The Quantum Approximate Optimization Algorithm (QAOA) aims to leverage quantum hardware to solve computationally complex problems efficiently, particularly those considered hard for classical computers. While QAOA is a promising candidate for achieving quantum supremacy in the NISQ era, demonstrating a clear quantum advantage over classical algorithms for practical NP problems remains a challenge. This is primarily because there's no algorithmic guarantee that QAOA will outperform classical methods without proving BQP ≠ NP. The Boolean satisfiability problem (SAT) provides a valuable testing ground for exploring this gap, given its varying computational hardness depending on the number of variables (k) in each clause. Specifically, while 2-SAT is solvable in polynomial time (class P), 3-SAT is NP-complete. Classical studies on random 3-SAT instances have revealed a computational phase transition at a critical clause-to-variable ratio (m/n), where instances become hardest to solve classically. Prior work on QAOA's performance on Max-2-SAT and Max-3-SAT showed a general worsening of performance with increasing density, termed 'reachability deficits'. This paper seeks to understand if and how this phenomenon relates to the classical phase transition, and whether a quantum advantage might be observed. The paper focuses on positive 1-in-k SAT problems (1-k-SAT+), a specific variant of SAT, alongside standard k-SAT problems, to explore the interplay between classical and quantum computational hardness.

Existing literature on QAOA has largely focused on its application to various optimization problems, but a comprehensive understanding of its computational complexity and the relationship to classical hardness remains limited. While there is evidence of quantum speed-ups on specific instances, a generalized quantum advantage has not been established. Several studies have investigated the performance of QAOA on Max-k-SAT problems, but these studies often lack a detailed analysis of the transition region. Moreover, the relationship between the trainability of QAOA and the complexity of the underlying problem has not been thoroughly examined. Prior work has noted 'reachability deficits' where QAOA performance degrades with increasing problem density, without fully connecting this to classical computational phase transitions. This paper builds upon this prior work by examining a specific type of SAT problem, 1-k-SAT+, to gain a deeper understanding of the connection between QAOA performance and classical computational phase transitions.

The study employs QAOA to solve k-SAT and 1-k-SAT+ problems. QAOA encodes the cost function into the Hamiltonian of a spin system, and approximates the ground state through a variational circuit. The circuit involves alternating applications of a problem Hamiltonian (Hc) and a mixing Hamiltonian (Hm), with variational parameters (γ, β) optimized to minimize the cost function. The authors utilize numerical finite-difference methods to calculate the gradient of the cost function with respect to the variational parameters. The standard deviation (SD) of the gradients is used as a measure of the trainability of QAOA, with smaller SD indicating harder training. To understand the connection between trainability and controllability, the authors examine the dimension of the dynamical Lie algebra (DLA) generated by the problem and mixing Hamiltonians. The DLA dimension provides a measure of the controllability of the quantum system. They analyze the decay of typical gradients with the number of qubits using the ratio of average gradient variances for different system sizes. The closeness of the QAOA unitary ensemble to a 2-design is assessed using the 4-point out-of-time-order correlator (OTOC). For accuracy assessment, the authors use the approximation ratio, comparing QAOA's performance against classical greedy algorithms for Max-1-k-SAT+, and calculating success probabilities in determining SAT/UNSAT for all problems. The quantum adiabatic algorithm (QAA) is used as a comparative algorithm, measuring the minimum energy gap during adiabatic evolution as an indicator of hardness. The authors derive an upper bound for the dimension of the DLA for the 1-k-SAT+ problems in the limit of m=n.

The paper's key findings include the identification of a trainability phase transition in QAOA for solving SAT problems. This transition manifests as a minimum in the typical gradient amplitude at a critical clause-to-variable ratio (m/n). This critical density generally differs from the classical SAT-UNSAT phase transition. The trainability transition is linked to the controllability of the QAOA circuit, as measured by the dimension of the dynamical Lie algebra (DLA). The authors observe that the DLA dimension exhibits a similar behavior to the inverse gradient, demonstrating a clear connection between trainability and controllability. The appearance of barren plateaus, characterized by exponentially decaying gradients, is confirmed, and linked to the complexity of the QAOA circuit and its proximity to a 2-design. Despite 'reachability deficits' at high problem densities, QAOA shows a surprisingly robust approximation ratio that decays slowly with increasing m/n. In the high density region, QAOA demonstrates a clear quantum advantage over classical approximate algorithms for both Max-1-3-SAT+ and Max-1-2-SAT+, indicating that the algorithm is particularly effective in regions where classical algorithms struggle. Interestingly, when reinterpreted as a decision problem, QAOA's performance shows a remnant of the classical SAT-UNSAT phase transition, with a valley of low success probability near the classical critical point. Comparison with the QAA reveals a similar computational phase transition tied to the minimum energy gap, with QAA performance also exhibiting robustness at high problem densities, mirroring the QAOA findings.

The findings highlight a fundamental difference between the empirical hardness of QAOA and classical algorithms, although both exhibit a form of phase transition. The discrepancy between the trainability transition and the classical SAT-UNSAT transition suggests that quantum algorithms might have different hardness landscapes than classical algorithms. The observed quantum advantage at high problem densities demonstrates the potential of QAOA for solving practically relevant, hard instances of combinatorial optimization problems. This work underscores the importance of investigating problem-specific hardness landscapes for quantum algorithms, rather than solely relying on worst-case complexity analyses. The connection between controllability and trainability offers valuable insights into the design and optimization of variational quantum algorithms, suggesting strategies for mitigating barren plateaus and improving performance. The robustness of QAOA in high-density regimes, where the algorithm faces reachability deficits, points to a potential resilience to problem complexity that warrants further exploration.

This paper provides significant insights into the empirical hardness of quantum approximate optimization algorithms, particularly in the context of SAT problems. The identification of a trainability phase transition, its connection to controllability and complexity, and the demonstration of quantum advantage in a practically relevant regime represent important contributions. Future research should explore the generalization of these findings to other combinatorial optimization problems and investigate strategies for further enhancing the performance of QAOA in the identified high-density regime. Additionally, further theoretical analysis could help to explain the observed phenomena more rigorously.

The study is primarily based on empirical observations, with limited analytical results. The numerical simulations were performed on relatively small system sizes, which may not fully capture the behavior of larger instances. The choice of specific classical benchmark algorithms may influence the comparison with QAOA. The upper bound derived for the DLA dimension is rather loose, and tighter bounds would be beneficial. The exact mechanism underlying the observed robustness of QAOA at high problem densities requires further investigation.