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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… (More)

We present an algorithm for computing depth-optimal decompositions of logical operations, leveraging a meet-in-the-middle technique to provide a significant speed-up over simple brute force algorithms. As an illustration of our method we implemented this algorithm and found factorizations of the commonly used quantum logical operations into elementary gates… (More)

A classical computer does not allow to calculate a discrete cosine transform on AE 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 This trivial lower bound is no longer valid for a computer that takes advantage of quantum mechanical superposition, entanglement, and interference principles.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… (More)

- R F Werner, G Alber, T Beth, R M Horodecki, H Rötteler, R F Weinfurter +6 others
- 2000

This text is part of a volume entitled " Quantum information — an introduction to basic theoretical concepts and experiments " , to be published in Springer Tracts in Modern Physics. Authors will be From the foundations of quantum theory to quantum technology-an introduction Mixed-state entanglement and quantum communication Joint index Joint list of… (More)

Clifford codes are a class of quantum error control codes that form a natural generalization of stabilizer codes. These codes were introduced in 1996 by Knill, but only a single Clifford code was known, which is not already a stabilizer code. We derive a necessary and sufficient condition that allows to decide when a Clifford code is a stabilizer code, and… (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 an algorithm to construct quantum circuits for encoding and inverse encoding of quantum convo-lutional 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 Ollivier and Tillich… (More)

— 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, which yields nice error bases, and a combinato-rial construction by Werner, which yields shift-and-multiply bases. An open problem posed by Schlingemann and Werner relates these two… (More)

Recently, it was shown that Repeat-Until-Success (RUS) circuits can achieve a 2.5 times reduction in expected depth over ancilla-free techniques for single-qubit unitary decomposition. However, the previously best-known algorithm to synthesize RUS circuits requires exponential classical runtime. In this work we present an algorithm to synthesize an RUS… (More)

We present a family of non-abelian groups for which the hidden subgroup problem can be solved efficiently on a quantum computer.