On a seven-compound drug-response model, warm-start quantum approximate optimization (QAOA) on IonQ Forte-1 returned valid assignments more often than random bitstrings, but this alone did not show effective optimization. Ideal QAOA raised optimum probability above uniform feasible sampling in only four of twelve reference circuits. Hardware often fell below its own noiseless circuits, while greedy search solved all hardware models within 200 objective evaluations. Expanded simulations showed a gain over feasible sampling in 28 of 35 CAMA-1 panels and none of four 647-V panels. Annealing solved all these panels in every seed. A Grover mixer preserved feasibility and improved optimum probability over feasible sampling in all ten tested models. We analyzed 25 completed tasks containing 5,300 shots from 18 circuits and 14 instances. The encodings use 6-35 qubits and at most 4,900 feasible assignments, which we enumerated to establish exact optima. Noiseless references now cover both the original six circuits and six wider circuits. At 35 qubits, with 4,900 feasible assignments, ideal feasibility was 35.85%, compared with 7 of 200 valid hardware outputs. Its ideal optimum probability was below both sampling controls. The circuits sample assignments in a model built from measured single-agent and pairwise responses.
On a seven-compound drug-response model, warm-start quantum approximate optimization (QAOA) on IonQ Forte-1 returned valid assignments more often than random bitstrings, but this alone did not show effective optimization. Ideal QAOA raised optimum probability above uniform feasible sampling in only four of twelve reference circuits. Hardware often fell below its own noiseless circuits, while greedy search solved all hardware models within 200 objective evaluations. Expanded simulations showed a gain over feasible sampling in 28 of 35 CAMA-1 panels and none of four 647-V panels. Annealing solved all these panels in every seed. A Grover mixer preserved feasibility and improved optimum probability over feasible sampling in all ten tested models. We analyzed 25 completed tasks containing 5,300 shots from 18 circuits and 14 instances. The encodings use 6-35 qubits and at most 4,900 feasible assignments, which we enumerated to establish exact optima. Noiseless references now cover both the original six circuits and six wider circuits. At 35 qubits, with 4,900 feasible assignments, ideal feasibility was 35.85%, compared with 7 of 200 valid hardware outputs. Its ideal optimum probability was below both sampling controls. The circuits sample assignments in a model built from measured single-agent and pairwise responses.
The Quantum Approximate Optimization Algorithm (QAOA) is a leading hybrid heuristic for combinatorial optimization, but efficiently handling hard constraints remains a significant challenge. XY-mixers successfully confine quantum state evolution to a feasible subspace, such as the Hamming-weight-1 sector for one-hot constraints. On the contrary, warm-starting biases the search toward promising regions based on preliminary solutions. Combining these two techniques requires maintaining the essential alignment between the initial state and the mixer Hamiltonian to preserve convergence guarantees. Previous work demonstrated warm-starting with XY-mixers via a biased initial state, but relying only on standard mixer Hamiltonians. Consequently, the initial state is no longer a ground state of the mixer. In this work, we overcome these limitations by formulating a warm-started XY-mixer Hamiltonian for one-hot constraints and proving its ground-state properties. Furthermore, we provide a shallow circuit implementation suitable for NISQ implementations. We embed the warm-starting into a classical heuristic that iteratively updates the bias based on previous samples, called Iterative Warm-Starting (IWS). Extensive numerical simulations on Max-k-Cut and Traveling Salesperson Problem instances demonstrate that IWS-QAOA significantly accelerates the solution-finding process, increasing the probability of sampling optimal solutions by orders of magnitude compared to standard XY-QAOA. Finally, we validate our approach on the ibm_boston QPU using hardware-tailored 144-qubit problem instances. By coupling IWS-QAOA with a greedy steepest-descent post-processing strategy to repair infeasible measurements caused by hardware noise, we successfully identify optimal solutions on actual quantum devices.
Full-pipeline benchmark of the Quantum Approximate Optimization Algorithm (QAOA) against classical combinatorial solvers on a weighted Max-Cut formulation of exam-scheduling conflict minimization. Compares random assignment, greedy heuristic, 2-opt local search, exact ILP (OR-Tools), and QAOA — first via noisy local simulation on an 18-qubit validation subset, then via live execution on IBM's ibm_marrakesh (Heron r2) processor at the full 42-qubit scale. Uses shot-weighted expected-value scoring throughout rather than best-observed-sample reporting. At 18 qubits QAOA reaches 95.4% of the proven optimum; at 42 qubits the hardware run's expected value falls below random guessing (49.05% vs 50.00%), with all 4,096 shots returning distinct bitstrings — the signature of decoherence overwhelming the circuit. Includes causal analysis of the failure mode and outlines planned comparison against quantum annealing hardware.
Combinatorial Optimization problems play an important role in various domains such as logis- tics, life science or the financial sector. As their complexity in general grows exponentially with their problem size, finding the optimal solution classically becomes infeasible very fast. This raises the central question of whether quantum computers are in principle suitable for solving combinatorial optimization problems better and faster than classical computers. In order to find answers to that question, one needs to study quantum algorithms designed for those problems under ideal conditions (i.e., simulations, which are however limited due to exponential scaling) to investigate their potential, but one also needs real quantum hardware to test its viability and evaluate whether practical advantages are reasonable. In this regard, we consider variational quantum algorithms, especially the Quantum Approximate Optimization Algorithm (QAOA) consisting of a problem and a mixer Hamiltonian with freely selectable parameters, originally proposed as variational hybrid ansatz with classical optimization. This algorithm structure of- fers flexible and relatively shallow circuits, which makes it suitable for current NISQ hardware. However, optimizing the variational circuit parameters can be an NP-hard problem at worst. In this thesis, we study different versions of QAOA, especially with a focus on limitations, perfor- mance and potential advantages opposite classical methods. The first part of the thesis concerns the investigation of various aspects regarding the imple- mentation of the variational QAOA protocol such as: the incorporation of a problem constraint either by adding a suitable penalty term of by choosing a suitable mixer (that restricts the search space), the scaling of the problem Hamiltonian in comparison to the mixer Hamiltonian, the choice of the initial values for the classical optimization of the circuit parameters, and finally the choice of the function to be minimized (i.e., CVaR vs. expectation value). We apply this proto- col to both the portfolio optimization and the feature selection and evaluate the performance. In order to improve QAOA, we consider the warm-start QAOA for different problem instances and analyze its improved performance with regard to involved quantum effects. For this purpose, we propose a classical pre-processing method and demonstrate that the warm-start results can be reproduced or even exceeded by this method. The second part addresses the issue that the standard variational form requires a high number of circuit parameters that have to be optimized at a sufficiently large depth, which is a major limitationforachievingpotentialscalingadvantagescomparedtoclassicalmethods. Toovercome this limitation, we investigate and further develop fixed schedules for QAOA, which only rely on two free parameters. We start with introducing the continuous adiabatic theorem as it motivates the form of QAOA and show its limitations, which we address by applying a discrete version of the theorem. With our findings, a new version of QAOA based on linear-ramps (LR-QAOA) is introduced and further developed. In order to derive suitable parameter values for this schedule, we propose a method based on extrapolation from smaller towards larger problem sizes. With our results obtained on a noiseless emulator, we evaluate the quantum runtime scaling and discuss whether a scaling advantage can be achieved compared to classical methods.
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, Fest. For MDKP and MIS, an empirical operating point near Fest≈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!/2100 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 Fest=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.
Demonstrating quantum heuristics that outperform strong classical solvers on large-scale optimization remains an open challenge. Here we introduce Regularized Warm-Started QAOA (RWS-QAOA), which initializes qubits by minimizing expected energy with a regularizer that penalizes near-bitstring states, preventing QAOA from stalling. We further propose a protocol that yields fixed, instance-independent parameters, enabling RWS-QAOA to operate as a non-variational algorithm in which the quantum circuit parameters are fixed and only a classical warm starting step is instance-dependent. We evaluate RWS-QAOA on the Max-Cut problem for random regular graphs, where this protocol yields a constant-depth quantum circuit, across three complementary settings. First, on Quantinuum's trapped-ion processor, RWS-QAOA outperforms the classical algorithms with the best provable guarantees for Max-Cut on 3-regular graphs, namely Goemans-Williamson and Halperin-Livnat-Zwick, on 96-node instances. Second, tensor-network simulations on graphs with up to N=10,000 nodes show that depth-6 RWS-QAOA, achieving an average cut fraction of 0.9167, surpasses the best classical heuristics under matched restrictions (no local-search post-processing and no iterative refinement). Third, we remove these restrictions and benchmark against the strongest unrestricted classical heuristics, including an optimized parallel Burer-Monteiro solver that improves upon the MQLib implementation. Even against this stronger baseline, we project that surface-code RWS-QAOA reaches a quantum-classical runtime crossover below 0.2 seconds on 3,000-node graphs with fewer than 1.3 million physical qubits. Our results show that constant-depth quantum circuits combined with a classical warm start have a credible potential to surpass classical solvers on the Max-Cut problem when executed on future quantum computers.
Abstract Approximate optimization algorithms are commonly evaluated through exact optimum recovery or objective-space approximation, although these criteria need not coincide with the quality of the resulting decision in its downstream operating environment. This paper develops an end-to-end evaluation framework that separates three notions of solver success: exact recovery of the mathematical optimum, near-optimality with respect to the optimization surrogate, and operational acceptability under stochastic downstream evaluation. We study the framework using a small binary inventory-control benchmark represented by a calibrated and held-out-validated QUBO surrogate and solved with exact enumeration, greedy search, simulated annealing, uniform feasible sampling, and ideal QAOA at depths p=1,2,3. The results show that mathematical near-optimality and downstream operational acceptability define distinct solution sets. For QAOA, the probability mass assigned to operationally acceptable solutions can differ materially from both exact-recovery and QUBO-near-optimal probability, producing substantial differences in the number of measurements required to observe an acceptable solution. For n=8 at p=2, the median number of shots required for 95% observation probability decreases from 392 under exact recovery to 84 under QUBO near-optimality and 31 under the operational criterion. However, operational success alone can be misleading when acceptable states are abundant. We therefore use a descriptive enrichment comparison against uniform feasible sampling. Statistically supported enrichment is observed only in selected regimes, while classical greedy and simulated-annealing baselines outperform QAOA on these small instances. The results therefore do not indicate quantum advantage; rather, they show that solver evaluation can change materially when success is defined in terms of downstream decision consequences rather than surrogate-objective recovery alone. The QAOA statevector implementation and selected frozen confirmatory cases were independently reproduced with Qiskit and Qiskit Aer to floating-point precision.