Complexity of commuting Hamiltonians on a square lattice of qubits

We consider the computational complexity of Hamiltonians which are sums of commuting terms acting on plaquettes in a square lattice of qubits, and we show that deciding whether the ground state minimizes the energy of each local term individually is in the complexity class NP. That is, if the ground states has this property, this can be proven using a classical certificate which can be efficiently verified on a classical computer. Different to previous results on commuting Hamiltonians, our certificate proves the existence of such a state without giving instructions on how to prepare it.Comment: 16 pages, 12 figures. v2: Minor corrections. Accepted version, Journal-Ref adde

Emergence of an European innovation system and its impact on Austria

The concept of a European Innovation System (EIS), defined as common effort of the EU as a whole and not merely as sum on national undertakings of the EU member states, has never been far below the surface for those seeking to create a united Europe. However, it needed a lot of years to emerge on the surface, still these days mor an - partly already operationalised - idea than an elaborated conceptual policy. In a brief historical perspective the emergence of the EIS wil be elaborated. At the time being, the 4th Framework Programme for European RTD (FP4) is running. It covers the period from 1994 until 1998 and encompasses an overall budget of approx. 15 billion ECU, which is considerably higher than the respective budgets in the late 1980s. However, the budget for FP4 and FP5 as well is still just around 4 % of the sum of all national public RTD-budgets of the 15 member states. Moreover, in terms of R&D expenditure, the EU is still lacking behind its main global competitors. While there is a slight but steady decreasing trend in those EU countries which already spend the largest proportion on R&D expenditure, most growth can be stated either in those countries starting from a relatively low base or the Nordic countries, while the Austrian value stagnates. The differences in the distribution of R&D expenditures by socio-economic objectives between the various national governments and the European Commission are remarkable. The objectives laid down in FP4 can be regarded as an additional value for the Austrian innovation system. Due to its specific nature, FP4 has some substantial advantages for the Austrian innovation system.First, Austria contributes roughly three per cent of the total FP4 budget, but has access to considerably more know-how. Second, the EU RTD programme is based on inter-institutional networking. This forces and facilitates the entry of industrial enterprises in research consortia and thus stimulates the co-operation between academic and enrepreneurial research efforts. Third, there are a lot of EU RTD efforts which focus directly on the active participation of SMEs, which form the overall industrial structure in Austria. The fourth advantage of the EU RTD programmes for Austria lies in its obvious and highly necessary concentration on high-tech sectors. In Austria the rate of export specialisation on goods with high R&D-input is twice as low as in the EU. First results of the Austrian participation in FP4 show some remarkable features. Out of 3972 submitted project proposals with Austrian participation, 1053 were funded by the EC. Especially successful were proposals with Austrian participants from the business sector (40 % of successful proposers), followed by participants from the universities (32 %) and non-university research institutions (16 %), both below their respective share in terms of application. Concerning the different technological programmes, Austria performed especially well in the non-nuclear energy programmes, in some of the environmental targeted research programmes, in information technologies and telematics as well as in transport targeted research.

Alternative descriptions of wave and particle aspects of the harmonic oscillator

The dynamical properties of the wave and particle aspects of the harmonic oscillator can be studied with the help of the time-dependent Schroedinger equation (SE). Especially the time-dependence of maximum and width of Gaussian wave packet solutions allow to show the evolution and connections of those two complementary aspects. The investigation of the relations between the equations describing wave and particle aspects leads to an alternative description of the considered systems. This can be achieved by means of a Newtonian equation for a complex variable in connection with a conservation law for a nonclassical angular momentum-type quantity. With the help of this complex variable, it is also possible to develop a Hamiltonian formalism for the wave aspect contained in the SE, which allows to describe the dynamics of the position and momentum uncertainties. In this case the Hamiltonian function is equivalent to the difference between the mean value of the Hamiltonian operator and the classical Hamiltonian function

Wigner Distribution Functions and the Representation of Canonical Transformations in Time-Dependent Quantum Mechanics

For classical canonical transformations, one can, using the Wigner transformation, pass from their representation in Hilbert space to a kernel in phase space. In this paper it will be discussed how the time-dependence of the uncertainties of the corresponding time-dependent quantum problems can be incorporated into this formalism.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

Simplex Z_2 spin liquids on the Kagome lattice with Entangled Pair States: spinon and vison coherence lengths, topological entropy and gapless edge modes

Gapped Z_2 spin liquids have been proposed as candidates for the ground-state of the S=1/2 quantum antiferromagnet on the Kagome lattice. We extend the use of Projected Entangled Pair States to construct (on the cylinder)Resonating Valence Bond (RVB) states including both nearest-neighbor and next-nearest neighbor singlet bonds. Our ansatz -- dubbed "simplex spin liquid" -- allows for an asymmetry between the two types of triangles (of order 2-3% in the energy density after optimization) leading to the breaking of inversion symmetry. We show that the topological Z_2 structure is still preserved and, by considering the presence or the absence of spinon and vison lines along an infinite cylinder, we explicitly construct four orthogonal RVB Minimally Entangled States. The spinon and vison coherence lengths are extracted from a finite size scaling w.r.t the cylinder perimeter of the energy splittings of the four sectors and are found to be of the order of the lattice spacing. The entanglement spectrum of a partitioned (infinite) cylinder is found to be gapless suggesting the occurrence, on a cylinder with {\it real} open boundaries, of gapless edge modes formally similar to Luttinger liquid (non-chiral) spin and charge modes. When inversion symmetry is spontaneously broken, the RVB spin liquid exhibits an extra Ising degeneracy, which might have been observed in recent exact diagonalisation studies.Comment: 5 pages, 6 figures - v2 with moderate revision of abstract, text and conclusio

Gapped Z2Z_2 spin liquid in the breathing kagome Heisenberg antiferromagnet

We investigate the spin-1/2 Heisenberg antiferromagnet on the kagome lattice with breathing anisotropy (i.e. with weak and strong triangular units), constructing an improved simplex Resonating Valence Bond (RVB) ansatz by successive applications (up to three times) of local quantum gates which implement a filtering operation on the bare nearest-neighbor RVB state. The resulting Projected Entangled Pair State involves a small number of variational parameters (only one at each level of application) and preserves full lattice and spin-rotation symmetries. Despite its simple analytic form, the simplex RVB provides very good variational energies at strong and even intermediate breathing anisotropy. We show that it carries $Z_2$ topological order which does not fade away under the first few applications of the quantum gates, suggesting that the RVB topological spin liquid becomes a competing ground state candidate for the kagome antiferromagnet at large breathing anisotropy

Quantum proofs can be verified using only single qubit measurements

QMA (Quantum Merlin Arthur) is the class of problems which, though potentially hard to solve, have a quantum solution which can be verified efficiently using a quantum computer. It thus forms a natural quantum version of the classical complexity class NP (and its probabilistic variant MA, Merlin-Arthur games), where the verifier has only classical computational resources. In this paper, we study what happens when we restrict the quantum resources of the verifier to the bare minimum: individual measurements on single qubits received as they come, one-by-one. We find that despite this grave restriction, it is still possible to soundly verify any problem in QMA for the verifier with the minimum quantum resources possible, without using any quantum memory or multiqubit operations. We provide two independent proofs of this fact, based on measurement based quantum computation and the local Hamiltonian problem, respectively. The former construction also applies to QMA$_1$, i.e., QMA with one-sided error.Comment: 7 pages, 1 figur

Semionic resonating valence bond states

The nature of the kagome Heisenberg antiferromagnet (HAFM) is under ongoing debate. While recent evidence points towards a Z_2 topological spin liquid, the exact nature of the topological phase is still unclear. In this paper, we introduce semionic Resonating Valence Bond (RVB) states, this is, Resonating Valence Bond states which are in the Z_2 ordered double-semion phase, and study them using Projected Entangled Pair States (PEPS). We investigate their physics and study their suitability as an ansatz for the HAFM, as compared to a conventional RVB state which is in the Toric Code Z_2 topological phase. In particular, we find that a suitably optimized "semionic simplex RVB" outperforms the equally optimized conventional "simplex RVB" state, and that the entanglement spectrum (ES) of the semionic RVB behaves very differently from the ES of the conventional RVB, which suggests to use the ES to discriminate the two phases. Finally, we also discuss the possible relevance of space group symmetry breaking in valence bond wavefunctions with double-semion topological order.Comment: 14 pages, 21 figures. v2: minor correction