xy-mixer 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.

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.

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.

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.

For NP-hard ground-state preparation of diagonal Hamiltonians (e.g., MaxCut), this work proposes Imaginary-Time Evolution with Block Encoding (ITE-BE)—the first method integrating imaginary-time evolution and block encoding to deterministically prepare ground states without variational optimization: all circuit parameters are analytically determined from Hamiltonian coupling coefficients, eliminating classical optimization overhead. Theoretically, ITE-BE achieves asymptotically lower quantum resource complexity than QAOA. Experimentally, shallow QAOA circuits augmented with ITE-BE-based layer selection outperform deeper QAOA in success probability and solution quality. Moreover, ITE-BE enables deterministic execution of the first circuit layer even under transverse initial states—a capability previously unattained. By bypassing parameter optimization and offering analytical circuit construction, ITE-BE establishes a new paradigm for quantum optimization, with broad applicability across finance, condensed-matter physics, and computer science.

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.

During the Qiskit 1.x→2.x migration, the default shot count for QAOA was drastically reduced, causing insufficient state-space coverage (only 23%) and leading to output distribution shifts and significant accuracy degradation—undermining reproducibility. Method: We developed a standardized QAOA implementation based on Qiskit 2.x v2 primitives, rigorously controlling circuit construction, optimizer selection, and Hamiltonian encoding, and systematically quantified performance decay across varying shot counts. Contribution/Results: We identified the shot reduction as the root cause and proposed a reproducibility-preserving shot count of 250,000, which fully restores original accuracy. This work is the first to expose the critical impact of implicit parameters in the quantum-classical interface layer on hybrid algorithm performance. It establishes a parameter calibration paradigm and empirical benchmark for quantum software version migration, enabling robust cross-version algorithm deployment.

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.