higher-order qaoa

This work addresses the challenges faced by the standard Quantum Approximate Optimization Algorithm (QAOA) when applied to combinatorial optimization problems with hard constraints—namely, the complexity of mixer circuits and poor noise resilience. Focusing on constraints defined by linear functions, the authors propose a simplified hypercube mixer architecture that leverages restricted mixing operators and circuit optimization techniques to substantially reduce gate count while strictly confining the evolution to the feasible solution space. Theoretical analysis establishes an upper bound on the number of binary variables beyond which the simplification may fail. Numerical experiments demonstrate that the proposed approach maintains constraint satisfaction and improves solution accuracy under realistic noise conditions, thereby enhancing the practicality of QAOA on near-term noisy intermediate-scale quantum (NISQ) devices.

This work addresses industrial-scale combinatorial optimization problems in the NISQ era, focusing on enhancing the practicality and constraint handling of the Quantum Approximate Optimization Algorithm (QAOA). Method: For NP-hard problems such as Max-Cut, we propose a Grover-mixer-based constraint-encoding scheme that rigorously embeds the feasible solution space into the QAOA variational circuit, eliminating sampling of infeasible solutions. We further generalize QAOA to a constrained Variational Quantum Eigensolver (c-VQE), supporting high-order Ising Hamiltonian modeling and analytical gradient computation via the parameter-shift rule. Contribution/Results: Implemented end-to-end on PennyLane, our approach demonstrates significant improvements in solution quality and convergence speed over unconstrained baselines. Systematic evaluation confirms c-VQE’s robustness under realistic noise models and its scalability to larger problem instances, establishing a foundation for near-term quantum optimization with hard constraints.

Addressing the challenge of parameter optimization for the Quantum Approximate Optimization Algorithm (QAOA) under few-shot constraints, this work introduces an end-to-end optimization protocol integrating multi-start initialization, a linear-response optimizer, and numerical pre-optimization, validated in closed-loop on real noisy hardware. For the first time, instance-level fine-tuning of a 5-layer QAOA is demonstrated on a 32-qubit trapped-ion processor—setting a record for two-qubit gate count. The linear-response optimizer is shown to be both computationally efficient and robust to noise at low shot budgets (<10⁴). Compared to conventional approaches, our method significantly reduces the number of shots required for convergence, enhances optimization stability, and improves solution quality. This establishes a scalable, hardware-efficient optimization paradigm for QAOA deployment on intermediate-scale noisy quantum processors.

To address the challenge that the Quantum Approximate Optimization Algorithm (QAOA) struggles to outperform classical algorithms on noisy, intermediate-scale quantum (NISQ) hardware, this work introduces the [[k+2,k,2]] Iceberg quantum error-detection code—first applied to QAOA. We implement a 20-logical-qubit MaxCut optimization on a real trapped-ion platform, achieving the largest-scale partially fault-tolerant universal quantum optimization experiment to date. The encoded QAOA significantly improves solution quality and state fidelity. We further propose a calibratable error–performance extrapolation model that quantifies the hardware threshold: QAOA is projected to surpass the Goemans–Williamson classical approximation ratio when single-gate error rates fall below ∼10⁻⁴. This work establishes a scalable error-detection pathway and a predictive performance framework for practical quantum optimization in realistic noisy environments.

This work addresses the fundamental gap in theoretical understanding of the time complexity of the Quantum Approximate Optimization Algorithm (QAOA), specifically determining the minimal number of layers (i.e., depth) required to guarantee a constant approximation ratio. Method: We establish the first general, rigorous lower-bound framework linking circuit depth to approximation performance, unifying analysis for both Grover-type and transverse-field mixers by connecting QAOA parameter optimization to quantum annealing evolution time. Results: We prove that for most combinatorial optimization problems, Grover-type QAOA requires Ω(poly(n)) layers to achieve a constant approximation ratio. The bound depends only on statistical properties of the k-local cost Hamiltonian—quantities efficiently computable from its coefficients. Our results generalize the Grover search lower bound to broad classes of QAOA-based search protocols and provide the first systematic theoretical criterion for assessing QAOA’s feasibility and resource requirements.

This work exposes a fundamental limitation of the Quantum Approximate Optimization Algorithm (QAOA) in solving constrained combinatorial optimization problems—particularly permutation-based NP-hard problems—where feasible solutions form a low-dimensional manifold within the Boolean hypercube; standard QAOA suffers from exponentially suppressed sampling of feasible configurations. To address this, we propose Constraint-Enhanced QAOA (CE-QAOA), which directly encodes constraints into the Hamiltonian: it employs block-local XY mixers within the parity-preserving subspace, combined with unary encoding and graph-theoretic analysis. This design ensures angular robustness and depth–problem-size alignment, overcoming conventional feasibility bottlenecks. We prove that, for circuit depth $O(n)$, CE-QAOA achieves an $n^2$-dependent exponential gain in probability mass over feasible solutions relative to standard QAOA—yielding provable exponential speedup. CE-QAOA thus establishes a scalable, constraint-aware quantum algorithmic paradigm for constrained combinatorial optimization.

This work addresses the challenge of solving large-scale combinatorial optimization problems on noisy intermediate-scale quantum (NISQ) devices, where limited qubit counts hinder practical applicability. The authors propose a compression method for the Quantum Approximate Optimization Algorithm (QAOA) that exploits problem symmetries and conserved quantities to identify invariant subspaces within the Hilbert space. By constructing an isometric mapping that preserves equivalence, the approach drastically reduces the required number of qubits while strictly maintaining the original QAOA dynamics and optimization performance. Theoretical analysis and numerical experiments on Max-Cut instances demonstrate, for the first time, lossless compression of quantum resources, enabling the method’s application to a broad class of large-scale constrained combinatorial optimization problems.

This work addresses the poor feasibility ratio and constraint violation issues of the standard Quantum Approximate Optimization Algorithm (QAOA) when applied to the Vehicle Routing Problem (VRP), where conventional Pauli-X mixers often disrupt local constraint structures. To overcome these limitations, the authors propose a constraint-aware QAOA framework that incorporates a lightweight initialization strategy to encode one-hot constraints, thereby reducing the initial state space, and introduces an XY-X mixing Hamiltonian that preserves constraint satisfaction while maintaining exploratory freedom over feasible configurations. Empirical results demonstrate that the proposed approach consistently outperforms standard QAOA across ideal, finite-sampling, and noisy settings, yielding significantly higher proportions of feasible solutions and improved solution quality. Although hardware noise partially diminishes this advantage, the performance gain is expected to increase as error rates decrease with advancing quantum hardware.

This work addresses the challenge of parameter space explosion and optimization difficulty in deep-layer Quantum Approximate Optimization Algorithm (QAOA) circuits, where conventional linear dimensionality reduction methods fail to capture the intrinsic nonlinear structure of the parameter manifold. The study introduces kernel principal component analysis (KPCA) with a radial basis function kernel for dimensionality reduction in QAOA parameter spaces, validated on Erdős–Rényi, Barabási–Albert, and Watts–Strogatz graph instances. Experimental results on 12-node Max-Cut problems at circuit depth 8 demonstrate that KPCA achieves approximation ratios exceeding 0.86—significantly outperforming standard PCA, which yields 0.81–0.83—and reduces the number of required quantum circuit evaluations by over 93%, thereby substantially enhancing both the efficiency and performance of deep QAOA optimization.

This work addresses the lack of reusable, multi-QPU-cooperative simulation tools for the Quantum Approximate Optimization Algorithm (QAOA) in engineering design and decision-making involving Quadratic Unconstrained Binary Optimization (QUBO) problems. We present the first distributed QAOA simulation framework that supports user-defined numbers and capacities of quantum processing units (QPUs). Built on Qiskit, the framework fully integrates QUBo modeling, distributed variable allocation, cross-QPU coupling handling, parameterized quantum circuit generation, and a Streamlit-based graphical interface. Runtime optimizations—including circuit reuse, batched evaluation, and parallel multi-start strategies—are incorporated to enhance efficiency. Experiments on standard QUBO benchmarks and the unit commitment problem demonstrate that both distributed and monolithic QAOA implementations recover optimal solutions, with staged optimization significantly reducing runtime while maintaining consistency with classical single-QPU QAOA results.