arXiv subjects

Franz G. Fuchs

Publications and source records attributed to Franz G. Fuchs.

13 recordsLinked to original sources

arXiv2026

Iterative quantum algorithms for the minimum vertex cover problem based on continuous-time quantum walks

We introduce a constraint-preserving hybrid quantum-classical greedy framework for the minimum vertex cover problem, which extends directly to maximum independent set by bitwise complementation. The framework uses projected Pauli-X terms whose sum preserves the feasible subspace and acts within it exactly as the adjacency matrix of a layered graph of feasible covers. This graph is connected, so every feasible cover is linked to the configuration containing all vertices by a sequence of allowed single-vertex flips. Starting from this configuration, the corresponding continuous-time quantum walk propagates amplitude into layers containing progressively smaller covers. We rank vertices using either their marginal cover probabilities or the expected cover size obtained after fixing each candidate vertex in the cover, and use these rankings to guide recursive greedy reductions. Across several random-graph families, with walk times fixed using independent calibration ensembles, the quantum-informed algorithms achieve lower mean approximation ratios and solve a larger fraction of instances optimally than their corresponding classical greedy baselines. The conditioned-energy strategy performs best on the tested instances and retains algorithmic performance close to the exact continuous-time limit under low-depth Trotterisation. For bounded-degree graphs, each Trotter layer has circuit depth independent of system size, and the framework requires neither penalty terms nor variational training.

quant-ph↗

arXiv2026

Per-Shot Evaluation of QAOA on Max-Cut: A Black-Box Implementation Comparison with Goemans-Williamson

The Quantum Approximate Optimization Algorithm (QAOA) has emerged as a promising approach for addressing combinatorial optimization problems on near-term quantum hardware. In this work, we conduct an empirical evaluation of QAOA on the Max-Cut problem, using the Goemans-Williamson (GW) algorithm as a classical baseline for comparison. Unlike many prior studies, our methodology treats QAOA implementations as black-box optimizers, relying solely on default parameter settings without manual fine-tuning. We evaluate specific off-the-shelf QAOA implementations under default settings, not the algorithmic potential of QAOA with optimized parameters. This reflects a more realistic use case for end users who may lack the resources or expertise for instance-specific optimization. To facilitate fair and informative evaluation, we construct benchmark instances using well-known graph generation models that emulate practical graph structures, avoiding synthetic constructions tailored to either quantum or classical algorithms. A central component of our analysis is a per-shot statistical framework, which tracks the quality of QAOA outputs as a function of the number of circuit executions. This enables probabilistic comparisons with the GW algorithm by examining when and how frequently QAOA surpasses classical performance baselines such as the GW expectation and lower bound. Our results provide insight into the practical applicability of QAOA for Max-Cut and highlight its current limitations, offering a framework that can guide the assessment and development of future QAOA implementations.

quant-ph↗

arXiv2025

An Exact Branch and Bound Algorithm for the generalized Qubit Mapping Problem

Quantum circuits are typically represented by a (ordered) sequence of gates over a set of virtual qubits. During compilation, the virtual qubits of the gates are assigned to the physical qubits of the underlying quantum hardware, a step often referred to as the qubit assignment problem. To ensure that the resulting circuit respects hardware connectivity constraints, additional SWAP gates are inserted as needed, which is known as the qubit routing problem. Together, they are called the Qubit Mapping Problem (QMP), which is known to be NP-hard. A very common way to deal with the complexity of the QMP is to partition the sequence of gates into a sequence of gate groups (or layers). However, this imposes a couple of important restrictions: (1) SWAP gates can only be added between pairs of consecutive groups, and (2) all the gates belonging to a certain group have to be executed (in parallel) in the same time slot. The first one prevents gates to be re-arranged optimally, while the second one imposes a time discretization that practically ignores gate execution time. While this clearly reduces the size of the feasible space, little is still known about how much is actually lost by imposing a fixed layering when looking at the minimization of either the number of SWAPs or the makespan of the compiled circuit. In this paper, we present a flexible branch and bound algorithm for a generalized version of the QMP that either considers or ignores the gate layering and the gate execution time. The algorithm can find find proven optimal solutions for all variations of the QMP, but also offers a great platform for different heuristic algorithms. We present results on several benchmark sets of small quantum circuits, and we show how ignoring the layering can significantly improve some key performance indicators of the compiled circuit.

quant-ph↗

arXiv2025

Compact Circuits for Constrained Quantum Evolutions of Sparse Operators

We introduce a general framework for constructing compact quantum circuits that implement the real-time evolution of Hamiltonians of the form $H = \sigma P_B$, where $\sigma$ is a Pauli string commuting with a projection operator $P_B$ onto a subspace of the computational basis. Such Hamiltonians frequently arise in quantum algorithms, including constrained mixers in QAOA, fermionic and excitation operators in VQE, and lattice gauge theory applications. Additionally, we construct transposition gates, widely used in quantum computing, that scale more efficiently than the best known constructions in literature. Our method emphasizes the minimization of non-transversal gates, particularly T-gates, critical for fault-tolerant quantum computing. We construct circuits requiring $\mathcal{O}(n|B|)$ CX gates and $\mathcal{O}(n |B| + \log(|B|) \log (1/\epsilon))$ T-gates, where $n$ is the number of qubits, $|B|$ the dimension of the projected subspace, and $\epsilon$ the desired approximation precision. For subspaces that are generated by Pauli X-orbits we further reduce complexity to $\mathcal{O}(n \log |B|)$ CX gates and $\mathcal{O}(n+\log(\frac{1}{\epsilon}))$ T gates. Our constructive proofs yield explicit algorithms and include several applications, such as improved transposition circuits, efficient implementations of fermionic excitations, and oracle operators for combinatorial optimization. In the sparse case, i.e. when $|B|$ is small, the proposed algorithms scale favourably when compared to direct Pauli evolution.

quant-ph↗

arXiv2024

LX-mixers for QAOA: Optimal mixers restricted to subspaces and the stabilizer formalism

We present a novel formalism to both understand and construct mixers that preserve a given subspace. The method connects and utilizes the stabilizer formalism that is used in error correcting codes. This can be useful in the setting when the quantum approximate optimization algorithm (QAOA), a popular meta-heuristic for solving combinatorial optimization problems, is applied in the setting where the constraints of the problem lead to a feasible subspace that is large but easy to specify. The proposed method gives a systematic way to construct mixers that are resource efficient in the number of controlled not gates and can be understood as a generalization of the well-known X and XY mixers and a relaxation of the Grover mixer: Given a basis of any subspace, a resource efficient mixer can be constructed that preserves the subspace. The numerical examples provided show a dramatic reduction of CX gates when compared to previous results. We call our approach logical X-Mixer or logical X QAOA ($\textbf{LX-QAOA}$), since it can be understood as dividing the subspace into code spaces of stabilizers S and consecutively applying logical rotational X gates associated with these code spaces. Overall, we hope that this new perspective can lead to further insight into the development of quantum algorithms.

quant-ph↗

arXiv2024

Encodings of the weighted MAX k-CUT on qubit systems

The weighted MAX k-CUT problem involves partitioning a weighted undirected graph into k subsets, or colors, to maximize the sum of the weights of edges between vertices in different subsets. This problem has significant applications across multiple domains. This paper explores encoding methods for MAX k-CUT on qubit systems, utilizing quantum approximate optimization algorithms (QAOA) and addressing the challenge of encoding integer values on quantum devices with binary variables. We examine various encoding schemes and evaluate the efficiency of these approaches. The paper presents a systematic and resource efficient method to implement phase separation for diagonal square binary matrices. When encoding the problem into the full Hilbert space, we show the importance of encoding the colors in a balanced way. We also explore the option to encode the problem into a suitable subspace, by designing suitable state preparations and constrained mixers (LX- and Grover-mixer). Numerical simulations on weighted and unweighted graph instances demonstrate the effectiveness of these encoding schemes, particularly in optimizing circuit depth, approximation ratios, and computational efficiency.

quant-ph↗

arXiv2024

Quantum reservoir computing using the stabilizer formalism for encoding classical data

Utilizing a quantum system for reservoir computing has recently received a lot of attention. Key challenges are related to how on can optimally en- and decode classical information, as well as what constitutes a good reservoir. Our main contribution is a generalization of the standard way to robustly en- and decode time series into subspaces defined by the cosets of a given stabilizer. A key observation is the necessity to perform the decoding step, which in turn ensures a consistent way of encoding. This provides a systematic way to encode classical information in a robust way. We provide a numerical analysis on a discrete time series given by two standard maps, namely the logistic and the Hénon map. Our numerical findings indicate that the system's performance is increasing with the length of the training data.

quant-ph↗

arXiv2024

QuantumReservoirPy: A Software Package for Time Series Prediction

In recent times, quantum reservoir computing has emerged as a potential resource for time series prediction. Hence, there is a need for a flexible framework to test quantum circuits as nonlinear dynamical systems. We have developed a software package to allow for quantum reservoirs to fit a common structure, similar to that of reservoirpy which is advertised as "a python tool designed to easily define, train and use (classical) reservoir computing architectures". Our package results in simplified development and logical methods of comparison between quantum reservoir architectures. Examples are provided to demonstrate the resulting simplicity of executing quantum reservoir computing using our software package.

quant-ph↗

arXiv2022

Constrained mixers for the quantum approximate optimization algorithm

The quantum approximate optimization algorithm/quantum alternating operator ansatz (QAOA) is a heuristic to find approximate solutions of combinatorial optimization problems. Most literature is limited to quadratic problems without constraints. However, many practically relevant optimization problems do have (hard) constraints that need to be fulfilled. In this article, we present a framework for constructing mixing operators that restrict the evolution to a subspace of the full Hilbert space given by these constraints; We generalize the "XY"-mixer designed to preserve the subspace of "one-hot" states to the general case of subspaces given by a number of computational basis states. We expose the underlying mathematical structure which reveals more of how mixers work and how one can minimize their cost in terms of number of CX gates, particularly when Trotterization is taken into account. Our analysis also leads to valid Trotterizations for "XY"-mixer with fewer CX gates than is known to date. In view of practical implementations, we also describe algorithms for efficient decomposition into basis gates. Several examples of more general cases are presented and analyzed.

quant-ph↗

arXiv2020

Quantum Poker A game for quantum computers suitable for benchmarking error mitigation techniques on NISQ devices

Quantum computers are on the verge of becoming a commercially available reality. They represent a paradigm shift in computing, with a steep learning gradient. The creation of games is a way to ease the transition for beginners. We present a game similar to the Poker variant Texas hold 'em with the intention to serve as an engaging pedagogical tool to learn the basics rules of quantum computing. The concepts of quantum states, quantum operations and measurement can be learned in a playful manner. The difference to the classical variant is that the community cards are replaced by a quantum register that is "randomly" initialized, and the cards for each player are replaced by quantum gates, randomly drawn from a set of available gates. Each player can create a quantum circuit with their cards, with the aim to maximize the number of $1$'s that are measured in the computational basis. The basic concepts of superposition, entanglement and quantum gates are employed. We provide a proof-of-concept implementation using Qiskit. A comparison of the results for the created circuits using a simulator and IBM machines is conducted, showing that error rates on contemporary quantum computers are still very high. For the success of noisy intermediate scale quantum (NISQ) computers, improvements on the error rates and error mitigation techniques are necessary, even for simple circuits. We show that quantum error mitigation (QEM) techniques can be used to improve expectation values of observables on real quantum devices.

quant-ph↗

arXiv2018

Direct interactive visualization of locally refined spline volumes for scalar and vector fields

We present a novel approach enabling interactive visualization of volumetric Locally Refined B-splines (LR-splines). To this end we propose a highly efficient algorithm for direct visualization of scalar and vector fields given by an LR-spline. In both cases, our main contribution to achieve interactive frame rates is an acceleration structure for fast element look-up and a change of basis for efficient evaluation. To further improve the efficiency, we present a heuristic for adaptive sampling distance for the numerical integration. A comparison with existing adaptive approaches is performed. The algorithms are designed to fully utilize modern graphics processing unit (GPU) capabilities. Important applications where LR-spline volumes emerge are given for instance by approximation of large-scale simulation and sensor data, and Isogeometric Analysis (IGA). We showcase interactive rendering achieved by our approach on different representative use cases, stemming from simulations of wind flow around a telescope, Magnetic Resonance (MR) imaging of a human brain, and simulations of a fluidized bed used for mixing and coating particles in industrial processes.

cs.GR↗

arXiv2017

Uncertainty Quantification for Linear Hyperbolic Equations with Stochastic Process or Random Field Coefficients

In this paper hyperbolic partial differential equations with random coefficients are discussed. Such random partial differential equations appear for instance in traffic flow problems as well as in many physical processes in random media. Two types of models are presented: The first has a time-dependent coefficient modeled by the Ornstein--Uhlenbeck process. The second has a random field coefficient with a given covariance in space. For the former a formula for the exact solution in terms of moments is derived. In both cases stable numerical schemes are introduced to solve these random partial differential equations. Simulation results including convergence studies conclude the theoretical findings.

math.AP↗

arXiv2015

Interactive Isogeometric Volume Visualization with Pixel-Accurate Geometry

A recent development, called isogeometric analysis, provides a unified approach for design, analysis and optimization of functional products in industry. Traditional volume rendering methods for inspecting the results from the numerical simulations cannot be applied directly to isogeometric models. We present a novel approach for interactive visualization of isogeometric analysis results, ensuring correct, i.e., pixel-accurate geometry of the volume including its bounding surfaces. The entire OpenGL pipeline is used in a multi-stage algorithm leveraging techniques from surface rendering, order-independent transparency, as well as theory and numerical methods for ordinary differential equations. We showcase the efficiency of our approach on different models relevant to industry, ranging from quality inspection of the parametrization of the geometry, to stress analysis in linear elasticity, to visualization of computational fluid dynamics results.

cs.GR↗