator matrix. We present a simple greedy algorithm which surprisingly yields many codes improving the previously known lower bounds on the minimum distance. We also discuss variations of the algorithm that succeed when the greedy algorithm is not feasible or fails.

We present new constructions of codes for asymmetric channels for both binary and nonbinary alphabets, based on methods of generalized code concatenation. For the binary asymmetric channel, our methods construct nonlinear single-error-correcting codes from ternary outer codes. We show that some of the Varshamov-Tenengol'ts-Constantin-Rao codes, a class of binary nonlinear codes for this channel, have a nice structure when viewed as ternary codes. In many cases, our ternary construction yields even better codes. For the nonbinary asymmetric channel, our methods construct linear codes for many lengths and distances which are superior to the linear codes of the same length capable of correcting the same number of symmetric errors. In the binary case, Varshamov has shown that almost all good linear codes for the asymmetric channel are also good for the symmetric channel. Our results indicate that Varshamov's argument does not extend to the nonbinary case, i.e., one can find better linear codes for asymmetric channels than for symmetric ones.

The weights in maximum distance separable (MDS) codes of length n and dimension k over the finite field GF(q) are studied. Up to some explicit exceptional cases, the MDS codes with parameters given by the MDS conjecture are shown to contain all k weights in the range n - k + 1 to n. The proof uses the covering radius of the dual code.

It is shown that the generator matrix of a putative linear binary code [36, 10, 14] contains the generator matrix of a [32, 7, 14] or a [33, 8, 14] binary linear code. In a first step, all nonisomorphic [32, 7, 14] and [33, 8, 14] binary codes are enumerated. Then it is shown that none of these codes yields a [35, 10, 13] linear binary code.

We construct new families of multi-error-correcting quantum codes for the amplitude damping channel. Our key observation is that, with proper encoding, two uses of the amplitude damping channel simulate a quantum erasure channel. This allows us to use concatenated codes with quantum erasure-correcting codes as outer codes for correcting multiple amplitude damping errors. Our new codes are degenerate stabilizer codes and have parameters which are better than the amplitude damping codes obtained by any previously known construction.

Codeword stabilized (CWS) codes form a general class of quantum codes that includes stabilizer codes and many families of nonadditive codes with good parameters. Similar to classical nonlinear codes, a CWS code can be decoded by screening all possible errors. For an n-qubit quantum code correcting up to t errors, this brute-force approach consecutively tests different errors of weight t or less, and employs a separate n-qubit measurement in each test. To simplify decoding, we propose an algorithm that employs a single measurement to process all errors located on a given cluster of t qubits. Compared to an exhaustive error screening, this reduces the total number of measurements required for error correction about 3ttimes.

We consider the problem of computing an encoding circuit for a quantum convolutional code given by a polynomial stabilizer matrix S(D) = (X(D) | Z(D)). We present an algorithm that is very similar to a polynomial-time algorithm for computing the Smith form of a polynomial matrix. This is a step towards the conjecture that any quantum convolutional code has an encoder with polynomially bounded depth.

We show how good quantum error-correcting codes can be constructed using generalized concatenation. The inner codes are quantum codes, the outer codes can be linear or nonlinear classical codes. Many new good codes are found, including both stabilizer codes as well as so-called non-additive codes.

We consider the problem for which lengths a self-dual MDS code over Fqexists.We show that for q = 2m, there are self-dual MDS codes for all even lengths up to 2m. Furthermore, self-dual MDS codes of length q + 1 over Fqexist for all odd prime powers q. Additionally, we present some new self-dual MDS codes.

We present a family of non-additive quantum codes based on Goethals and Preparata codes with parameters ((2m, 22m-5m+1, 8)). The dimension of these codes is eight times higher than the dimension of the best known additive quantum codes of equal length and minimum distance.

We extend the stabilizer formalism to a class of nonadditive quantum codes which are constructed from non-linear classical codes. As an example, we present infinite families of nonadditive codes which are derived from Goethals and Preparata codes.

It is shown that chains of cyclic codes in conjunction with constructions X and XX can be used to efficiently construct sequences of linear codes for application in incremental redundancy communications. In addition to using a CRC for error detection, a novel CRC-less approach to error detection, which is based on the confidence level of the soft decision decoding output, is introduced. This novel approach provides an attractive trade-off between error rate performance and throughput for incremental redundancy communications.

This paper deals with the problem of increasing the minimum distance of a linear code by adding one or more columns to the generator matrix. Several methods to compute extensions of linear codes are presented. Many codes improving the previously known lower bounds on the minimum distance have been found.

We address the problems of constructing quantum convolutional codes (QCCs) and of encoding them. The first construction is a CSS-type construction which allows us to find QCCs of rate 2/4. The second construction yields a quantum convolutional code by applying a product code construction to an arbitrary classical convolutional code and an arbitrary quantum block code. We show that the resulting codes have highly structured and efficient encoders. Furthermore, we show that the resulting quantum circuits have finite depth, independent of the lengths of the input stream, and show that this depth is polynomial in the degree and frame size of the code.

Rate-(n-2)/n unrestricted and CSS-type quantum convolutional codes with up to 4096 states and minimum distances up to 10 are constructed as stabilizer codes from classical self-orthogonal rate-1/n F4-linear and binary linear convolutional codes, respectively. These codes generally have higher rate and less decoding complexity than comparable quantum block codes or previous quantum convolutional codes. Rate-(n-2)/n block stabilizer codes with the same rate and error-correction capability and essentially the same decoding complexity are derived from these convolutional codes via tail-biting

We present an algorithm to construct quantum circuits for encoding and inverse encoding of quantum convolutional codes. We show that any quantum convolutional code contains a subcode of finite index which has a non-catastrophic encoding circuit. Our work generalizes the conditions for non-catastrophic encoders derived in a paper by Oliver and Tillich (quant-ph/0401134) which are applicable only for a restricted class of quantum convolutional codes We also show that the encodes and their inverse constructed by our method naturally can be applied online, i.e., qubits can be sent and received with constant delay

A new algorithm is presented for computing the minimum weight of additive codes. It is superior to the existing method, achieving performance similar to the Brouwer-Zimmermann algorithm applied to linear codes of the same cardinality

We present a construction of self-orthogonal codes using product codes. From the resulting codes, one can construct both block quantum error-correcting codes and quantum convolutional codes. We show that from the examples of convolutional codes found, we can derive ordinary quantum error-correcting codes using tail-biting with parameters [42N, 24N, 3]2. While it is known that the product construction cannot improve the rate in the classical case, we show that this can happen for quantum codes: we show that a code [15, 7, 3]2is obtained by the product of a code [5,1,3]2with a suitable code

Well known constructions are applied to chains of quasi-cyclic codes in order to achieve new codes that improve on best known bounds on the minimum distance. These techniques have previously been successfully applied to chains of cyclic and algebraic-geometric codes. In conjunction with an improved algorithm for computing the minimum weight of quasi-cyclic codes, the constructions have so far yielded 274 new codes improving the lower bounds

We construct maximum distance separable quantum error-correcting codes. The codes are defined over q-dimensional quantum systems, where q is any prime power. The construction yields quantum MDS codes of length up to q+1 for all possible dimensions and some quantum MDS codes of length up to q2+1. In particular, those families contain codes [[6,2,3]]pand [[7,3,3]]pfor pges3 (cf. [K. Feng (2002)])

Starting with a chain of cyclic linear binary codes of length 127, linear binary codes of lengths 129-167, and dimensions 30-50 are constructed. Some of these codes have a minimum distance exceeding the lower bound given in Brouwer's table.

Owing to the high sensitivity of quantum mechanical systems to even small perturbations, means of error protection are essential for any computation or communication process based on quantum mechanics. After a short introduction to quantum registers and operations as well as quantum channels, different approaches to the problem of protecting quantum information are presented.