arXiv subjects

Franz Georg Fuchs

Publications and source records attributed to Franz Georg Fuchs.

3 recordsLinked to original sources

arXiv2026

A Quantum Encoding of Traveling Salesperson Tours via Route Generation, Cost Phases, and a Reversible Valid-Permutation Oracle

For a traveling salesperson problem (TSP) of $n$ cities, we present a compact quantum encoding based on a time-register representation of tours. A candidate route is represented as a sequence of $n-1$ city labels over discrete time steps, with one fixed start city and the remaining cities encoded in binary registers. We describe three ingredients of the construction: uniform route generation over the route register, a reversible validity oracle, and a phase oracle that encodes the total tour cost. The validity oracle checks both that the non-start city labels form a permutation and, for incomplete graphs, that every directed edge used by the route exists. The cost oracle then accumulates the start-edge, intermediate-transition, and return-edge costs into a tour-dependent phase for valid routes. This yields a coherent superposition of candidate routes with feasibility and tour-length information embedded directly in the quantum state. The complete construction uses $\mathcal{O}(n\log n)$ qubits, while a naive implementation requires $\mathcal{O}(n^3\log_2 n)$ CX gates and $\mathcal{O}\!\left(n^3[\log_2 n+\log_2(1/\epsilon)]\right)$ $T$ gates, where $\epsilon$ denotes the target approximation precision. The encoding is compatible with amplitude amplification or spectral filtering techniques such as the quantum singular value transform (QSVT) or Grover's algorithm. However, due to the exponentially small fraction of valid tours, the overall complexity remains exponential even when combined with amplitude amplification.

quant-ph↗

arXiv2020

Efficient encoding of the weighted MAX k-CUT on a quantum computer using QAOA

The weighted MAX k-CUT problem consists of finding a k-partition of a given weighted undirected graph G(V,E) such that the sum of the weights of the crossing edges is maximized. The problem is of particular interest as it has a multitude of practical applications. We present a formulation of the weighted MAX k-CUT suitable for running the quantum approximate optimization algorithm (QAOA) on noisy intermediate scale quantum (NISQ)-devices to get approximate solutions. The new formulation uses a binary encoding that requires only |V|log_2(k) qubits. The contributions of this paper are as follows: i) A novel decomposition of the phase separation operator based on the binary encoding into basis gates is provided for the MAX k-CUT problem for k >2. ii) Numerical simulations on a suite of test cases comparing different encodings are performed. iii) An analysis of the resources (number of qubits, CX gates) of the different encodings is presented. iv) Formulations and simulations are extended to the case of weighted graphs. For small k and with further improvements when k is not a power of two, our algorithm is a possible candidate to show quantum advantage on NISQ devices.

quant-ph↗

arXiv2016

Uncertainty quantification for hyperbolic conservation laws with flux coefficients given by spatiotemporal random fields

In this paper hyperbolic partial differential equations with random coefficients are discussed. We consider the challenging problem of flux functions with coefficients modeled by spatiotemporal random fields. Those fields are given by correlated Gaussian random fields in space and Ornstein-Uhlenbeck processes in time. The resulting system of equations consists of a stochastic differential equation for each random parameter coupled to the hyperbolic conservation law. We define an appropriate solution concept in his setting and analyze errors and convergence of discretization methods. A novel discretization framework, based on Monte Carlo Finite Volume methods, is presented for the robust computation of moments of solutions to those random hyperbolic partial differential equations. We showcase the approach on two examples which appear in applications: The magnetic induction equation and linear acoustics, both with a spatiotemporal random background velocity field.

math.AP↗