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Shallow and Robust QAOA: Improving Feasibility and Hardware Performance via Linear-Chain and Ramp Schedules

Talha Azfar, Ruimin Ke

Published Jan 5, 2026• 0 citations
Publisher

Abstract

Abstract The Quantum Approximate Optimization Algorithm (QAOA) is known to struggle on near-term hardware due to constraint handling, parameter search overhead, and tight depth limits. We address this with a hardware-aware co-design: a linear-ramp parameter schedule paired with a Linear-Chain (LC) ansatz that restricts $ZZ$ interactions to nearest neighbors, lowering swaps and depth. We benchmark Linear-Ramp LC-QAOA against standard QAOA, Two-Step QAOA, and Grover-Mixer QAOA on small VRP instances across ideal simulation, Aer simulation, and IBM Eagle/Heron runs, measuring feasibility, solution quality, and hardware efficiency. Linear-ramp improves convergence and feasibility; LC reduces two-qubit gates and boosts noise robustness. Grover-Mixer and Two-Step can aid constraints in theory but are hampered in practice by deeper circuits. On hardware, overall feasibility remains $<1$\%, yet LC-QAOA with $XpXm$ dynamical decoupling more than doubles feasibility and recovers the optimum in several trials. We also show efficient scaling, including a larger case with ramp-initialized LC-QAOA plus CVaR. The results underscore hardware-aligned ansätze and structured schedules as a promising path.

DOI
10.21203/rs.3.rs-8297477/v1

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arXiv:2302.04479•Quantum Science and Technology•Jan 18, 2024•37 citations
Abstract The Quantum Approximate Optimization Algorithm (QAOA) is known to struggle on near-term hardware due to constraint handling, parameter search overhead, and tight depth limits. We address this with a hardware-aware co-design: a linear-ramp parameter schedule paired with a Linear-Chain (LC) ansatz that restricts ZZZZZZ interactions to nearest neighbors, lowering swaps and depth. We benchmark Linear-Ramp LC-QAOA against standard QAOA, Two-Step QAOA, and Grover-Mixer QAOA on small VRP instances across ideal simulation, Aer simulation, and IBM Eagle/Heron runs, measuring feasibility, solution quality, and hardware efficiency. Linear-ramp improves convergence and feasibility; LC reduces two-qubit gates and boosts noise robustness. Grover-Mixer and Two-Step can aid constraints in theory but are hampered in practice by deeper circuits. On hardware, overall feasibility remains <1<1<1%, yet LC-QAOA with XpXmXpXmXpXm dynamical decoupling more than doubles feasibility and recovers the optimum in several trials. We also show efficient scaling, including a larger case with ramp-initialized LC-QAOA plus CVaR. The results underscore hardware-aligned ansätze and structured schedules as a promising path.
The Quantum Approximate Optimization Algorithm (QAOA) is difficult to deploy on near-term hardware for constrained routing problems because feasibility constraints produce dense interaction graphs, deep transpiled circuits, and costly parameter search. We study this challenge for Vehicle Routing Problem (VRP) instances using a hardware-aware QAOA workflow that combines linear-ramp parameter initialization, Conditional Value-at-Risk (CVaR) optimization, and a Linear-Chain QAOA (LC-QAOA) surrogate that retains local ZZ interactions to reduce two-qubit depth. A matched full-QAOA versus LC-QAOA feasibility comparison is performed for 4-node instances, while the 5- and 6-node experiments characterize circuit executability and low-probability optimum recovery at the tested depths. In 4-node hardware runs, LC-QAOA increases mean feasible-sample probability from 0.182% to 0.917% and mean optimum-sample probability from 0.045% to 0.239%, giving approximately 5×5\times5× improvement over full QAOA. At 6 nodes, a full-QAOA circuit at p=1p=1p=1 already has large transpiled two-qubit depth, while LC-QAOA remains executable and samples CPLEX-certified optima in selected hardware runs at depths p =10–13, albeit with low optimum probability. Overall, the results identify LC-QAOA as a hardware-efficient surrogate heuristic that trades Hamiltonian completeness for lower depth and improved feasible-solution sampling on noisy quantum hardware.
Most available quantum algorithms address constrained optimization problems by treating constraints as soft penalty terms within a QUBO formulation. This approach requires careful adjustment of the penalty coefficients, which scales poorly with the number of constraints and lacks a proper strategy to balance feasibility and solution quality. In this work, we introduce two extensions of standard linear-ramp QAOA (lr-QAOA) tailored to problems with multiple heterogeneous constraints. We first construct ΛΛΛ-lr-QAOA, in which each penalty term is assigned its own linear-ramp schedule, promoting penalty weights from external hyperparameters to internal variational parameters of QAOA, similarly to the objective and mixer parameters. By optimizing all schedules jointly in a single run, this approach eliminates nested penalty tuning and scales more efficiently to multiple constraints. The optimization is guided by a feasibility-driven loss function that pushes the quantum state towards high-quality feasible solutions. As a further refinement, we introduce piecewise-ramp QAOA, in which the linear ramps are replaced by two-segment piecewise schedules, enhancing the expressiveness of the Ansatz at the cost of a small parameter overhead independent of the circuit depth. We benchmark both methods on Earth-observation satellite mission planning tasks formulated as budget-constrained Maximum Weight Independent Set problems. Numerical results show that piecewise-ramp QAOA consistently outperforms lr-QAOA and ΛΛΛ-lr-QAOA across circuit depths and system sizes. Furthermore, both ΛΛΛ-lr-QAOA and piecewise-ramp QAOA exhibit a high feasibility rate, which is crucial in industrial applications. Our analysis highlights an intrinsic feasibility-optimality trade-off, which we address by introducing a filtered variant of the loss providing a single hyperparameter to tune this balance.
Abstract The quantum approximate optimization algorithm (QAOA) is a promising algorithm for solving combinatorial optimization problems (COPs), with performance governed by variational parameters {γi,βi}i=0p−1{\{{\gamma }_{i},{\beta }_{i}\}}_{i = 0}^{p-1}{γi​,βi​}i=0p−1​ { γ i , β i } i = 0 p − 1 . While most prior work has focused on classically optimizing these parameters, we demonstrate that fixed linear ramp schedules, linear ramp QAOA (LR-QAOA), can efficiently approximate optimal solutions across diverse COPs. Simulations with up to N q = 42 qubits and p = 400 layers suggest that the success probability scales as P(x∗)≈2−η(p)Nq+CP({x}^{* })\approx {2}^{-\eta (p){N}_{q}+C}P(x∗)≈2−η(p)Nq​+C P ( x * ) ≈ 2 − η ( p ) N q + C , where η ( p ) decreases with increasing p . For example, in Weighted Maxcut instances, η (10) = 0.22 improves to η (100) = 0.05. Comparisons with classical algorithms, including simulated annealing, Tabu Search, and branch-and-bound, show a scaling advantage for LR-QAOA. We show results of LR-QAOA on multiple QPUs (IonQ, Quantinuum, IBM) with up to N q = 109 qubits, p = 100, and circuits requiring 21,200 CNOT gates. Finally, we present a noise model based on two-qubit gate counts that accurately reproduces the experimental behavior of LR-QAOA.
Constraint-preserving mixers keep QAOA within a feasible subspace, but feasibility and global mixer connectivity do not determine which feasible configurations are available to a particular shallow circuit. We study constrained QAOA at the level of the circuit actually executed: a specified feasible initial state, finite depth, and finite schedule of mixer interactions. We define the finite-depth accessible set Rp(M,x0)R_p(M,x_0)Rp​(M,x0​) and show that the computational-basis support of the ideal QAOA circuit is contained within it. This gives a parameter-independent objective ceiling fR⋆=max⁡x∈Rpf(x)f_R^\star=\max_{x\in R_p}f(x)fR⋆​=maxx∈Rp​​f(x) and a normalized reachable-quality diagnostic QRQ_RQR​, separating accessible volume from the objective quality contained within it. We evaluate this framework using resource-matched block-local XY mixer schedules for constrained portfolio optimization and synthetic block-constrained quadratic problems. Across 207 favorable-tail CVaR QAOA configurations, QRQ_RQR​ shows the strongest association with realized shallow-QAOA performance among the structural diagnostics considered, with Pearson and Spearman correlations of 0.70030.70030.7003 and 0.80570.80570.8057; the relationship remains positive under controlled and dependence-aware analyses. The distinction persists across changes in feasible initialization, portfolio size, and objective family. A compilation-only study further shows that, for the dense portfolio Hamiltonians considered here, phase-separator routing dominates the compiled two-qubit resource scale, while mixer schedules with comparable compiled two-qubit cost can exhibit markedly different finite-depth accessible quality. These results motivate objective-relevant finite-depth accessibility as a complementary diagnostic for resource-bounded constrained QAOA.
A challenge with the Quantum Approximate Optimisation Algorithm (QAOA), and variational algorithms in general, is finding good variational parameters, a task which in itself can be NP-hard. Recent work has sought to de-variationalise QAOA by picking well-informed guesses for the variational parameters. The Linear Ramp QAOA (LR-QAOA) achieves this by using parameter schedules inspired by the quantum adiabatic algorithm. In this work, we propose Spectral Gap Informed Ramp QAOA (SGIR-QAOA), a new QAOA variant that incorporates spectral gap information from an adiabatic Hamiltonian, with the QAOA mixer Hamiltonian as the initial Hamiltonian, to construct smooth parameter schedules. SGIR-QAOA performs slow evolution where the spectral gap of the adiabatic Hamiltonian is small. We show that SGIR-QAOA has performance improvements over the LR-QAOA on Grover's problem at constant depth and that SGIR-QAOA requires shorter depths to achieve the same optimal solution probability. We then show that these performance benefits extend to a problem with potential practical applications - the Maximum Independent Set (MIS) problem. Finally, we demonstrate the scalability of the SGIR-QAOA method using extrapolated spectral gap information for scales that the spectral gap cannot be exactly evaluated, and show that the advantage appears to persist under mild depolarising noise.
Abstract The quantum approximate optimisation algorithm (QAOA) is a hybrid quantum–classical algorithm used to approximately solve combinatorial optimisation problems. It involves multiple iterations of a parameterised ansatz that consists of a problem and mixer Hamiltonian, with the parameters being classically optimised. While QAOA can be implemented on near-term quantum hardware, physical limitations such as gate noise, restricted qubit connectivity, and state-preparation-and-measurement (SPAM) errors can limit circuit depth and decrease performance. To address these limitations, this work introduces the eXpressive QAOA (XQAOA), an overparameterised variant of QAOA that assigns more classical parameters to the ansatz to improve its performance at low depths. XQAOA also introduces an additional Pauli-Y component in the mixer Hamiltonian, allowing the mixer to implement arbitrary unitary transformations on each qubit. To benchmark the performance of XQAOA at unit depth, we derive its closed-form expression for the MaxCut problem and compare it to QAOA, Multi-Angle QAOA (MA-QAOA) (Herrman et al 2022 Sci. Rep. 12 6781), a classical-relaxed algorithm, and the state-of-the-art Goemans–Williamson algorithm on a set of unweighted regular graphs with 128 and 256 nodes for degrees ranging from 3 to 10. Our results indicate that at unit depth, XQAOA has benign loss landscapes with local minima concentrated near the global optimum, allowing it to consistently outperform QAOA, MA-QAOA, and the classical-relaxed algorithm on all graph instances and the Goemans–Williamson algorithm on graph instances with degrees greater than 4. Small-scale simulations also reveal that unit-depth XQAOA invariably surpasses both QAOA and MA-QAOA on all tested depths up to five. Additionally, we find an infinite family of graphs for which XQAOA solves MaxCut exactly and analytically show that for some graphs in this family, special cases of XQAOA are capable of achieving a much larger approximation ratio than QAOA. Overall, XQAOA is a more viable choice for variational quantum optimisation on near-term quantum devices, offering competitive performance at low depths.