quantum optimization certification

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.

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 low efficiency of detecting critical security vulnerabilities—such as use-after-free, null-pointer dereference, and division-by-zero—in classical formal program verification. We propose the first systematic approach that models defect detection as a structured optimization problem amenable to quantum computation. Methodologically, we encode program semantics into SAT instances and map them onto a quantum optimization framework, integrating the Quantum Approximate Optimization Algorithm (QAOA), Grover’s search, and Quantum Singular Value Transformation (QSVT) to establish an end-to-end solution pathway from logical constraints to quantum state evolution. Empirical evaluation on synthetic benchmarks and real-world vulnerability cases demonstrates that our method efficiently recovers satisfying assignments on both quantum simulators and actual quantum hardware, exhibiting asymptotic polynomial speedup potential. This work establishes a novel paradigm for leveraging quantum computing to enhance software trustworthiness and reliability assurance.

This work addresses the Total Dominating Set Problem (TDP)—an NP-hard combinatorial optimization problem in graph theory—requiring the smallest vertex subset ( D subseteq V ) such that ( D ) has no isolated vertices and every vertex in ( V setminus D ) is adjacent to at least one vertex in ( D ). Method: We propose the first systematic quantum approximate optimization framework for TDP, introducing a quantum encoding scheme and a parameterized quantum circuit implementation based on the Quantum Approximate Optimization Algorithm (QAOA). Contribution/Results: Our experiments reveal a markedly skewed distribution of effective QAOA parameters, offering new insights into parameter landscape structure and informing future parameter-learning strategies. We validate correctness across diverse graph topologies, confirming QAOA’s capability to solve TDP instances; however, solution quality critically depends on high-fidelity parameter optimization. This work establishes a novel quantum-approximate pathway for tackling NP-hard combinatorial optimization problems, advancing the application of variational quantum algorithms to graph-theoretic challenges.

This work investigates the computational complexity of evaluating expectation values in the Quantum Approximate Optimization Algorithm (QAOA) for the MaxCut problem. By employing deterministic polynomial-time Turing reductions, Laurent polynomial analysis, and #P-hardness proof techniques, it establishes that for circuit depth \( p \geq 2 \), computing exact or exponentially precise expectation values—and their gradients and Hessians—is #P-hard, even when restricted to a single two-body correlation term or a constrained parameter set. This result demonstrates that the complexity transition from \( p = 1 \) to \( p \geq 2 \) in QAOA transcends mere NP-hardness, ascending to the #P level associated with counting optimal solutions. Moreover, the reduction simultaneously recovers both the maximum cut value and the number of optimal solutions, thereby revealing a fundamental computational barrier inherent to this task.

This work addresses the lack of general performance guarantees in existing quantum optimization algorithms and the difficulty of balancing efficiency with error control in circuit partitioning. It introduces, for the first time, a quantum–classical moment duality framework applicable to arbitrary quantum states. By leveraging a second-order sum-of-squares (SoS) semidefinite program, the approach establishes a duality between two-qubit Pauli-Z correlation matrices and the Goemans–Williamson relaxation, yielding certifiable lower bounds on Max-Cut values. Simultaneously, this correlation matrix reveals the tensor structure of quantum circuits, enabling efficient partitioning with rigorous error bounds. Experimental results demonstrate that near-optimal lower bounds can be achieved using only two-point correlation data, and the theoretical error bounds are empirically validated to hold tightly in practice.

This work resolves the decade-old Farhi–Goldstone–Gutmann (FGG) conjecture in quantum optimization, which posits whether depth-$p$ QAOA can exactly achieve an approximation ratio of $(2p+1)/(2p+2)$ on the “disagrees ring.” We present the first approach that integrates a large language model (Claude Fable 5) with the formal verification system Lean 4, employing an agent-based toolchain to automatically uncover hidden dynamical symmetries. This insight transforms the existential challenge into an explicit constructive proof, yielding the first machine-generated and formally verified complete proof of the FGG conjecture. Human intervention is limited to confirming the formal problem statement, thereby significantly advancing AI-driven research in quantum information theory.

This work addresses the challenge of providing performance guarantees for noisy quantum samplers by introducing the FPRASq framework—a novel end-to-end polynomial-time algorithm that offers conditional approximation guarantees for NP-hard quantum optimization problems. The approach integrates a constraint-augmented Heavy-Hitter variant of QAOA, polynomial-time feasibility repair, and an objective-scoring mechanism, requiring only shallow-depth QAOA circuits and independent sampling, followed by classical post-processing. Empirical evaluations on IBM’s Eagle r3 processor demonstrate superior performance over existing QOptlib benchmark solutions on instances with hundreds of variables. Theoretical analysis further shows that classically simulating the output of this method efficiently would imply NP ⊆ BPP, thereby suggesting a potential quantum advantage.

This work investigates the approximation limits of the max-LINSAT problem with bounded variable degree over arbitrary finite fields. By integrating computational complexity theory, approximation algorithm analysis, linear algebra over finite fields, and quantum information theory, it establishes—for the first time—the NP-hardness of achieving an approximation ratio beyond the random assignment baseline of \(r/q + O(1/\sqrt{D})\) for general finite fields. Furthermore, the study reveals a fundamental information-theoretic bottleneck in decoding quantum interferometry (DQI): classical decoders are inherently limited to an error scaling of \(1/\sqrt{D \log D}\), whereas quantum decoders achieve the optimal \(1/\sqrt{D}\) scaling. This result underscores the essential role of quantum decoding in aligning with the intrinsic complexity-theoretic structure of DQI.