Andreas Klappenecker

Nonbinary Stabilizer Codes Over Finite Fields

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 Fq in terms of classical codes over Fq 2 is provided that generalizes the well-known notion of additive codes over F4 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

Constructions of Mutually Unbiased Bases

Two orthonormal bases B and B′ of a d-dimensional complex inner-product space are called mutually unbiased if and only if |〈b|b′ 〉|2 = 1/d holds for all b ∈ B and b′ ∈ B′. The size of any set containing pairwise mutually unbiased bases of ℂ d cannot exceed d + 1. If d is a power of a prime, then extremal sets containing d+1 mutually unbiased bases are known to exist. We give a simplified proof of this fact based on the estimation of exponential sums. We discuss conjectures and open problems concerning the maximal number of mutually unbiased bases for arbitrary dimensions.

On Quantum and Classical BCH Codes

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(radicn), 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

Mutually unbiased bases are complex projective 2-designs

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)}

Graphs, quadratic forms, and quantum codes

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

Discrete cosine transforms on quantum computers

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 a computer that takes advantage of quantum mechanical superposition, entanglement, and interference principles. In fact, we show that it is possible to realize the discrete cosine transforms and the discrete sine transforms of size N/spl times/N and types I, II, III and IV with as little as O(log/sup 2/N) operations on a quantum computer; whereas the known fast algorithms on a classical computer need O(N logN) operations.

Asymmetric quantum codes: constructions, bounds and performance

Recently, quantum error-correcting codes have been proposed that capitalize on the fact that many physical error models lead to a significant asymmetry between the probabilities for bit- and phase-flip errors. An example for a channel that exhibits such asymmetry is the combined amplitude damping and dephasing channel, where the probabilities of bit and phase flips can be related to relaxation and dephasing time, respectively. We study asymmetric quantum codes that are obtained from the Calderbank–Shor–Steane (CSS) construction. For such codes, we derive upper bounds on the code parameters using linear programming. A central result of this paper is the explicit construction of some new families of asymmetric quantum stabilizer codes from pairs of nested classical codes. For instance, we derive asymmetric codes using a combination of Bose–Chaudhuri–Hocquenghem (BCH) and finite geometry low-density parity-check (LDPC) codes. We show that the asymmetric quantum codes offer two advantages, namely to allow a higher rate without sacrificing performance when compared with symmetric codes and vice versa to allow a higher performance when compared with symmetric codes of comparable rates. Our approach is based on a CSS construction that combines BCH and finite geometry LDPC codes.

Beyond stabilizer codes .I. Nice error bases

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.

Finding available parking spaces made easy

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

Asymmetric quantum LDPC 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.