Evaluating the Limits of QAOA Parameter Transfer at High-Rounds on Sparse Ising Models With Geometrically Local Cubic Terms

TL;DR

The paper addresses the challenge of scaling QAOA by exploring non-variational parameter transfer: angles learned on small, hardware-friendly Ising instances are fixed and applied to much larger problems with geometrically local cubic terms. Using JuliQAOA, the authors train angles up to $p=49$ on small instances and validate transfers with statevector simulations, PEPS/MPS tensor-network methods, and LOWESA, complemented by extensive IBM QPU experiments up to $156$ qubits. The results show general trends of improvement with depth across large instances and even near-ground-state energies in some cases, but also highlight nonmonotonic behavior and instance-dependent variability, indicating both promise and limitations of transfer-based QAOA. Overall, the work demonstrates that non-variational angle transfer can enable high-depth QAOA on hardware-friendly graphs and provides a framework for further scalable, learning-free QAOA deployment, supported by robust tensor-network validations and hardware measurements.

Abstract

The emergent practical applicability of the Quantum Approximate Optimization Algorithm (QAOA) for approximate combinatorial optimization is a subject of considerable interest. One of the primary limitations of QAOA is the task of finding a set of good parameters. Parameter transfer is a phenomenon where QAOA angles trained on problem instances that are self-similar tend to perform well for other problem instances from that similar class. This suggests a potentially highly efficient and scalable non-variational learning method for QAOA angle finding. We systematically study QAOA parameter transferability from small problems (16, 27 qubits) onto large problem instances (up to 156 qubits) for heavy-hex graph Ising models with geometrically local higher order terms using the Julia based QAOA simulation tool JuliQAOA to perform classical angle finding for up to 49 QAOA layers. Parameter transfer of the fixed angles is validated using a combination of full statevector, Projected Entangled Pair States, Matrix Product State, and LOWESA numerical simulations. We find that the QAOA parameter transfer from single instances applied to unseen problem instances does not in general provide monotonically improving performance as a function of p - there are many cases where the performance temporarily decreases as a function of p - but despite this the transferred angles have a general trend of improved expectation value as the QAOA depth increases, in many cases converging close to the true ground-state energy of the 100+ qubit instances. We also sample the hardware-compatible Ising models using the ensemble of fixed QAOA angles on several superconducting qubit IBM Quantum processors with 127, 133, and 156 qubits. We find continuous solution quality improvement of the hardware-compatible QAOA circuits run on the IBM NISQ processors up to p=5 on ibm_fez, p=9 on ibm_torino, and p=10 on ibm_pittsburgh.

Figures (18)

• Figure 1: $16$ qubit instance QAOA angles learned up to $p=49$ or close to the ground-state energy. Each row corresponds to a different problem instance (first 10 are random-coefficient, bottom two are uniform positive and negative coefficients).
• Figure 2: $27$ qubit instance QAOA angles learned up to $p=7$ for all instances. Each row corresponds to a different problem instance (first 10 are random-coefficient, bottom two are uniform positive and negative coefficients).
• Figure 3: Approximation ratio (y-axis) vs $p$ (x-axis) for the $12$ distinct $16$ qubit Ising model (left) and the $12$ distinct $27$ qubit Ising models (right). The legends show the specific instance type -- $10$ random coefficient models and one model with entirely positive coefficients and one model with entirely negative coefficients. The $27$ qubit instance training is cutoff at $p=7$; note that for all instances the approximation ratio is above $0.90$ at $p=7$.
• Figure 4: Numerical simulation of parameter transfer of one of the $16$ qubit problem instances onto the other $11$$16-$qubit problem instances (left) and onto the $12$$27-$qubit problem instances (right). These simulations are all noiseless (no shot noise). y-axis is the approximation ratio for the given target problem instances, x-axis is $p$ QAOA steps. Each line corresponds to a different problem instance (which were not using in the training of this set of fixed angles), which is indexed in the legend numerically and by the all positive and negative coefficient Ising models. Red asterisk markers denote the $p$ step where the expectation value decreases in quality (increases in energy) for that given problem instance, if that occurs. $8$ out the $11$ instances in the left plot have at least one non-improving $p$ step, and $9$ out of $12$ of the instances in the right plot have at least one non-improving $p$ step. However, there is an overall trend for all problem instances shown in these plots that the higher $p$ angles do give consistently improving energies, even if there are some blips of non-monotonic improvements.
• Figure 5: IBM quantum processor QAOA performance results for the $156$ qubit IBM quantum computers in terms of objective function (Ising model) energy on the y-axis as a function of the QAOA depth $p$ on the log scale x-axis. Each processor is sampling a single $156$-spin hardware-defined Ising model instance, whose global minimum energy is $-246$. Each plotted point (defined by a $p$ and a fixed set of transferred QAOA parameters) is showing the mean expectation value estimated using $10^5$ samples, except ibm_fez where $10^6$ samples are used. The classically trained angles are denoted in the legend by the qubit count of the problem followed by an index integer of the random instance (or all positive or negative coefficients), resulting in a total of $24$ sets of QAOA angles. Within each sub-plot, the red and blue lines each show the two sets of transferred QAOA angles, for $27$ and $16$-qubit source instances, which resulted in the best (lowest) expectation value found on that particular QPU. Although these $5$ sub-plots show sampling results from the same fixed Ising model, the y-axis are not scaled to be the same for each device so as to highlight any small differences between the noisy computations. Note that each figure legend contains multiple lines that are either solid ($16$-qubit training instance) or dashed ($27$-qubit training instance) gray -- these lines are not visually differentiable, and are intended to show the collective sampling rates for the sets of fixed QAOA angles. The minimum (mean) energies, with continuous energy improvement as a function of $p$, are marked with $\times$ symbols and vertical lines at that $p$ index.