arXiv subjects

Thorge Müller

Publications and source records attributed to Thorge Müller.

4 recordsLinked to original sources

arXiv2026

Beyond Quantum Advantage: Improved Classical Algorithms for the Binary Paint Shop Problem

The binary paint shop problem (BPSP) is an APX-hard optimization problem in which, given $n$ car models that occur twice in a sequence of length $2n$, the objective is to find a colouring sequence such that each car model pair is painted differently while minimizing the number of times the paint is swapped along the sequence. A recent classical heuristic, known as the recursive star greedy (RSG) algorithm, is conjectured to achieve an expected paint swap ratio of $0.361$, thereby outperforming the Quantum Approximate Optimization Algorithm (QAOA) with circuit depth $p=7$. Since the performance of the QAOA with logarithmic circuit depth is instance independent, the average paint swap-ratio is upper-bounded by the QAOA. We provide an improved upper-bound of the BPSP by extending the QAOA to depth $p=17$, outputting an expected paint swap ratio of $0.334$ via an exact computation while numerical extrapolation suggests a further reduction to a value of $0.295$. To provide hardware-relevant comparisons, we additionally implement the BPSP on a D-Wave Quantum Annealer Advantage 2, obtaining a minimum paint swap ratio of $0.329$. Given that the QAOA with logarithmic circuit depth does not exhibit a quantum advantage for sparse optimization problems such as the BPSP, this implies the existence of a classical algorithm that outperforms both the RSG algorithm and logarithmic depth QAOA. We provide numerical evidence that the Mean-Field Approximate Optimization Algorithm (MF-AOA) is one such algorithm, yielding a paint swap ratio of approximately $0.280$ beating all known classical and quantum algorithms for the BPSP.

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arXiv2026

Hamiltonian simulation with explicit formulas for Digital-Analog Quantum Computing

Digital-analog is a quantum computational paradigm that employs the natural interaction Hamiltonian of a system as the entangling resource, combined with single qubit gates, to implement universal quantum operations. As in the case of its digital gate-based counterpart, designing digital-analog circuits that employ optimal quantum resources often requires an exceedingly large classical computational time. In this work we find a suboptimal solution to this exponentially large problem, showing that it can be solved within polynomial computational time. In particular, we provide an exact solution for the problem of expressing arbitrary two-body Hamiltonians as the sum of local unitary transformations of an arbitrary Ising Hamiltonian, with the total number of required terms being at most quadratic in system size. This allows us to design a digital-analog simulation protocol that avoids employing numerical optimization over a large parameter space at the preprocessing stage, minimizing computational resources and allowing for further scaling.

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arXiv2024

Limitations of Quantum Approximate Optimization in Solving Generic Higher-Order Constraint-Satisfaction Problems

The ability of the Quantum Approximate Optimization Algorithm (QAOA) to deliver a quantum advantage on combinatorial optimization problems is still unclear. Recently, a scaling advantage over a classical solver was postulated to exist for random 8-SAT at the satisfiability threshold. At the same time, the viability of quantum error mitigation for deep circuits on near-term devices has been put in doubt. Here, we analyze the QAOA's performance on random Max-$k$XOR as a function of $k$ and the clause-to-variable ratio. As a classical benchmark, we use the Mean-Field Approximate Optimization Algorithm (MF-AOA) and find that it performs better than or equal to the QAOA on average. Still, for large $k$ and numbers of layers $p$, there may remain a window of opportunity for the QAOA. However, by extrapolating our numerical results, we find that reaching high levels of satisfaction would require extremely large $p$, which must be considered rather difficult both in the variational context and on near-term devices.

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arXiv2022

Approximating the quantum approximate optimization algorithm with digital-analog interactions

The quantum approximate optimisation algorithm was proposed as a heuristic method for solving combinatorial optimisation problems on near-term quantum computers and may be among the first algorithms to perform useful computations in the post-supremacy, noisy, intermediate scale era of quantum computing. In this work, we exploit the recently proposed digital-analog quantum computation paradigm, in which the versatility of programmable universal quantum computers and the error resilience of quantum simulators are combined to improve platforms for quantum computation. We show that the digital-analog paradigm is suited to the variational quantum approximate optimisation algorithm, due to its inherent resilience against coherent errors, by performing large-scale simulations and providing analytical bounds for its performance in devices with finite single-qubit operation times. We observe regimes of single-qubit operation speed in which the considered variational algorithm provides a significant improvement over non-variational counterparts.

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