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arXiv:2506.03241 · Physical Review Research

Universal resources for quantum approximate optimization algorithm and quantum annealing

Pablo Díez-Valle, Fernando J. Gómez-Ruiz, Diego Porras, Juan José García-Ripoll

quant-phcond-mat.stat-mechSubmitted Jun 3, 2025• Announced Jun 5, 2025• 3 citations• Revised Oct 30, 2026 (v3)
v1 · Jun 3, 2025v2 · Mar 26, 2026v3 · Oct 30, 2026
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Abstract

The Quantum Approximate Optimization Algorithm (QAOA) is a variational ansatz that resembles the Trotterized dynamics of a Quantum Annealing (QA) protocol. This work formalizes this connection formally and empirically, showing the angles of a multilayer QAOA circuit converge to universal QA trajectories. Furthermore, the errors in both QAOA circuits and QA paths act as thermal excitations in pseudo-Boltzmann probability distributions whose temperature decreases with the invested resource -- i.e. integrated angles or total time -- and which in QAOA also contain a higher temperature arising from the Trotterization. This also means QAOA and QA are cooling protocols and simulators of partition functions whose target temperature can be tuned by rescaling the universal trajectory. The average cooling power of both methods exhibits favorable algebraic scalings with respect to the target temperature and problem size, whereby in QAOA the coldest temperature is inversely proportional to the number of layers, $T\sim 1/p$, and to the integrated angles -- or integrated interactions in QA.

Journal reference
Phys. Rev. Research 8, 013211 (2026)
DOI
10.1103/hxv2-sbr7
Comments
12 pages, 9 Figures. Title updated to reflect journal version

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The Quantum Approximate Optimization Algorithm (QAOA) is a variational ansatz that resembles the Trotterized dynamics of a Quantum Annealing (QA) protocol. This work formalizes this connection formally and empirically, showing the angles of a multilayer QAOA circuit converge to universal QA trajectories. Furthermore, the errors in both QAOA circuits and QA paths act as thermal excitations in pseudo-Boltzmann probability distributions whose temperature decreases with the invested resource -- i.e. integrated angles or total time -- and which in QAOA also contain a higher temperature arising from the Trotterization. This also means QAOA and QA are cooling protocols and simulators of partition functions whose target temperature can be tuned by rescaling the universal trajectory. The average cooling power of both methods exhibits favorable algebraic scalings with respect to the target temperature and problem size, whereby in QAOA the coldest temperature is inversely proportional to the number of layers, T∼1/pT\sim 1/pT∼1/p, and to the integrated angles -- or integrated interactions in QA.
The quantum approximate optimization algorithm (QAOA) and quantum annealing are two of the most popular quantum optimization heuristics. While QAOA is known to be able to approximate quantum annealing, the approximation requires QAOA angles to vanish with the problem size nnn, whereas optimized QAOA angles are observed to be size-independent for small nnn and constant in the infinite-size limit. This fact led to a folklore belief that QAOA has a mechanism that is fundamentally different from quantum annealing. In this work, we provide evidence against this by analytically showing that QAOA energy approximates that of quantum annealing under two conditions, namely that angles vary smoothly from one layer to the next and that the sum is bounded by a constant. These conditions are known to hold for near-optimal QAOA angles empirically. Our proof relies on a series expansion of QAOA energy in sum of angles, which we show converges to quantum annealing limit as QAOA depth grows for constant sum of angles even if angles do not vanish with problem size nnn. A corollary of our results is a quadratic improvement for the bound on depth required to compile Trotterized quantum annealing of the SK model in the average case.
The quantum approximate optimization algorithm (QAOA) was originally proposed to find approximate solutions to combinatorial optimization problems on quantum computers. However, the algorithm has also attracted interest for sampling purposes since it was theoretically demonstrated under reasonable complexity assumptions that one layer of the algorithm already engineers a probability distribution beyond what can be simulated by classical computers. In this regard, a recent study has also shown that, in universal Ising models, this global probability distribution resembles pure but thermal-like distributions at a temperature that depends on the internal correlations of the spin model. In this work, through an interferometric interpretation of the algorithm, we extend the theoretical derivation of the amplitudes of the eigenstates and the Boltzmann distributions generated by a single-layer QAOA. We also review the implications of this behavior from practical and fundamental perspectives.
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The quantum approximate optimization algorithm (QAOA) generates an approximate solution to combinatorial optimization problems using a variational ansatz circuit defined by parameterized layers of quantum evolution. In theory, the approximation improves with increasing ansatz depth but gate noise and circuit complexity undermine performance in practice. Here, we investigate a multi-angle ansatz for QAOA that reduces circuit depth and improves the approximation ratio by increasing the number of classical parameters. Even though the number of parameters increases, our results indicate that good parameters can be found in polynomial time for a test dataset we consider. This new ansatz gives a 33% increase in the approximation ratio for an infinite family of MaxCut instances over QAOA. The optimal performance is lower bounded by the conventional ansatz, and we present empirical results for graphs on eight vertices that one layer of the multi-angle anstaz is comparable to three layers of the traditional ansatz on MaxCut problems. Similarly, multi-angle QAOA yields a higher approximation ratio than QAOA at the same depth on a collection of MaxCut instances on fifty and one-hundred vertex graphs. Many of the optimized parameters are found to be zero, so their associated gates can be removed from the circuit, further decreasing the circuit depth. These results indicate that multi-angle QAOA requires shallower circuits to solve problems than QAOA, making it more viable for near-term intermediate-scale quantum devices.
One of the leading candidates for near-term quantum advantage is the class of Variational Quantum Algorithms, but these algorithms suffer from classical difficulty in optimizing the variational parameters as the number of parameters increases. Therefore, it is important to understand the expressibility and power of various ansätze to produce target states and distributions. To this end, we apply notions of emulation to Variational Quantum Annealing and the Quantum Approximate Optimization Algorithm (QAOA) to show that QAOA is outperformed by variational annealing schedules with equivalent numbers of parameters. Our Variational Quantum Annealing schedule is based on a novel polynomial parameterization that can be optimized in a similar gradient-free way as QAOA, using the same physical ingredients. In order to compare the performance of ansätze types, we have developed statistical notions of Monte-Carlo methods. Monte-Carlo methods are computer programs that generate random variables that approximate a target number that is computationally hard to calculate exactly. While the most well-known Monte-Carlo method is Monte-Carlo integration (e.g. Diffusion Monte-Carlo or path-integral quantum Monte-Carlo), QAOA is itself a Monte-Carlo method that finds good solutions to NP-complete problems such as Max-cut. We apply these statistical Monte-Carlo notions to further elucidate the theoretical framework around these quantum algorithms.
In this letter, we provide analytical and numerical evidence that the single-layer Quantum Approximate Optimization Algorithm (QAOA) on universal Ising spin models produces thermal-like states with Gaussian perturbations. We find that these pseudo-Boltzmann states can not be efficiently simulated on classical computers according to state-of-art techniques, and we relate this distribution to the optimization potential of QAOA. Moreover, we observe that the temperature depends on a hidden universal correlation between the energy of a state and the covariance of other energy levels and the Hamming distances of the state to those energies.