Quantum measurements and the Abelian Stabilizer Problem

We present a polynomial quantum algorithm for the Abelian stabilizer problem which includes both factoring and the discrete logarithm. Thus we extend famous Shor's results. Our method is based on a procedure for measuring an eigenvalue of a unitary operator. Another application of this procedure is a polynomial quantum Fourier transform algorithm for an arbitrary finite Abelian group. The paper also contains a rather detailed introduction to the theory of quantum computation.Comment: 22 pages, LATE

Crucial Words and the Complexity of Some Extremal Problems for Sets of Prohibited Words

We introduced the notation of a set of prohibitions and give definitions of a complete set and a crucial word with respect to a given set of prohibitions. We consider 3 particular sets which appear in different areas of mathematics and for each of them examine the length of a crucial word. One of these sets is proved to be incomplete. The problem of determining lengths of words that are free from a set of prohibitions is shown to be NP-complete, although the related problem of whether or not a given set of prohibitions is complete is known to be effectively solvable.Comment: 16 page

Exact results for spin dynamics and fractionization in the Kitaev Model

We present certain exact analytical results for dynamical spin correlation functions in the Kitaev Model. It is the first result of its kind in non-trivial quantum spin models. The result is also novel: in spite of presence of gapless propagating Majorana fermion excitations, dynamical two spin correlation functions are identically zero beyond nearest neighbor separation, showing existence of a gapless but short range spin liquid. An unusual, \emph{all energy scale fractionization}of a spin -flip quanta, into two infinitely massive $\pi$-fluxes and a dynamical Majorana fermion, is shown to occur. As the Kitaev Model exemplifies topological quantum computation, our result presents new insights into qubit dynamics and generation of topological excitations.Comment: 4 pages, 2 figures. Typose corrected, figure made better, clarifying statements and references adde

Universal Quantum Computation with the nu=5/2 Fractional Quantum Hall State

We consider topological quantum computation (TQC) with a particular class of anyons that are believed to exist in the Fractional Quantum Hall Effect state at Landau level filling fraction nu=5/2. Since the braid group representation describing statistics of these anyons is not computationally universal, one cannot directly apply the standard TQC technique. We propose to use very noisy non-topological operations such as direct short-range interaction between anyons to simulate a universal set of gates. Assuming that all TQC operations are implemented perfectly, we prove that the threshold error rate for non-topological operations is above 14%. The total number of non-topological computational elements that one needs to simulate a quantum circuit with $L$ gates scales as $L(\log L)^3$.Comment: 17 pages, 12 eps figure