One formidable difficulty in quantum communication and computation is to protect information-carrying states against undesired interactions with the environment. To address this difficulty, many good error-correcting codes have been derived as binary stabilizer codes. Fault-tolerant prompted study of nonbinary codes, but theory such not advanced that This paper describes basic over finite fields. The relation between general clarified by introducing a Galois for these objects. A...

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> Classical Bose–Chaudhuri–Hocquenghem (BCH) codes that contain their (Euclidean or Hermitian) dual can be used to construct quantum stabilizer codes; this correspondence studies the properties of such codes. It is shown a BCH code length <emphasis><formula formulatype="inline"> <tex>$n$</tex></formula></emphasis> its only if designed distance formulatype="inline"><tex>$\delta =O(\sqrt...

Recently, quantum error-correcting codes have been proposed that capitalize on the fact many physical error models lead to a significant asymmetry between probabilities for bit- and phase-flip errors. An example channel exhibits such is combined amplitude damping dephasing channel, where of bit phase flips can be related relaxation time, respectively. We study asymmetric are obtained from Calderbank–Shor–Steane (CSS) construction. For codes, we derive upper bounds code parameters using...

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

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

We show that any stabilizer code over a finite field is equivalent to graphical quantum code. Furthermore we prove The technique used in the proof establishes new connection between codes and quadratic forms. provide some simple examples illustrate our results.

We address the problem of constructing positive operator-valued measures (POVMs) in finite dimension n consisting n2 operators rank one which have an inner product close to uniform. This is motivated by related question symmetric informationally complete POVMs (SIC-POVMs) for products are perfectly However, SIC-POVMs notoriously hard construct and, despite some success them numerically, there no analytic construction known. present two constructions approximate versions SIC-POVMs, where a...

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

We construct nonbinary quantum codes from classical generalized Reed-Muller and derive the conditions under which these can be punctured. provide a partial answer to question raised by Grassl, Beth Rotteler on existence of q-ary MDS length n with q les <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> - 1

Recently, quantum error-correcting codes were proposed that capitalize on the fact many physical error models lead to a significant asymmetry between probabilities for bit flip and phase errors. An example channel which exhibits such is combined amplitude damping dephasing channel, where of flips can be related relaxation time, respectively. We give systematic constructions asymmetric stabilizer exploit this asymmetry. Our approach based CSS construction combines BCH finite geometry LDPC codes.

In this paper we investigate the use of quantum information to share classical secrets. While every secret sharing scheme is a error correcting code, converse not true. Motivated by sought find codes which can be converted schemes. If are interested in secrets using information, then show that class pure $[[n,1,d]]_q$ CSS perfect These schemes sense unauthorized parties do learn anything about secret. Gottesman had given conditions test whether subset an authorized or set; they enable us...

A classical computer does not allow the calculation of a discrete cosine transform on N points in less than linear time. This trivial lower bound is no longer valid for that takes advantage quantum mechanical superposition, entanglement, and interference principles. In fact, we show it possible to realize transforms sine size N/spl times/N types I, II, III IV with as little O(log/sup 2/N) operations computer; whereas known fast algorithms need O(N logN) operations.

Quantum convolutional codes can be used to protect a sequence of qubits arbitrary length against decoherence. We introduce two new families quantum codes. Our construction is based on an algebraic method which allows construct classical from block codes, in particular BCH These have the property that they contain their Euclidean, respectively Hermitian, dual Hence, define by stabilizer code construction. compute BCH-like bounds free distances controlled as case and establish non-catastrophic...

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

Instantaneous noise-based logic can avoid time-averaging, which implies significant potential for low-power parallel operations in beyond-Moore-law-chips. However, its random-telegraph-wave representation, the complete uniform superposition (superposition of all N-bit binary numbers) will be zero with high probability, that is, non-zero exponentially low thus would require exponential time-complexity. To fix this deficiency, we modify amplitudes signals L and H states achieve an speedup...

We discuss the problem of predicting number available parking spaces in a lot. The lot is modeled by continuous-time Markov chain, following Caliskan, Barthels, Scheuermann, and Mauve. regularly communicates occupied spaces, capacity, arrival rate through vehicular ad-hoc network. navigation system vehicle has to compute from this data probability an space upon arrival. derive structural result that considerably simplifies computation transition probabilities vehicle.

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 codes can be superior to We show that are provided the abstract error group has an Abelian index group. In particular, if errors modeled by tensor products Pauli matrices, then associated necessarily

Subsystem codes are a generalization of noiseless subsystems, decoherence-free subspaces and stabilizer codes. We generalize the quantum Singleton bound to q -linear subsystem It follows that no code over prime field can beat bound. On other hand, we show remarkable fact there exist impure beating Hamming A number open problems concern comparison in performance One suggested by Poulin's work asks whether use fewer syndrome measurements than an optimal maximum distance separable while...

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

The controlled-not gate and the single qubit gates are considered elementary in quantum computing. It is natural to ask how many such needed implement more elaborate or circuits. Recall that a controlled-U can be realized with two four gates. We prove this implementation optimal if only matrix U satisfies conditions trU\neq 0, tr(UX)\neq detU\neq 1. also derive implementations remaining non-generic cases. realizations of controlled unitary (pp139-155) G. Song A. Klappenecker