Improving QAOA to find approximate QUBO solutions in O(1) shots

TL;DR

These results provide the first systematic evidence that fpQAOA can achieve scalable approximate performance with polynomial-depth circuits, and remove any single component of the scheme restores exponential growth in the number of required shots.

Abstract

We study a modified fixed-point version of the Quantum Approximate Optimization Algorithm (fpQAOA), in which parameters are trained on small instances and transferred to larger problems. Our scheme combines three key ingredients: (i) targeting approximate rather than exact solutions through the success probability at a prescribed approximation ratio (AR), (ii) scaling the circuit depth linearly with the problem size using a two-parameter sin-cos angle encoding, and (iii) normalizing QUBO Hamiltonians by their Frobenius norm. Across several ensembles of random QUBO instances, we observe that these modifications yield a non-increasing (and often decreasing) median number of quantum circuit runs ("shots") required to achieve AR $α=0.95$, while the per-shot complexity remains polynomial. Extrapolation indicates an effectively constant $O(1)$ sampling complexity under this combined fpQAOA construction. Strikingly, removing any single component of the scheme restores exponential growth in the number of required shots, highlighting the synergistic nature of the three modifications. Our results provide the first systematic evidence that fpQAOA can achieve scalable approximate performance with polynomial-depth circuits.

Figures (4)

• Figure 1: Schematic overview of the three modifications introduced in our fpQAOA scheme. (M1) Approximate solutions are accepted based on an AR threshold $\alpha$. (M2) The circuit depth is scaled with problem size ($p=n$). (M3) QUBO instances are normalized by the Frobenius norm. Combined with a low-dimensional Fourier encoding of angles and a fixed-point training procedure that transfers parameters from small to large instances, these ingredients change the sampling complexity from exponential scaling (blue) to an effectively constant $O(1)$ behavior (orange).
• Figure 2: Empirical sampling complexity of fpQAOA trained on 200 $n=6$ instances with sin–-cos encoding, Frobenius normalization, and depth scaling $p=n$. For each problem size $n$, 1000 test instances are generated. (a) Weighted MaxCut problem. (b) Sherrington--Kirkpatrick Hamiltonian minimization problem. Orange dashed line: median; blue boxes: Q1–Q3 interquartile range (25%–75%); "error bars": 1st and 99th percentiles; red dots: outliers.
• Figure 3: Comparison of the scaling of median STS values for fpQAOA at $\alpha=0.95$, with all modifications enabled (same curves as in Fig. \ref{['fig:main_res']}) and with one modification (M1, M2, or M3) removed. The complexity of brute-force search is included as a baseline. Exponential fits are shown in brackets when applicable. (a) Weighted MaxCut problem. (b) Sherrington--Kirkpatrick Hamiltonian minimization problem.
• Figure 4: Behavior of median STS values for different ARs $\alpha$, including $\alpha=0.95$ (considered in Fig. \ref{['fig:main_res']}) and $\alpha=1$ (considered in Fig. \ref{['fig:fp-comparison']}). (a) Weighted MaxCut problem. (b) Sherrington--Kirkpatrick Hamiltonian minimization problem.