Abstract Hodge decomposition and minimal models for cyclic algebras

We show that an algebra over a cyclic operad supplied with an additional linear algebra datum called Hodge decomposition admits a minimal model whose structure maps are given in terms of summation over trees. This minimal model is unique up to homotopy

Feynman diagrams and minimal models for operadic algebras

We construct an explicit minimal model for an algebra over the cobar-construction of a differential graded operad. The structure maps of this minimal model are expressed in terms of sums over decorated trees. We introduce the appropriate notion of a homotopy equivalence of operadic algebras and show that our minimal model is homotopy equivalent to the original algebra. All this generalizes and gives a conceptual explanation of well-known results for A∞-algebras. Furthermore, we show that these results carry over to the case of algebras over modular operads; the sums over trees get replaced by sums over general Feynman graphs. As a by-product of our work we prove gauge-independence of Kontsevich's ‘dual construction’ producing graph cohomology classes from contractible differential graded Frobenius algebras

Free resolutions of algebras

Given an algebra A, presented by generators and relations, i.e. as a quotient of a tensor algebra by an ideal, we construct a free algebra resolution of A, i.e. a differential graded algebra which is quasi-isomorphic to A and which is itself a tensor algebra. The construction rests combinatorially on the set of bracketings that arise naturally in the description of a free contractible differential graded algebra with given generators

Cocommutative coalgebras: homotopy theory and Koszul duality

We extend a construction of Hinich to obtain a closed model category structure on all differential graded cocommutative coalgebras over an algebraically closed field of characteristic zero. We further show that the Koszul duality between commutative and Lie algebras extends to a Quillen equivalence between cocommutative coalgebras and formal coproducts of curved Lie algebras.Comment: 38 page

Holographic data storage in a DX-center material

We report on the optical storage of digital data in a semiconductor sample containing DX centers. The diffraction efficiency and the bit-error-rate performance of multiplexed data images are shown to agree well with a simple model of the material. Uniform storage without an exposure schedule is demonstrated. The volume sensitivity is found to be ~10^3 times that of LiNBO3:Fe. The importance of coherent addition of scattered light with diffracted light in holographic data storage is discussed

Quantum parallelism of the controlled-NOT operation: an experimental criterion for the evaluation of device performance

It is shown that a quantum controlled-NOT gate simultaneously performs the logical functions of three distinct conditional local operations. Each of these local operations can be verified by measuring a corresponding truth table of four local inputs and four local outputs. The quantum parallelism of the gate can then be observed directly in a set of three simple experimental tests, each of which has a clear intuitive interpretation in terms of classical logical operations. Specifically, quantum parallelism is achieved if the average fidelity of the three classical operations exceeds 2/3. It is thus possible to evaluate the essential quantum parallelism of an experimental controlled-NOT gate by testing only three characteristic classical operations performed by the gate.Comment: 6 pages, no figures, added references and discussio

Deutsch-Jozsa algorithm as a test of quantum computation

A redundancy in the existing Deutsch-Jozsa quantum algorithm is removed and a refined algorithm, which reduces the size of the register and simplifies the function evaluation, is proposed. The refined version allows a simpler analysis of the use of entanglement between the qubits in the algorithm and provides criteria for deciding when the Deutsch-Jozsa algorithm constitutes a meaningful test of quantum computation.Comment: 10 pages, 2 figures, RevTex, Approved for publication in Phys Rev

Solution of steady and unsteady transonic-vortex flows using Euler and full-potential equations

Two methods are presented for inviscid transonic flows: unsteady Euler equations in a rotating frame of reference for transonic-vortex flows and integral solution of full-potential equation with and without embedded Euler domains for transonic airfoil flows. The computational results covered: steady and unsteady conical vortex flows; 3-D steady transonic vortex flow; and transonic airfoil flows. The results are in good agreement with other computational results and experimental data. The rotating frame of reference solution is potentially efficient as compared with the space fixed reference formulation with dynamic gridding. The integral equation solution with embedded Euler domain is computationally efficient and as accurate as the Euler equations

Programmable purification of type-I polarization-entanglement

We suggest and demonstrate a scheme to compensate spatial and spectral decoherence effects in the generation of polarization entangled states by type-I parametric downconversion. In our device a programmable spatial light modulator imposes a polarization dependent phase-shift on different spatial sections of the overall downconversion output and this effect is exploited to realize an effective purification technique for polarization entanglement.Comment: published versio

Property testing of unitary operators

In this paper, we systematically study property testing of unitary operators. We first introduce a distance measure that reflects the average difference between unitary operators. Then we show that, with respect to this distance measure, the orthogonal group, quantum juntas (i.e. unitary operators that only nontrivially act on a few qubits of the system) and Clifford group can be all efficiently tested. In fact, their testing algorithms have query complexities independent of the system's size and have only one-sided error. Then we give an algorithm that tests any finite subset of the unitary group, and demonstrate an application of this algorithm to the permutation group. This algorithm also has one-sided error and polynomial query complexity, but it is unknown whether it can be efficiently implemented in general