Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2508.05749 · Physical Review A

Expressivity limits in quantum walk-based optimization

Guilherme Adamatti Bridi, Debbie Lim, Lirandë Pira, Raqueline A. M. Santos, Franklin de Lima Marquezino, Soumik Adhikary

Published Jan 20, 2026• 1 citation
arXivHTMLPDF

Abstract

Quantum algorithms have emerged as a promising tool to solve combinatorial optimization problems. The quantum walk optimization algorithm (QWOA) is one such variational approach that has recently gained attention. In the broader context of variational quantum algorithms (VQAs), understanding the expressivity of the ansatz has proven critical for evaluating their performance. A key method to study this aspect involves analyzing the dimension of the dynamic Lie algebra (DLA). In this work, we derive novel upper bounds on the DLA dimension for QWOA applied to arbitrary optimization problems. Specifically, we show that the DLA dimension scales at most quadratically with the number of distinct eigenvalues of the problem Hamiltonian. As a consequence, our bound guarantees a polynomial DLA dimension with respect to the input size for optimization problems in the class $\mathsf{NPO}\text{-}\mathsf{PB}$. This result, coupled with recently established performance bounds for QWOA, allows us to identify complexity-theoretic conditions under which QWOA must be overparameterized to obtain optimal or approximate solutions for $\mathsf{NPO}\text{-}\mathsf{PB}$ problems.

DOI
10.1103/8l54-9g97

Related papers

On the dynamical Lie algebras of quantum approximate optimization algorithms

Jonathan Allcock, Miklós Sántha, Yuan Pei +1 more

Dynamical Lie algebras (DLAs) have emerged as a valuable tool in the study of parameterized quantum circuits, helping to characterize both their expressiveness and trainability. In particular, the absence or presence of barren plateaus (BPs) – flat regions in parameter space that prevent the efficient training of variational quantum algorithms – has recently been shown to be intimately related to quantities derived from the associated DLA.In this work, we investigate DLAs for the quantum approximate optimization algorithm (QAOA), one of the most studied variational quantum algorithms for solving graph MaxCut and other combinatorial optimization problems. While DLAs for QAOA circuits have been studied before, existing results have either been based on numerical evidence, or else correspond to DLA generators specifically chosen to be universal for quantum computation on a subspace of states. We initiate an analytical study of barren plateaus and other statistics of QAOA algorithms, and give bounds on the dimensions of the corresponding DLAs and their centers for general graphs. We then focus on the n -vertex cycle and complete graphs. For the cycle graph we give an explicit basis, identify its decomposition into the direct sum of a 2 -dimensional center and a semisimple component isomorphic to n − 1 copies of s u ( 2 ) . We give an explicit basis for this isomorphism, and a closed-form expression for the variance of the cost function, proving the absence of BPs. For the complete graph we prove that the dimension of the DLA is O ( n 3 ) and give an explicit basis for the DLA.

arXiv:2407.12587•Quantum•May 29, 2026•1 citation

Quantum optimisation applied to the Quadratic Assignment Problem

Andrew Freeland, Jingbo Wang

This paper investigates the performance of the emerging non-variational Quantum Walk-based Optimisation Algorithm (NV-QWOA) for solving small instances of the Quadratic Assignment Problem (QAP). NV-QWOA is benchmarked against classical heuristics, the MaxMin Ant System (MMAS) and Greedy Local Search (GLS), as well as the Grover quantum search algorithm, which serves as a quantum baseline. Performance is evaluated using two metrics: the number of objective function evaluations and the number of algorithm iterations required to consistently reach optimal or near optimal solutions across QAP instances with 5 to 10 facilities. The motivation for this study stems from limitations of both classical exact methods and current quantum algorithms. Variational Quantum Algorithms (VQAs), such as QAOA and VQE, while widely studied, suffer from costly parameter tuning and barren plateaus that hinder convergence. By adopting a non-variational approach, this work explores a potentially more efficient and scalable quantum strategy for combinatorial optimisation. The results provide a direct comparative analysis between classical and quantum frameworks, characterising the average case performance of NV-QWOA. Our findings highlight the practical utility of quantum walks for complex combinatorial problems and establish a foundation for future quantum optimisation algorithms.

quant-pharXiv:2601.01104•Jan 6, 2026•0 citations

Beyond Single Trajectories: Optimal Control and Jordan-Lie Algebra in Hybrid Quantum Walks for Combinatorial Optimization

Tianen Chen, Yun Shang

The Quantum Approximate Optimization Algorithm (QAOA) follows a single, fixed evolution path, overlooking the potential computational advantage of coherently superposing multiple trajectories. Here we overcome this limitation with a hybrid quantum walk (HQW) ansatz that super poses multiple Hamiltonian-driven paths coherently within each circuit layer via a dynamical coin operator. QAOA emerges as a special case of this framework with a static Pauli-X coin. Using Pontryagin's minimum principle, we derive the optimal form of the coin operator, demonstrating that it generally differs from a constant gate. A dynamical Lie algebra analysis reveals that HQW generates a strictly larger Jordan-Lie algebra, providing an algebraic foundation for its enhanced expressivity. Especially, we reveal the connection between the unique Jordan product negativity in HQW's DLA and its performance advantages. Numerical experiments on Max-Cut and Maximum Independent Set problems show that HQW systematically outperforms QAOA in convergence speed, solution accuracy, and robustness. Our work establishes a path-superposition paradigm for quantum optimization, combining optimal control theory with algebraic structure to guide the design of advanced quantum algorithms.

quant-pharXiv:2604.25760•Apr 29, 2026•0 citations

Benchmarking Quantum Heuristics: Non-Variational QWOA for Weighted Maxcut

Tavis Bennett, Aidan Smith, Edric Matwiejew +1 more

We present benchmarking results for the non-variational Quantum Walk Optimisation Algorithm (non-variational QWOA) applied to the weighted maxcut problem, using classical simulations for problem sizes up to $n = 31$. The amplified quantum state, prepared using a quadratic number of alternating unitaries, achieves a constant average-case measurement probability for globally optimal solutions across these problem sizes. This behaviour contrasts with that of classical heuristics, which, for NP-hard optimisation problems, typically exhibit solve probabilities that decay as problem size increases. Performance comparisons with two local-search heuristics on the same benchmark instances suggest that the non-variational QWOA may offer a meaningful advantage by scaling more favourably with problem size. These results provide supporting evidence for the potential of this quantum heuristic to achieve quantum advantage, though further work is needed to assess whether the observed performance scaling persists at larger problem sizes, and to confirm whether similar performance trends are observed for the other problem classes to which the non-variational QWOA is designed to generalise.

arXiv:2505.24191•May 30, 2025•0 citations

Benchmarking Lie-Algebraic Pretraining and Non-Variational QWOA for the MaxCut Problem

Matthaus Zering, Jolyon Joyce, Tal Gurfinkel +1 more

The Quantum Approximate Optimization Algorithm (QAOA) is a leading candidate for achieving quantum advantage in combinatorial optimization on Near-Term Intermediate-Scale Quantum (NISQ) devices. However, random initialization of the variational parameters typically leads to vanishing gradients, rendering standard variational optimization ineffective. This paper provides a comparative performance analysis of two distinct strategies designed to improve trainability: Lie algebraic pretraining framework that uses Lie-algebraic classical simulation to find near-optimal initializations, and non-variational QWOA (NV-QWOA) that targets a restrict parameter subspace covered by 3 hyperparameters. We benchmark both methods on the unweighted Maxcut problem using a circuit depth of $p = 256$ across 200 Erdős-Rényi and 200 3-regular graphs, each with 16 vertices. Both approaches significantly improve upon the standard randomly initialized QWOA. NV-QWOA attains a mean approximation ratio of 98.9\% in just 60 iterations, while the Lie-algebraic pretrained QWOA improves to 77.71\% after 500 iterations. That optimization proceeds more quickly for NV-QWOA is not surprising given its significantly smaller parameter space, however, that an algorithm with so few tunable parameters reliably finds near-optimal solutions is remarkable. These findings suggest that the structured parameterization of NV-QWOA offers a more robust training approach than pretraining on lower-dimensional auxiliary problems. Future work is needed to confirm scaling to larger problem sizes and to asses generalization to other problem classes.

quant-pharXiv:2512.22856•Dec 30, 2025•0 citations

Efficient Estimation of Reduced QAOA Expressibility on Acyclic Graphs

Bao Bach, Boris Tsvelikhovskiy, Jose Falla +1 more

Classically equivalent formulations of an optimization problem need not lead to equivalent quantum algorithms. For MaxCut, fixing the value of a single vertex removes a simple global symmetry without changing the underlying optimization problem, yet it can substantially alter the quantum dynamics generated by the Quantum Approximate Optimization Algorithm (QAOA). These dynamics are captured by the circuit's dynamical Lie algebra (DLA), whose direct construction can become exponentially expensive. Here we show that, for symmetry-reduced QAOA on trees, substantial information about the reduced DLA can instead be obtained efficiently from the graph alone. We introduce a polynomial classical algorithm that recursively distinguishes vertices using shortest path structure and degree parity. When all vertices are resolved individually, the method determines the complete reduced DLA and, under the corresponding graph conditions, certifies maximal expressibility of the reduced QAOA ansatz. Even when full resolution is not achieved, the algorithm identifies embedded subalgebras, provides rigorous lower bounds on DLA dimension, and certifies controllable subsystems. Our theoretical and experimental results show how classical graph structure can be used to diagnose and potentially guide the quantum dynamics of variational algorithms before running them on quantum hardware.

quant-pharXiv:2609.03317•Sep 4, 2026•0 citations
Quantum algorithms have emerged as a promising tool to solve combinatorial optimization problems. The quantum walk optimization algorithm (QWOA) is one such variational approach that has recently gained attention. In the broader context of variational quantum algorithms (VQAs), understanding the expressivity of the ansatz has proven critical for evaluating their performance. A key method to study this aspect involves analyzing the dimension of the dynamic Lie algebra (DLA). In this work, we derive novel upper bounds on the DLA dimension for QWOA applied to arbitrary optimization problems. Specifically, we show that the DLA dimension scales at most quadratically with the number of distinct eigenvalues of the problem Hamiltonian. As a consequence, our bound guarantees a polynomial DLA dimension with respect to the input size for optimization problems in the class NPO-PB\mathsf{NPO}\text{-}\mathsf{PB}NPO-PB. This result, coupled with recently established performance bounds for QWOA, allows us to identify complexity-theoretic conditions under which QWOA must be overparameterized to obtain optimal or approximate solutions for NPO-PB\mathsf{NPO}\text{-}\mathsf{PB}NPO-PB problems.
Dynamical Lie algebras (DLAs) have emerged as a valuable tool in the study of parameterized quantum circuits, helping to characterize both their expressiveness and trainability. In particular, the absence or presence of barren plateaus (BPs) – flat regions in parameter space that prevent the efficient training of variational quantum algorithms – has recently been shown to be intimately related to quantities derived from the associated DLA.In this work, we investigate DLAs for the quantum approximate optimization algorithm (QAOA), one of the most studied variational quantum algorithms for solving graph MaxCut and other combinatorial optimization problems. While DLAs for QAOA circuits have been studied before, existing results have either been based on numerical evidence, or else correspond to DLA generators specifically chosen to be universal for quantum computation on a subspace of states. We initiate an analytical study of barren plateaus and other statistics of QAOA algorithms, and give bounds on the dimensions of the corresponding DLAs and their centers for general graphs. We then focus on the n -vertex cycle and complete graphs. For the cycle graph we give an explicit basis, identify its decomposition into the direct sum of a 2 -dimensional center and a semisimple component isomorphic to n − 1 copies of s u ( 2 ) . We give an explicit basis for this isomorphism, and a closed-form expression for the variance of the cost function, proving the absence of BPs. For the complete graph we prove that the dimension of the DLA is O ( n 3 ) and give an explicit basis for the DLA.
This paper investigates the performance of the emerging non-variational Quantum Walk-based Optimisation Algorithm (NV-QWOA) for solving small instances of the Quadratic Assignment Problem (QAP). NV-QWOA is benchmarked against classical heuristics, the MaxMin Ant System (MMAS) and Greedy Local Search (GLS), as well as the Grover quantum search algorithm, which serves as a quantum baseline. Performance is evaluated using two metrics: the number of objective function evaluations and the number of algorithm iterations required to consistently reach optimal or near optimal solutions across QAP instances with 5 to 10 facilities. The motivation for this study stems from limitations of both classical exact methods and current quantum algorithms. Variational Quantum Algorithms (VQAs), such as QAOA and VQE, while widely studied, suffer from costly parameter tuning and barren plateaus that hinder convergence. By adopting a non-variational approach, this work explores a potentially more efficient and scalable quantum strategy for combinatorial optimisation. The results provide a direct comparative analysis between classical and quantum frameworks, characterising the average case performance of NV-QWOA. Our findings highlight the practical utility of quantum walks for complex combinatorial problems and establish a foundation for future quantum optimisation algorithms.
The Quantum Approximate Optimization Algorithm (QAOA) follows a single, fixed evolution path, overlooking the potential computational advantage of coherently superposing multiple trajectories. Here we overcome this limitation with a hybrid quantum walk (HQW) ansatz that super poses multiple Hamiltonian-driven paths coherently within each circuit layer via a dynamical coin operator. QAOA emerges as a special case of this framework with a static Pauli-X coin. Using Pontryagin's minimum principle, we derive the optimal form of the coin operator, demonstrating that it generally differs from a constant gate. A dynamical Lie algebra analysis reveals that HQW generates a strictly larger Jordan-Lie algebra, providing an algebraic foundation for its enhanced expressivity. Especially, we reveal the connection between the unique Jordan product negativity in HQW's DLA and its performance advantages. Numerical experiments on Max-Cut and Maximum Independent Set problems show that HQW systematically outperforms QAOA in convergence speed, solution accuracy, and robustness. Our work establishes a path-superposition paradigm for quantum optimization, combining optimal control theory with algebraic structure to guide the design of advanced quantum algorithms.
We present benchmarking results for the non-variational Quantum Walk Optimisation Algorithm (non-variational QWOA) applied to the weighted maxcut problem, using classical simulations for problem sizes up to n=31n = 31n=31. The amplified quantum state, prepared using a quadratic number of alternating unitaries, achieves a constant average-case measurement probability for globally optimal solutions across these problem sizes. This behaviour contrasts with that of classical heuristics, which, for NP-hard optimisation problems, typically exhibit solve probabilities that decay as problem size increases. Performance comparisons with two local-search heuristics on the same benchmark instances suggest that the non-variational QWOA may offer a meaningful advantage by scaling more favourably with problem size. These results provide supporting evidence for the potential of this quantum heuristic to achieve quantum advantage, though further work is needed to assess whether the observed performance scaling persists at larger problem sizes, and to confirm whether similar performance trends are observed for the other problem classes to which the non-variational QWOA is designed to generalise.
The Quantum Approximate Optimization Algorithm (QAOA) is a leading candidate for achieving quantum advantage in combinatorial optimization on Near-Term Intermediate-Scale Quantum (NISQ) devices. However, random initialization of the variational parameters typically leads to vanishing gradients, rendering standard variational optimization ineffective. This paper provides a comparative performance analysis of two distinct strategies designed to improve trainability: Lie algebraic pretraining framework that uses Lie-algebraic classical simulation to find near-optimal initializations, and non-variational QWOA (NV-QWOA) that targets a restrict parameter subspace covered by 3 hyperparameters. We benchmark both methods on the unweighted Maxcut problem using a circuit depth of p=256p = 256p=256 across 200 Erdős-Rényi and 200 3-regular graphs, each with 16 vertices. Both approaches significantly improve upon the standard randomly initialized QWOA. NV-QWOA attains a mean approximation ratio of 98.9\% in just 60 iterations, while the Lie-algebraic pretrained QWOA improves to 77.71\% after 500 iterations. That optimization proceeds more quickly for NV-QWOA is not surprising given its significantly smaller parameter space, however, that an algorithm with so few tunable parameters reliably finds near-optimal solutions is remarkable. These findings suggest that the structured parameterization of NV-QWOA offers a more robust training approach than pretraining on lower-dimensional auxiliary problems. Future work is needed to confirm scaling to larger problem sizes and to asses generalization to other problem classes.
Classically equivalent formulations of an optimization problem need not lead to equivalent quantum algorithms. For MaxCut, fixing the value of a single vertex removes a simple global symmetry without changing the underlying optimization problem, yet it can substantially alter the quantum dynamics generated by the Quantum Approximate Optimization Algorithm (QAOA). These dynamics are captured by the circuit's dynamical Lie algebra (DLA), whose direct construction can become exponentially expensive. Here we show that, for symmetry-reduced QAOA on trees, substantial information about the reduced DLA can instead be obtained efficiently from the graph alone. We introduce a polynomial classical algorithm that recursively distinguishes vertices using shortest path structure and degree parity. When all vertices are resolved individually, the method determines the complete reduced DLA and, under the corresponding graph conditions, certifies maximal expressibility of the reduced QAOA ansatz. Even when full resolution is not achieved, the algorithm identifies embedded subalgebras, provides rigorous lower bounds on DLA dimension, and certifies controllable subsystems. Our theoretical and experimental results show how classical graph structure can be used to diagnose and potentially guide the quantum dynamics of variational algorithms before running them on quantum hardware.