Johannes Bausch

The Complexity of Translationally Invariant Spin Chains with Low Local Dimension

We prove that estimating the ground state energy of a translationally invariant, nearest-neighbour Hamiltonian on a 1D spin chain is

Size-driven quantum phase transitions

\textsf {QMA}_{{\textsf {EXP}}}

Classifying Data with Local Hamiltonians.

QMAEXP-complete, even for systems of low local dimension (

The complexity of divisibility

\approx 40

Increasing the quantum UNSAT penalty of the circuit-to-Hamiltonian construction

≈40). This is an improvement over the best previously known result by several orders of magnitude, and it shows that spin-glass-like frustration can occur in translationally invariant quantum systems with a local dimension comparable to the smallest-known non-translationally invariant systems with similar behaviour. While previous constructions of such systems rely on standard models of quantum computation, we construct a new model that is particularly well-suited for encoding quantum computation into the ground state of a translationally invariant system. This allows us to shift the proof burden from optimizing the Hamiltonian encoding a standard computational model, to proving universality of a simple model. Previous techniques for encoding quantum computation into the ground state of a local Hamiltonian allow only a linear sequence of gates, hence only a linear (or nearly linear) path in the graph of all computational states. We extend these techniques by allowing significantly more general paths, including branching and cycles, thus enabling a highly efficient encoding of our computational model. However, this requires more sophisticated techniques for analysing the spectrum of the resulting Hamiltonian. To address this, we introduce a framework of graphs with unitary edge labels. After relating our Hamiltonian to the Laplacian of such a unitary labelled graph, we analyse its spectrum by combining matrix analysis and spectral graph theory techniques.

Perturbation Gadgets: Arbitrary Energy Scales from a Single Strong Interaction

Linear combinations of chi square random variables occur in a wide range of fields. Unfortunately, a closed, analytic expression for the probability density function is not yet known. Starting out from an analytic expression for the density of the sum of two gamma variables, a computationally efficient algorithm to numerically calculate the linear combination of chi square random variables is developed. An explicit expression for the error bound is obtained. The proposed technique is shown to be computationally efficient, i.e. only polynomial in growth in the number of terms compared to the exponential growth of most other methods. It provides a vast improvement in accuracy and shows only logarithmic growth in the required precision. In addition, it is applicable to a much greater number of terms and currently the only way of computing the distribution for hundreds of terms. As an application, the exponential dependence of the eigenvalue fluctuation probability of a random matrix model for 4D supergravity with N scalar fields is found to be of the asymptotic form exp(−0.35N).