We show that one of the Shor-Laflamme weight enumerators of a codeword stabilized quantum code may be interpreted as the distance enumerator of an associated classical code.

Hybrid codes simultaneously encode both quantum and classical information into physical qubits. We give several general results about hybrid codes, most notably that the quantum codes comprising a genuine hybrid code must be impure and that hybrid codes can always detect more errors than comparable quantum codes. We also introduce the weight enumerators for general hybrid codes, which we then use to derive linear programming bounds. Finally, inspired by the construction of some families of nonadditive codes, we construct several infinite families of genuine hybrid codes with minimum distance two and three.

A hybrid code can simultaneously encode classical and quantum information into quantum digits such that the information is protected against errors when transmitted through a quantum channel. It is shown that a hybrid code has the remarkable feature that it can detect more errors than a comparable quantum code that is able to encode the classical and quantum information. Weight enumerators are introduced for hybrid codes that allow to characterize the minimum distance of hybrid codes. Surprisingly, the weight enumerators for hybrid codes do not obey the usual MacWilliams identity.

Transversal operations are an elegant way to realize fault-tolerant quantum gates. Fault-tolerant quantum computation has been studied in detail over a finite field. In this paper, we derive transversal Clifford operations for CSS codes over nice rings, including Fourier transforms, SUM gates, and phase gates. Transversal operations alone cannot provide a computationally universal set of gates. As an example of a non-transversal gate, we derive fault-tolerant implementations of doubly-controlled Z gates for triorthogonal stabilizer codes over nice rings.

Dynamic distributed systems allow processes to join and leave the system, so the number of processes participating in a computation varies over time. Examples of dynamic distributed systems include peer-to-peer networks, sensor networks, mobile ad-hoc networks, and many more. A fundamental problem in any distributed system is to find consensus among the processes on a common input value. For dynamic distributed systems, it is not entirely clear how the problem should be formulated, as processes can join and leave before consensus is reached. We formulate and solve a strong version of the consensus problem in dynamic distributed systems in the presence of Byzantine faulty processes. We show that one cannot improve upon our algorithm in terms of the bound on the number of processes. For stochastic dynamic distributed systems, we determine the probability that a set of processes can reach strong consensus.

Subsystem codes are quantum error correcting schemes unifying stabilizer codes, decoherence free subspaces and noiseless subsystems. Subsystem codes were most commonly based on the generalized Pauli basis with dimensions of a power of prime. Recently, a class of nice error bases indexed by a nearring were introduced by the second author. We give a construction of subsystem codes over nice nearrings. Furthermore, we show that free subsystem codes over a finite chain ring cannot perform better than those over a finite field.

Nice error bases are a fundamental primitive of quantum information processing. For example, they govern the discretization of errors in quantum error-correcting codes. It is show that the generalized Pauli basis, the most widely used example of nice error bases, has some remarkable structural properties. However, the generalized Pauli basis is limited to dimensions that are a power of a prime, since it is constructed with the help of a finite field. A wider class of nice error bases is introduced that shares many features of the generalized Pauli basis, yet allows one to remove the restriction to prime power dimensions. The nice error bases are indexed by nearrings. Nearrings that support the construction of nice error bases are called nice. It is shown that all finite nearfields are nice. It is shown that a finite ring is nice if and only if it finite Frobenius ring. Several fundamental properties of nice nearrings are established.

Quantum error-correcting codes over finite fields have been widely studied, but quantum codes over rings have been left largely unexplored. This paper introduces stabilizer codes over finite Frobenius rings and establishes their connection to classical code. Structural properties of stabilizer codes over finite Frobenius rings are established. It is proved that free stabilizer codes over finite commutative chain rings cannot outperform stabilizer codes over finite fields.

Many envisioned applications of multi-robot swarms involve the detection, production or maintenance of global structures through only local means. This paper introduces a scalable, distributed algorithm to approximately characterize important global geometric and topological properties. For a given spatial arrangement of robots, the algorithm estimates the longest network (geodesic) distance in any direction as well as the average Euclidean distance only using locally sensed information. In so doing, the robots need only to communicate with and sense (range and bearing) nearby robots. The algorithm uses a greedy method to approximate both distance metrics via parallel one-way message traversals. We provide a bound for the number of such traversals, showing a global characterization is produced in a running time that is sublinear in the total number of robots. Along with this analysis, we conduct simulations with hundreds of robots to validate the algorithm.

Subsystem codes are a generalization of decoherence free subspaces, noiseless subsystems, and quantum error-correcting codes. The known constructions of subsystem codes from classical codes are limited to quantum systems that all have the same dimension, and this dimension must be a power of a prime. It is shown that one can remove these restrictions and obtain subsystem codes in quantum systems of arbitrary finite dimension from classical codes that are subgroups of an abelian group. The constructions are derived from Clifford codes over abstract error groups with abelian index groups.

In this paper we give new decoding algorithms for the generalized Shor codes and a class of subsystem codes due to Bacon and Casaccino. Our interest in these codes stems from the fact these codes can allow us to construct quantum codes from non-dual containing codes. In this paper we show how to decode these codes efficiently.

Subsystem codes are the most versatile class of quantum error-correcting codes known to date that combine the best features of all known passive and active error-control schemes. The subsystem code is a subspace of the quantum state space that is decomposed into a tensor product of two vector spaces: the subsystem and the co-subsystem. A generic method to derive subsystem codes from existing subsystem codes is given that allows one to trade the dimensions of subsystem and co-subsystem while maintaining or improving the minimum distance. As a consequence, it is shown that all pure MDS subsystem codes are derived from MDS stabilizer codes. The existence of numerous families of MDS subsystem codes is established. Propagation rules are derived that allow one to obtain longer and shorter subsystem codes from given subsystem codes. Furthermore, propagation rules are derived that allow one to construct a new subsystem code by combining two given subsystem 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 such asymmetry is the combined amplitude damping and dephasing channel, where the probabilities of bit flips and phase flips can be related to relaxation and dephasing time, respectively. We give systematic constructions of asymmetric quantum stabilizer codes that exploit this asymmetry. Our approach is based on a CSS construction that combines BCH and finite geometry LDPC codes.

No abstract available.

Recently, operator quantum error-correcting codes have been proposed to unify and generalize decoherence free subspaces, noiseless subsystems, and quantum error-correcting codes. This correspondence introduces a natural construction of such codes in terms of Clifford codes, an elegant generalization of stabilizer codes due to Knill. Character-theoretic methods are used to derive a simple method to construct operator quantum error-correcting codes from any classical additive code over a finite field, which obviates the need for self-orthogonal codes.

Convolutional stabilizer codes promise to make quantum communication more reliable with attractive online encoding and decoding algorithms. This paper introduces a new approach to convolutional stabilizer codes based on direct limit constructions. A quantum Singleton bound for pure convolutional stabilizer codes is given. A familiy of quantum convolutional codes is derived from generalized Reed-Solomon codes. These codes are shown to be optimal with respect to the (quantum) Singleton bound.

Duadic group algebra codes are a generalization of quadratic residue codes. This paper addresses an open problem raised by Zhu concerning the existence of duadic group algebra codes. These codes can be used to construct degenerate quantum stabilizer codes that have the nice feature that many errors of small weight do not need error correction.

Classical Bose–Chaudhuri–Hocquenghem (BCH) codes that contain their (Euclidean or Hermitian) dual codes can be used to construct quantum stabilizer codes; this correspondence studies the properties of such codes. It is shown that a BCH code of length$n$can contain its dual code only if its designed distance$\delta =O(\sqrt {n})$, and the converse is proved in the case of narrow-sense codes. Furthermore, the dimension of narrow-sense BCH codes with small design distance is completely determined, and – consequently – the bounds on their minimum distance are improved. These results make it possible to determine the parameters of quantum BCH codes in terms of their design parameters.

Good quantum codes, such as quantum MDS codes, are typically nondegenerate, meaning that errors of small weight require active error-correction, which is -unfortunately- itself prone to errors. In this paper, examples of degenerate quantum codes are constructed that alleviate this problem in that they allow some errors of small weight that do not require active error correction. In particular, two new families of [[n, 1, ges radicn]]qdegenerate quantum codes are derived from classical duadic codes

An attractive feature of BCH codes is that one can infer valuable information from their design parameters (length, size of the finite field, and designed distance), such as bounds on the minimum distance and dimension of the code. In this paper, it is shown that one can also deduce from the design parameters whether or not a primitive, narrow-sense BCH contains its Euclidean or Hermitian dual code. This information is invaluable in the construction of quantum BCH codes. A new proof is provided for the dimension of BCH codes with small designed distance, and simple bounds on the minimum distance of such codes and their duals are derived as a consequence. These results allow us to derive the parameters of two families of primitive quantum BCH codes as a function of their design parameters

One formidable difficulty in quantum communication and computation is to protect information-carrying quantum states against undesired interactions with the environment. To address this difficulty, many good quantum error-correcting codes have been derived as binary stabilizer codes. Fault-tolerant quantum computation prompted the study of nonbinary quantum codes, but the theory of such codes is not as advanced as that of binary quantum codes. This paper describes the basic theory of stabilizer codes over finite fields. The relation between stabilizer codes and general quantum codes is clarified by introducing a Galois theory for these objects. A characterization of nonbinary stabilizer codes over$bf F_q$in terms of classical codes over$ bf F_q^2$is provided that generalizes the well-known notion of additive codes over$ bf F_4$of the binary case. This paper also derives lower and upper bounds on the minimum distance of stabilizer codes, gives several code constructions, and derives numerous families of stabilizer codes, including quantum Hamming codes, quadratic residue codes, quantum Melas codes, quantum Bose–Chaudhuri–Hocquenghem (BCH) codes, and quantum character codes. The puncturing theory by Rains is generalized to additive codes that are not necessarily pure. Bounds on the maximal length of maximum distance separable stabilizer codes are given. A discussion of open problems concludes this paper.

Mutually unbiased bases (MUBs) are a primitive used in quantum information processing to capture the principle of complementarity. While constructions of maximal sets of d+1 such bases are known for system of prime power dimension d, it is unknown whether this bound can be achieved for any non-prime power dimension. In this paper we demonstrate that maximal sets of MUBs come with a rich combinatorial structure by showing that they actually are the same objects as the complex projective 2-designs with angle set {0, 1/d}. We also give a new and simple proof that symmetric informationally complete POVMs are complex projective 2-designs with angle set {1/(d+1)}

We construct nonbinary quantum codes from classical generalized Reed-Muller codes and derive the conditions under which these quantum codes can be punctured. We provide a partial answer to a question raised by Grassl, Beth and Rotteler on the existence of q-ary quantum MDS codes of length n with q les n les q2- 1

This paper proposes an energy efficient protocol for sensor data management. The protocol employs replicated data sinks to achieve (1) resiliency to data sink failure, and (2) efficiency in storing and retrieving sensor data. A simple address assignment scheme is introduced that partitions the sensor field into cells, where each cell contains one data sink and all sensors that are closest to this data sink. It is shown that this scheme is scalable and resilient against data sink and sensor node failures. Furthermore, the scheme has a reasonably low message complexity and a high energy efficiency

Unitary error bases generalize the Pauli matrices to higher dimensional systems. Two basic constructions of unitary error bases are known: An algebraic construction by Knill that yields nice error bases, and a combinatorial construction by Werner that yields shift-and-multiply bases. An open problem posed by Schlingemann and Werner relates these two constructions and asks whether each nice error basis is equivalent to a shift-and-multiply basis. We solve this problem and show that the answer is negative. However, we find that nice error bases have more structure than one can anticipate from their definition. In particular, we show that nice error bases can be written in a form in which at least half of the matrix entries are 0.

Clifford codes are quantum error control codes that generalize stabilizer codes. These codes were introduced in 1996 by Knill, but only a single nonstabilizer Clifford code was known to date. We derive a necessary and sufficient condition that allows one to decide when a Clifford code is a stabilizer code. We compile a table of all true Clifford codes for error groups of small order

Nice error bases have been introduced by Knill (1996) as a generalization of the Pauli basis. These bases are shown to be projective representations of finite groups. We classify all nice error bases of small degree, and all nice error bases with Abelian index groups. We show that, in general, an index group of a nice error basis is necessarily solvable.

For pt. I see ibid., vol.48, no.8, p.2392-95 (2002). Knill (1996) introduced a generalization of stabilizer codes, called Clifford codes. It remained unclear whether or not Clifford codes can be superior to stabilizer codes. We show that Clifford codes are stabilizer codes provided that the abstract error group has an Abelian index group. In particular, if the errors are modeled by tensor products of Pauli matrices, then the associated Clifford codes are necessarily stabilizer codes.

We show that it is possible to define Hartley-like transforms for (generalized) character tables of finite groups. This large class of transforms include Hartley transforms for discrete Fourier transforms over abelian groups and Hartley-like transforms for the discrete cosine transform of type I.