arXiv subjects

Hoong Chuin Lau

Publications and source records attributed to Hoong Chuin Lau.

At least 19 recordsLinked to original sources

arXiv2026

Shared Lookups and Lazy Residues: Fewer Toffoli Gates for Factoring RSA-2048

Gidney's 2025 construction for factoring RSA-2048 with few logical qubits spends most of its Toffoli gates on discrete-log table lookups and residue uncomputation. We reduce both costs. Shared lookups return entries for several primes of a residue number system at once because they have the same address. Retained intermediate residues remove recomputation during cleanup and enable lazy arithmetic with a single final modular reduction. We prove output preservation and estimate resources with an open benchmark built on the reference simulator. Against the best reference setting within the same logical-qubit budget, using the published model for the reference and benchmark counts for our circuits, we obtain $15\%$ fewer Toffolis at $1{,}461$ qubits ($5.08 \times 10^{9}$ per factoring) and $46\%$ fewer at $1{,}806$ qubits ($2.55 \times 10^{9}$). A matched ablation finds $28.8\%$ fewer Toffolis from three-prime sharing at fixed window and checkpoint settings. Architecture-specific timing determines how much becomes a runtime saving.

quant-ph↗

arXiv2026

DEG: Diffusion-Enhanced GFlowNet for Solving Vehicle Routing Problems

Traditional neural solvers for solving vehicle routing problems (VRPs) often suffer from limited solution diversity, motivating the development of Generative Flow Network (GFlowNet)-based models. However, the effectiveness of these models is frequently constrained by insufficient flow expansion in high-reward regions, limiting their ability to distribute probability across promising solution routes. Deeper exploration could yield superior results. Diffusion models, in contrast, provide stronger structural guidance for exploration. These two paradigms are naturally complementary: GFlowNet can supply edge-level signals for diffusion to embed, while diffusion can guide broader exploration. Leveraging this synergy, we propose Diffusion-Enhanced GFlowNet (DEG), a novel framework that integrates GFlowNet with a diffusion model to encourage richer flow expansion toward high-reward regions and derive higher-quality solutions. Specifically, DEG exploits GFlowNet's inherent diversity to generate edge-specific backward signals, applies the stochastic noise schedule of diffusion to perturb these signals, and then denoises them within the GFlowNet paradigm. To further improve scalability, we introduce a specialized decoder capable of dynamically adapting to diverse problem scales. Extensive experimental evaluations on synthetic and real-world datasets, including instances with up to 10,000 nodes, demonstrate that DEG consistently achieves favorable performance compared to baseline methods.

cs.AI↗

arXiv2026

Securing the Flow: Strategic Storage and Corridor Design for Maritime Energy Security under Chokepoint Disruptions

Maritime disruption planning requires joint strategic storage and transport capacity decisions under uncertainty. We develop a two-stage stochastic multi-commodity network-design model for national energy security that integrates corridor commitments, port--commodity inventories, storage expansion, and scenario-dependent routing for crude oil, LNG, LPG, and fertilizer. The formulation incorporates decision-dependent probabilities, corridor concentration limits, and mean--CVaR risk, admitting an exact MILP reformulation. In a data-anchored India-inspired case study, stochastic modeling outperforms the expected-value policy by 1.35%, with decision-dependent probabilities worth 0.67%. Service levels are heterogeneous: crude and fertilizer maintain full coverage, while LNG and LPG drop to 34.6% and 33.3% under worst-case scenarios. Storage expansion adds 1,775 capacity units across seven port--commodity pairs and reduces active corridor commitments. Sensitivity analyses demonstrate that dependence alters economic exposure while preserving policy optimality, though cost stresses modify first-stage decisions with a maximum regret of 1.1673%.

math.OC↗

arXiv2026

Beyond QAOA: A Review of AI and Quantum Computing for Adaptive Combinatorial Optimization

Near-term quantum approaches to combinatorial optimization are limited by qubit counts, circuit fidelity, sampling cost, and the difficulty of encoding constraints, while machine learning is increasingly used to configure and control quantum optimization workflows. We call such workflows adaptive: decisions conventionally fixed in advance, from formulation and penalties to shot budgets, backends, and whether to invoke a quantum processor at all, are made by learned policies that respond to the instance, the progress of the solve, or the hardware. This review examines three paradigms, AI for quantum optimization, quantum for AI-driven optimization, and AI-quantum co-optimization, and organizes the literature by the decision being learned rather than by application. A structured search identified a 119-paper corpus, of which 67 papers form an evidence core coded in detail. The evidence is considerably stronger for AI-assisted quantum optimization than for the reverse direction: learning already reduces quantum evaluations, improves initialization, supports decomposition and penalty control, and mitigates noise, whereas evidence that quantum computation improves learned optimizers remains largely confined to small-scale simulation. Experimental controls are thin: of 52 studies in the three paradigms, 22 include no classical baseline, the quantum contribution is fully isolated in 10 of 24 where an ablation applies, and the median largest experiment uses 16 qubits. We introduce an M0-M5 evidence hierarchy, from simulation to practical quantum advantage, and find no broadly convincing result at the highest level. We argue that scaling is increasingly a systems problem: the question is not only whether a problem fits on a quantum processor, but how classical and quantum resources should be allocated across the optimization process.

quant-ph↗

arXiv2026

Slepian Bounds on the Success Probability of Virtual Distillation

Virtual distillation is a powerful near-term error-mitigation primitive, but it is also a spectral filter: it amplifies the dominant eigenvector component already present in the noisy density matrix. We show that, for a finite-band variational state, this filtering cannot create new concentration inside a set of accepted measurement outcomes. The asymptotic success probability after distillation is bounded by the leading eigenvalue of a Slepian concentration operator built from a specified variational band and that outcome window. Moreover, the number of robust high-success spectral components is limited by the Slepian active dimension. \rev{For bit-string outcome windows, an explicit Walsh-band realization has a Krawtchouk kernel on the Boolean hypercube. Finite-size noisy-QAOA calculations for 2-regular Max-Cut illustrate both the in-band improvement and the out-of-band failure modes predicted by the branch-resolved result.

quant-ph↗

arXiv2026

From Circuits to Hardware: Benchmarking Standard and Qubit-Efficient Quantum Optimization on Real Hardware

Despite rapid progress in quantum optimization, broad real-hardware benchmarks comparing multiple algorithmic families across diverse combinatorial problems under a common protocol remain limited. We benchmark gate-based quantum optimization on four NP-hard problems: multi-dimensional knapsack (MDKP), maximum independent set (MIS), quadratic assignment (QAP), and market-share (MSP). We study VQE, CVaR-VQE, standard, multi-angle, and warm-start QAOA, together with qubit-efficient PCE and QRAO, on IBM Heron r1/r2 processors using resilience-level-2 mitigation. To our knowledge, this includes the first real-hardware QRAO results and the first multi-problem PCE hardware benchmark. Across 247 method-instance combinations, we report transpiled circuit size, hardware outcomes, and an independent-error gate-count fidelity proxy, $F_{\mathrm{est}}$. For MDKP and MIS, an empirical operating point near $F_{\mathrm{est}}\approx 0.1$, corresponding to about 770 two-qubit gates at the median Heron-r2 CZ error rate, marks the onset of noise-dominated execution. QAP exposes a separate bottleneck: dense one-hot encodings and an exponentially sparse feasible manifold, with feasible fraction $10!/2^{100}$ at $n=10$; no tested hardware method produces a feasible assignment. Compiled QAOA-family circuits are generally noise dominated, and a matched uniform-random control shows that most feasible low-fidelity outcomes fall within the random range, apart from one finite-sample MIS warm-start exception. A SWAP-aware, fractional-gate, Nighthawk-topology compilation counterfactual reduces two-qubit counts but leaves all circuits below $F_{\mathrm{est}}=10^{-3}$. These conclusions apply to the tested implementations rather than QAOA in general. Qubit-efficient methods extend runnable instance sizes, but only within the empirical fidelity budget.

quant-ph↗

arXiv2026

CVaR-Assisted Custom Penalty Function for Constrained Optimization

We propose a slack-free penalty formulation for constrained binary optimization that eliminates auxiliary slack variables and preserves the feasibility structure of the original problem. The proposed approach introduces a nonlinear custom penalty function to enforce inequality constraints directly in the objective function. To address the computational challenges associated with evaluating nonlinear penalties in variational quantum algorithms, we employ the finite-sampling method that avoids the exponential complexity required by exact expectation computation. Furthermore, we integrate the Conditional Value-at-Risk (CVaR) objective to improve optimization robustness and guide the search toward high-quality solutions. The proposed framework is evaluated on instances of the multi-dimensional knapsack problem, a classical benchmark in combinatorial optimization. We showcase that the proposed custom-penalty formulation combined with CVaR sampling achieves improved optimality gaps and more consistent performance compared with conventional slack-based QUBO formulations. We also tested our method on the Quantinuum H2 quantum hardware, demonstrating the possibilities of training small-sized VQE on noisy trapped-ion quantum computers. The results suggest that careful penalty design can play a critical role in enabling quantum and hybrid quantum-classical algorithms for constrained optimization problems that arise in operations research.

quant-ph↗

arXiv2026

Diagnosing Simulation and Hardware Barriers to Cross-Size Transfer in Equivariant Quantum Reinforcement Learning

Equivariant quantum circuits (EQCs) parameterise reinforcement-learning policies for combinatorial optimisation with a size-independent parameter count, suggesting policies trained on small instances may transfer to larger ones. Whether such transfer survives realistic execution has not been measured end-to-end. We train EQC policies on Euclidean Travelling Salesman instances and evaluate identical checkpoints across statevector simulation, matrix-product-state simulation, noisy simulation, a protocol-matched noiseless emulator, and trapped-ion hardware. Within the validated regime, zero-shot five-to-ten-city transfer beats target-size training in all six evaluations. Beyond it, three barriers emerge: bond-dimension truncation destroys policy quality even without transfer; larger size jumps degrade performance consistently with a conditional diagnostic bound; and finite-shot execution inflates the transfer gap from ${\sim}5\%$ to $31.3\%$ (sampling noise alone) to $45.3\%$ hardware), because action margins lie below the shot-noise floor and collapse as $n^{-2.1}$. A cross-platform campaign across four hardware vendors confirms the penalty is set by native two-qubit gate count and error mitigation, not shot budget. A shot-complexity bound formalises the obstruction. We claim no quantum advantage; we provide the diagnostic standard such claims should meet.

quant-ph↗

arXiv2026

AutoQResearch: LLM-Guided Closed-Loop Policy Search for Adaptive Variational Quantum Optimization

Configuring variational quantum algorithms for combinatorial optimization remains a difficult, expert-driven process requiring coordinated choices over solver family, ansatz, objective, and optimizer. We present AutoQResearch, an LLM-guided closed-loop experimentation framework that casts this task as sequential policy search over a curated design space. Instead of a single static configuration, the framework searches for adaptive solver-control policies that condition future decisions on diagnostics such as feasibility, optimality gap, and convergence stagnation. The system operates through a structured workflow: an LLM agent edits a small policy surface under a fixed evaluation harness, candidate policies are screened using cheap scout evaluations, and only the strongest candidates are promoted to full confirmation. This enables controlled autonomous exploration while guarding against proxy overfitting and unstable selection. We evaluate the framework on Maximum Independent Set (MIS) and the Capacitated Vehicle Routing Problem (CVRP). On MIS instances (16--64 vertices), discovered policies substantially outperform static baselines and reveal scale-dependent behavior: CVaR objectives are effective at small scale, while QRAO-based qubit compression provides the most effective explored scaling path. On CVRP curricula (8--12 customers) and a held-out E-n13-k4 benchmark, the framework discovers adaptations involving sampling budget, penalty design, and hybrid repair protocols, yielding high-quality solutions. Methodologically, we find that staged confirmation is essential: cheap proxy evaluations can materially misestimate policy quality and even invert candidate rankings. Overall, the paper positions AutoQResearch as a benchmarked quantum--GenAI co-design workflow for autonomous solver discovery in variational quantum optimization.

quant-ph↗

arXiv2026

Qubit-Scalable CVRP via Lagrangian Knapsack Decomposition and Noise-Aware Quantum Execution

Hybrid quantum optimization for vehicle routing faces a practical bottleneck: direct QUBO encodings of CVRP quickly exceed near-term qubit and gate budgets, while quantum evaluations are expensive, noise-limited, and sensitive to backend and circuit configuration. We address this gap with an end-to-end decomposition pipeline that converts CVRP into bounded-width quantum subproblems and treats quantum execution as a decision problem within the optimization loop. Starting from a Fisher--Jaikumar assignment linearization, we apply Lagrangian relaxation to dualize customer-assignment couplers, yielding independent per-vehicle knapsack subproblems that admit QUBO/Ising evaluation. To replace brittle subgradient tuning, we learn a multiplier-update controller using expert-guided pretraining followed by reinforcement-learning fine-tuning, with rewards based on execution-realized progress and route reconstruction. We also introduce a constrained contextual bandit as a hardware-aware execution layer that selects backend and circuit configuration with feasibility screening, enabling adaptation across heterogeneous noisy resources and parallel multi-QPU scheduling. Computational results on multiple CVRPLIB families show that the decomposition yields stable bounded-width subproblems across instance sizes, learned multiplier updates improve end-to-end routing quality relative to classical subgradient control under matched budgets, and hardware-mode configuration reduces median optimality gaps relative to static execution choices in our test set. We do not claim quantum advantage. Instead, the contribution is a practical end-to-end framework for scaling hybrid quantum CVRP optimization through OR decomposition, learning-augmented dual control, and adaptive hardware-aware execution.

quant-ph↗

arXiv2025

Understanding the Nature of Depth-1 Equivariant Quantum Circuit

The Equivariant Quantum Circuit (EQC) for the Travelling Salesman Problem (TSP) has been shown to achieve near-optimal performance in solving small TSP problems (up to 20 nodes) using only two parameters at depth 1. However, extending EQCs to larger TSP problem sizes remains challenging due to the exponential time and memory for quantum circuit simulation, as well as increasing noise and decoherence when running on actual quantum hardware. In this work, we propose the Size-Invariant Grid Search (SIGS), an efficient training optimization for Quantum Reinforcement Learning (QRL), and use it to simulate the outputs of a trained Depth-1 EQC up to 350-node TSP instances - well beyond previously tractable limits. At TSP with 100 nodes, we reduce total simulation times by 96.4%, when comparing to RL simulations with the analytical expression (151 minutes using RL to under 6 minutes using SIGS on TSP-100), while achieving a mean optimality gap within 0.005 of the RL trained model on the test set. SIGS provides a practical benchmarking tool for the QRL community, allowing us to efficiently analyze the performance of QRL algorithms on larger problem sizes. We provide a theoretical explanation for SIGS called the Size-Invariant Properties that goes beyond the concept of equivariance discussed in prior literature.

quant-ph↗

arXiv2025

Probing Neural Combinatorial Optimization Models

Neural combinatorial optimization (NCO) has achieved remarkable performance, yet its learned model representations and decision rationale remain a black box. This impedes both academic research and practical deployment, since researchers and stakeholders require deeper insights into NCO models. In this paper, we take the first critical step towards interpreting NCO models by investigating their representations through various probing tasks. Moreover, we introduce a novel probing tool named Coefficient Significance Probing (CS-Probing) to enable deeper analysis of NCO representations by examining the coefficients and statistical significance during probing. Extensive experiments and analysis reveal that NCO models encode low-level information essential for solution construction, while capturing high-level knowledge to facilitate better decisions. Using CS-Probing, we find that prevalent NCO models impose varying inductive biases on their learned representations, uncover direct evidence related to model generalization, and identify key embedding dimensions associated with specific knowledge. These insights can be potentially translated into practice, for example, with minor code modifications, we improve the generalization of the analyzed model. Our work represents a first systematic attempt to interpret black-box NCO models, showcasing probing as a promising tool for analyzing their internal mechanisms and revealing insights for the NCO community. The source code is publicly available.

cs.LG↗

arXiv2025

Hybrid Learning and Optimization methods for solving Capacitated Vehicle Routing Problem

The Capacitated Vehicle Routing Problem (CVRP) is a fundamental NP-hard problem in logistics. Augmented Lagrangian Methods (ALM) for solving CVRP performance depends heavily on well-tuned penalty parameters. In this paper, we propose a hybrid optimization approach that integrates deep reinforcement learning (RL) to automate the selection of penalty parameter values within both classical (RL-C-ALM) and quantum-enhanced (RL-Q-ALM) ALM solvers. Using Soft Actor-Critic, our approach learns penalty values from CVRP instance features and constraint violations. In RL-Q-ALM, subproblems are encoded as QUBOs and solved using Variational Quantum Eigensolvers (VQE). The agent learns across episodes by maximizing solution feasibility and minimizing cost. Experiments show that RL-C-ALM outperforms manually tuned ALM on synthetic and benchmark CVRP instances, achieving better solutions with fewer iterations. Also, RL-Q-ALM matches classical solution quality on small instances but incurs higher runtimes due to quantum overhead. Our results highlight the potential of combining RL with classical and quantum solvers for scalable, adaptive combinatorial optimization.

physics.soc-ph↗

arXiv2025

Cutting Slack: Quantum Optimization with Slack-Free Methods for Combinatorial Benchmarks

Constraint handling remains a key bottleneck in quantum combinatorial optimization. While slack-variable-based encodings are straightforward, they significantly increase qubit counts and circuit depth, challenging the scalability of quantum solvers. In this work, we investigate a suite of Lagrangian-based optimization techniques including dual ascent, bundle methods, cutting plane approaches, and augmented Lagrangian formulations for solving constrained combinatorial problems on quantum simulators and hardware. Our framework is applied to three representative NP-hard problems: the Travelling Salesman Problem (TSP), the Multi-Dimensional Knapsack Problem (MDKP), and the Maximum Independent Set (MIS). We demonstrate that MDKP and TSP, with their inequality-based or degree-constrained structures, allow for slack-free reformulations, leading to significant qubit savings without compromising performance. In contrast, MIS does not inherently benefit from slack elimination but still gains in feasibility and objective quality from principled Lagrangian updates. We benchmark these methods across classically hard instances, analyzing trade-offs in qubit usage, feasibility, and optimality gaps. Our results highlight the flexibility of Lagrangian formulations as a scalable alternative to naive QUBO penalization, even when qubit savings are not always achievable. This work provides practical insights for deploying constraint-aware quantum optimization pipelines, with applications in logistics, network design, and resource allocation.

quant-ph↗

arXiv2025

Solving Constrained Combinatorial Optimization Problems with Variational Quantum Imaginary Time Evolution

Solving combinatorial optimization problems using variational quantum algorithms (VQAs) has emerged as a promising research direction. Since the introduction of the Quantum Approximate Optimization Algorithm (QAOA), numerous variants have been proposed to enhance its performance. QAOA was later extended to the Quantum Alternating Operator Ansatz (QAOA+), which generalizes the initial state, phase-separation operator, and mixer to address constrained problems without relying on the standard Quadratic Unconstrained Binary Optimization (QUBO) formulation. However, QAOA+ often requires additional ancilla qubits and a large number of multi-controlled Toffoli gates to prepare the superposition of feasible states, resulting in deep circuits that are challenging for near-term quantum devices. Furthermore, VQAs are generally hindered by issues such as barren plateaus and suboptimal local minima. Recently, Quantum Imaginary Time Evolution (QITE), a ground-state preparation algorithm, has been explored as an alternative to QAOA and its variants. QITE has demonstrated improved performance in quantum chemistry problems and has been applied to unconstrained combinatorial problems such as Max-Cut. In this work, we apply the variational form of QITE (VarQITE) to solve the Multiple Knapsack Problem (MKP), a constrained problem, using a Max-Cut-tailored ansatz. To the best of our knowledge, this is the first attempt to address constrained optimization using VarQITE. We show that VarQITE achieves significantly lower mean optimality gaps compared to QAOA and other conventional methods. Moreover, we demonstrate that scaling the Hamiltonian coefficients can further reduce optimization costs and accelerate convergence.

quant-ph↗

arXiv2025

Implementing Slack-Free Custom Penalty Function for QUBO on Gate-Based Quantum Computers

Solving NP-hard constrained combinatorial optimization problems using quantum algorithms remains a challenging yet promising avenue toward quantum advantage. Variational Quantum Algorithms (VQAs), such as the Variational Quantum Eigensolver (VQE), typically require constrained problems to be reformulated as unconstrained ones using penalty methods.A common approach introduces slack variables and quadratic penalties in the QUBO formulation to handle inequality constraints. However, this leads to increased qubit requirements and often distorts the optimization landscape, making it harder to find high-quality feasible solutions. To address these issues, we explore a slack-free formulation that directly encodes inequality constraints using custom penalty functions, specifically the exponential function and the Heaviside step function. These step-like penalties suppress infeasible solutions without introducing additional qubits or requiring finely tuned weights. Inspired by recent developments in quantum annealing and threshold-based constraint handling in gate-based algorithms, we implement and evaluate our approach on the Multiple Knapsack Problem (MKP). Experimental results show that the step-based formulation significantly improves feasibility and optimality rates compared to unbalanced penalization, while reducing overall qubit overhead.

quant-ph↗

arXiv2025

Adaptive Graph Shrinking for Quantum Optimization of Constrained Combinatorial Problems

A range of quantum algorithms, especially those leveraging variational parameterization and circuit-based optimization, are being studied as alternatives for solving classically intractable combinatorial optimization problems (COPs). However, their applicability is limited by hardware constraints, including shallow circuit depth, limited qubit counts, and noise. To mitigate these issues, we propose a hybrid classical--quantum framework based on graph shrinking to reduce the number of variables and constraints in QUBO formulations of COPs, while preserving problem structure. Our approach introduces three key ideas: (i) constraint-aware shrinking that prevents merges that will likely violate problem-specific feasibility constraints, (ii) a verification-and-repair pipeline to correct infeasible solutions post-optimization, and (iii) adaptive strategies for recalculating correlations and controlling the graph shrinking process. We apply our approach to three standard benchmark problems: Multidimensional Knapsack (MDKP), Maximum Independent Set (MIS), and the Quadratic Assignment Problem (QAP). Empirical results show that our approach improves solution feasibility, reduces repair complexity, and enhances quantum optimization quality on hardware-limited instances. These findings demonstrate a scalable pathway for applying near-term quantum algorithms to classically challenging constrained optimization problems.

quant-ph↗

arXiv2025

A Nurse Staffing and Scheduling Problem with Bounded Flexibility and Demand Uncertainty

Nurse staffing and scheduling are persistent challenges in healthcare due to demand fluctuations and individual nurse preferences. This study introduces the concept of bounded flexibility, balancing nurse satisfaction with strict rostering rules, particularly a real-world time regularity policy from a major hospital in Singapore. We model the problem as a multi-stage stochastic program to address evolving demand, optimizing both aggregate staffing and detailed scheduling decisions. A reformulation into a two-stage structure using block-separable recourse reduces computational burden without loss of accuracy. To solve the problem efficiently, we develop a Generative AI-guided algorithm. Numerical experiments with real hospital data show substantial cost savings and improved nurse flexibility with minimal compromise to schedule regularity. Numerical experiments based on real-world nurse profiles, nurse preferences, and patient demand data are conducted to evaluate the performance of the proposed methods. Our results demonstrate that the stochastic model achieves significant cost savings compared to the deterministic model. Notably, a slight reduction in the regularity level can remarkably enhance nurse flexibility.

math.OC↗