Martin Rötteler

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We present families of quantum error-correcting codes which are optimal in the sense that the minimum distance is maximal. These maximum distance separable (MDS) codes are defined over q-dimensional quantum systems, where q is an arbitrary prime power. It is shown that codes with parameters [[n,n − 2d + 2, d]]q exist for all 3 ≤ n ≤ q and 1 ≤ d ≤ n/2+1. We(More)
We present an algorithm for computing depth-optimal decompositions of logical operations, leveraging a meet-in-the-middle technique to provide a significant speedup over simple brute force algorithms. As an illustration of our method, we implemented this algorithm and found factorizations of commonly used quantum logical operations into elementary gates in(More)
Two orthonormal bases B andB′ of a d-dimensional complex inner-product space are called mutually unbiased if and only if |〈b|b〉| = 1/d holds for all b ∈ B and b′ ∈ B′. The size of any set containing pairwise mutually unbiased bases of C cannot exceed d + 1. If d is a power of a prime, then extremal sets containing d+1 mutually unbiased bases are known to(More)
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(More)
e?cient fault-tolerant quantum computing arxiv fault-tolerant quantum computing crcnetbase an introduction to quantum error correction and fault quantum error correction and fault tolerant quantum computing fault tolerance in quantum computation eceu fault-tolerant quantum computation world scientific fault -tolerant quantum computation versus realistic(More)
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 bitand 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(More)
It has been known for some time that graph isomorphism reduces to the hidden subgroup problem (HSP). What is more, most exponential speedups in quantum computation are obtained by solving instances of the HSP. A common feature of the resulting algorithms is the use of quantum coset states, which encode the hidden subgroup. An open question has been how hard(More)
We present two methods for the construction of quantum circuits for quantum errorcorrecting codes (QECC). The underlying quantum systems are tensor products of subsystems (qudits) of equal dimension which is a prime power. For a QECC encoding k qudits into n qudits, the resulting quantum circuit has O(n(n− k)) gates. The running time of the classical(More)