arXiv subjects

Lennart Binkowski

Publications and source records attributed to Lennart Binkowski.

14 recordsLinked to original sources

arXiv2026

One for All: Universal Quantum Conic Programming Framework for Hard-Constrained Combinatorial Optimization Problems

We present a unified quantum-classical framework for addressing NP-complete constrained combinatorial optimisation problems, generalising the recently proposed Quantum Conic Programming (QCP) approach. Accordingly, it inherits many favourable properties of the original proposal such as preventing barren plateaus and NP-hard parameter optimisation. By collecting the entire classical feasibility structure in a single constraint, we enlarge QCP's scope to arbitrary hard-constrained problems. Yet, we prove that the additional restriction is mild enough to still allow for an efficient parameter optimisation via the formulation of a generalised eigenvalue problem (GEP) of adaptable dimension. Our rigorous proof further fills some apparent gaps in prior derivations of GEPs from parameter optimisation problems. We further detail a measurement protocol for formulating the classical parameter optimisation that does not require us to implement any problem-specific objective Hamiltonian or a quantum feasibility oracle. Lastly, we prove that, even under the influence of noise, QCP's parameterised ansatz class always captures the optimum attainable within its generated subcone. All of our results hold true for arbitrarily-constrained combinatorial optimisation problems.

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arXiv2026

Symmetry-based quantum algorithms for open-shop scheduling with hard constraints

Encoding hard-constrained optimization problems into a variational quantum algorithm often turns out to be a challenging task. In this work, we provide a solution for the class of open-shop scheduling problems (OSSPs), which we achieve by rigorously employing the symmetries of the classical problem. An established approach for encoding the hard constraints of the closely related traveling salesperson problem (TSP) into mixer Hamiltonians was recently given by Hadfield et al.'s Quantum Alternating Operator Ansatz (QAOA). For the OSSP, which contains TSP as a special case, we show that desired properties of similarly constructed mixers can be directly linked to a purely classical object: the group of feasibility-preserving bit value permutations. We also outline a generic way to construct QAOA-like mixers for these problems. We further propose a new variational quantum algorithm that incorporates the underlying group structure more naturally and, as a proof of principle, implement our new algorithm for a small OSSP instance on an IBM Q System One. Unlike the generic QAOA, our algorithm allows for bounding the amount and the domain of parameters necessary to reach every feasible solution from above: Optimizing at most quadratically many parameters should suffice to reach the optimum with certainty.

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arXiv2026

Exhaustive and feasible parametrisation with applications to the travelling salesperson problem

This paper introduces the concept of exhaustively parametrised, feasibility-respecting quantum circuits for constrained combinatorial optimisation problems. Such circuits can reach, given the right parameter values, every feasible solution with certainty -- including the optimum -- with a fixed number of parameters, while avoiding infeasible solutions altogether. This is in sharp contrast to conventional quantum alternating operator ansatz schemes, which are merely guaranteed to reach the optimum asymptotically. We introduce an abstract pipeline for constructing exhaustively parametrised, feasibility-respecting circuits from a transitive group action on a problem's feasible set. Our constructions rely on the simple combination of the group action with group representation and the novel notion of generating sequences: group elements in fixed order, possibly with repetitions, that generate the entire group. That is, we trace expressivity of parametrised quantum circuits back to the most fundamental concepts of group theory. We apply this pipeline to two concrete examples for the travelling salesperson problem, thus showing that exhaustively parametrised, feasibility-respecting circuits are not an empty definition. Furthermore, we provide numerical proof-of-principles on instances with up to nine cities, comparing the suitability of our constructions for parameter optimisation purposes against established mixers.

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arXiv2026

Practical lower bounds for hybrid quantum interior point methods in linear programming

Quantum interior point methods (QIPMs) promise polynomial speed-ups over classical solvers for linear programming by outsourcing the solution of Newton linear systems to quantum linear solvers (QLSAs). However, asymptotic speed-ups do not necessarily translate to practical advantages on realistic problem instances. In this work, I evaluate whether practical advantage of a standard hybrid QIPM pipeline can already be excluded relative to the classical open-source solver HiGHS on a broad and diverse collection of LP instances spanning eight problem families, including public benchmark libraries, such as MIPlib, and relaxations of combinatorial optimisation problems. Following the hybrid benchmarking paradigm initiated by Cade et al., I derive rigorous lower bounds on the quantum runtime under a series of highly benevolent assumptions and compare them against classical runtimes. I equip the QIPMs with the best-performing functional QLSA, the Chebyshev-based method, as identified by Lefterovici et al., and evaluate two Newton system formulations proposed by Mohammadisiahroudi et al.: the modified normal equation system and the orthogonal subspace system. The exclusion analysis yields a consistent negative picture: across all instances and for any realistic quantum cycle duration, the quantum runtime lower bounds already exceed the classical runtimes, establishing that these hybrid QIPMs will offer no practical advantage over good classical solvers for realistic linear programming instances.

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arXiv2025

Deep-Circuit QAOA

Despite its popularity, several empirical and theoretical studies suggest that the quantum approximate optimization algorithm (QAOA) has persistent issues in providing a substantial practical advantage. Numerical results for few qubits and shallow circuits are, at best, ambiguous, and the well-studied barren plateau phenomenon draws a rather sobering picture for deeper circuits. However, as more and more sophisticated strategies are proposed to circumvent barren plateaus, it stands to reason which issues are actually fundamental and which merely constitute - admittedly difficult - engineering tasks. By shifting the scope from the usually considered parameter landscape to the quantum state space's geometry we can distinguish between problems that are fundamentally difficult to solve, independently of the parameterization, and those for which there could at least exist a favorable parameterization. Here, we find clear evidence for a 'no free lunch'-behavior of QAOA on a general optimization task with no further structure; individual cases have, however, to be analyzed more carefully. Based on our analysis, we propose and justify a performance indicator for the deep-circuit QAOA that can be accessed by solely evaluating statistical properties of the classical objective function. We further discuss the various favorable properties a generic QAOA instance has in the asymptotic regime of infinitely many gates, and elaborate on the immanent drawbacks of finite circuits. We provide several numerical examples of a deep-circuit QAOA method based on local search strategies and find that - in alignment with our performance indicator - some special function classes, like QUBOs, indeed admit a favorable optimization landscape.

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arXiv2025

Classification of coined quantum walks on the line and comparison to correlated classical random walks

We present a comprehensive classification of one-dimensional coined quantum walks on the infinite line, focusing on the spatial probability distributions they induce. Building on prior results, we identify all initial coin states that lead to symmetric quantum walks for arbitrary coins, and provide a bijective parametrisation of all symmetric quantum walks modulo distributional equivalence. Extending beyond the symmetric case, we also give a surjective parametrisation of all coined quantum walks under the same equivalence relation and a bijective parametrisation modulo equivalence of the walks' limiting distributions. Furthermore, we derive corrected closed-form expressions for the walk amplitudes, resolving inaccuracies in previous literature, and generalise the approach to the correlated classical random walk. This unified framework enables a direct comparison between quantum and classical dynamics. Additionally, we discuss the asymptotic scaling of variances for both models, identifying quadratic spreading as a hallmark of non-trivial quantum walks and contrasting it with the linear behaviour of classical walks, except at the extremal points of maximal correlation. Finally, we compare the limiting distributions arising from quantum walks with the ones in the classical case.

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arXiv2025

Fast generation of Pauli transfer matrices utilizing tensor product structure

Analysis of quantum processes, especially in the context of noise, errors, and decoherence is essential for the improvement of quantum devices. An intuitive representation of those processes modeled by quantum channels are Pauli transfer matrices. They display the action of a linear map in the $n$-qubit Pauli basis in a way, that is more intuitive, since Pauli strings are more tangible objects than the standard basis matrices. We set out to investigate classical algorithms that convert the various representations into Pauli transfer matrices. We propose new algorithms that make explicit use of the tensor product structure of the Pauli basis. They convert a quantum channel in a given representation (Chi or process matrix, Choi matrix, superoperator, or Kraus operators) to the corresponding Pauli transfer matrix. Moreover, the underlying principle can also be used to calculate the Pauli transfer matrix of other linear operations over $n$-qubit matrices such as left-, right-, and sandwich multiplication as well as forming the (anti-)commutator with a given operator. Finally, we investigate the runtime of these algorithms, derive their asymptotic scaling and demonstrate improved performance using instances with up to seven qubits.

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arXiv2025

Quantum tree generator improves QAOA state-of-the-art for the knapsack problem

This paper introduces a novel approach to the Quantum Approximate Optimization Algorithm (QAOA), specifically tailored to the knapsack problem. We combine the recently proposed quantum tree generator as an efficient state preparation circuit for all feasible solutions to the knapsack problem with the framework of Grover-mixer QAOA to form the first representative of Amplitude Amplification-mixer QAOA (AAM-QAOA). On hard benchmark sets with up to 20 knapsack items, we demonstrate our method's improved performance over the current state-of-the-art Copula-QAOA. However, for larger instance sizes, both approaches fail to deliver better outcomes than greedily packing items in descending value-to-weight ratio, at least for the considered circuit depths. For sufficiently high circuit depths, however, we can prove that AAM-QAOA will eventually be able to sample the optimal solution.

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arXiv2025

From barren plateaus through fertile valleys: Conic extensions of parameterised quantum circuits

Optimisation via parameterised quantum circuits is the prevalent technique of near-term quantum algorithms. However, the omnipresent phenomenon of barren plateaus - parameter regions with vanishing gradients - sets a persistent hurdle that drastically diminishes its success in practice. In this work, we introduce an approach - based on non-unitary operations - that favours jumps out of a barren plateau into a fertile valley. These operations are constructed from conic extensions of parameterised unitary quantum circuits, relying on mid-circuit measurements and a small ancilla system. We further reduce the problem of finding optimal jump directions to a low-dimensional generalised eigenvalue problem. As a proof of concept we incorporate jumps within state-of-the-art implementations of the Quantum Approximate Optimisation Algorithm (QAOA). We demonstrate the extensions' effectiveness on QAOA through extensive simulations, showcasing robustness against barren plateaus and highly improved sampling probabilities of optimal solutions.

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arXiv2025

A quantum search method for quadratic and multidimensional knapsack problems

Solving combinatorial optimization problems is a promising application area for quantum algorithms in real-world scenarios. In this work, we extend the "Quantum Tree Generator" (QTG), previously proposed for the 0-1 Knapsack Problem, to the 0-1 Quadratic Knapsack Problem (QKP) and the Multidimensional Knapsack Problem (MDKP). The QTG constructs a superposition of all feasible solutions for a given instance and can therefore be utilized as a promising state preparation routine within amplitude amplification to produce high-quality solutions. Previously, QTG-based search was tested on the 0-1 Knapsack Problem, where it demonstrated the potential for practical quantum advantage, once quantum computers with a few hundred logical and fully connected qubits are available. Here, we evaluate the algorithm's performance on QKP and MDKP against the classical solver Gurobi. To facilitate large-scale evaluations, we employ an advanced benchmarking technique that enables runtime predictions for instances with up to 2000 variables for QKP and up to 1500 variables and 100 constraints for MDKP. Our results indicate that QTG-based search can produce high-quality solutions with competitive runtimes for QKP. However, its performance declines for MDKP, highlighting the challenges quantum algorithms face when tackling highly constrained optimization problems.

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arXiv2025

Quantum Fisher-Yates shuffle: Unifying methods for generating uniform superpositions of permutations

Uniform superpositions over permutations play a central role in quantum error correction, cryptography, and combinatorial optimisation. We introduce a simple yet powerful quantisation of the classical Fisher-Yates shuffle, yielding a suite of efficient quantum algorithms for preparing such superpositions on composite registers. Our method replaces classical randomness with coherent control, enabling five variants that differ in their output structure and entanglement with ancillary systems. We demonstrate that this construction achieves the best known combination of asymptotic resources among all existing approaches, requiring only $\mathcal{O}(n \log(n))$ qubits and $\mathcal{O}(n^{2} \log(n))$ gates and circuit depth. These results position the quantum Fisher-Yates shuffle as a strong candidate for optimality within this class of algorithms. Our work unifies several prior constructions under a single, transparent framework and opens up new directions for quantum state preparation using classical combinatorial insights. Our implementation in Qiskit is available as open-source code, supporting reproducibility and future exploration of quantum permutation-based algorithms.

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arXiv2024

A quantum algorithm for solving 0-1 Knapsack problems

Here we present two novel contributions for achieving quantum advantage in solving difficult optimisation problems, both in theory and foreseeable practice. (1) We introduce the "Quantum Tree Generator", an approach to generate in superposition all feasible solutions of a given instance, yielding together with amplitude amplification the optimal solutions for 0-1 knapsack problems. The QTG offers massive memory savings and enables competitive runtimes compared to the classical state-of-the-art knapsack solvers (such as COMBO, Gurobi, CP-SAT, Greedy) already for instances involving as few as 100 variables. (2) By introducing a new runtime calculation technique that exploits logging data from the classical solver COMBO, we can predict the runtime of our method way beyond the range of existing quantum platforms and simulators, for various benchmark instances with up to 600 variables. Combining both of these innovations, we demonstrate the QTG's potential practical quantum advantage for large-scale problems, indicating an effective approach for combinatorial optimisation problems.

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arXiv2024

Tensorized Pauli decomposition algorithm

This paper introduces a novel general-purpose algorithm for Pauli decomposition that employs matrix slicing and addition rather than expensive matrix multiplication, significantly accelerating the decomposition of multi-qubit matrices. In a detailed complexity analysis, we show that the algorithm admits the best known worst-case scaling and more favorable runtimes for many practical examples. Numerical experiments are provided to validate the asymptotic speed-up already for small instance sizes, underscoring the algorithm's potential significance in the realm of quantum computing and quantum chemistry simulations.

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arXiv2023

Elementary Proof of QAOA Convergence

The Quantum Alternating Operator Ansatz (QAOA) and its predecessor, the Quantum Approximate Optimization Algorithm, are one of the most widely used quantum algorithms for solving combinatorial optimization problems. However, as there is yet no rigorous proof of convergence for the QAOA, we provide one in this paper. The proof involves retracing the connection between the Quantum Adiabatic Algorithm and the QAOA, and naturally suggests a refined definition of the `phase separator' and `mixer' keywords.

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