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arXiv:2503.09563

Quantum Approximate Optimization Algorithm in Finite Size and Large Depth and Equivalence to Quantum Annealing

Sami Boulebnane, James Sud, Ruslan Shaydulin, Marco Pistoia

Published Mar 12, 2025• 1 citation
arXivHTMLPDF

Abstract

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 $n$, whereas optimized QAOA angles are observed to be size-independent for small $n$ 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 $n$. 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.

DOI
10.48550/arxiv.2503.09563

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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) 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) is a hybrid quantum-classical variational algorithm designed to tackle combinatorial optimization problems.Despite its promise for near-term quantum applications, not much is currently understood about the QAOA's performance beyond its lowestdepth variant.An essential but missing ingredient for understanding and deploying the QAOA is a constructive approach to carry out the outer-loop classical optimization.We provide an in-depth study of the performance of the QAOA on MaxCut problems by developing an efficient parameter-optimization procedure and revealing its ability to exploit nonadiabatic operations.Building on observed patterns in optimal parameters, we propose heuristic strategies for initializing optimizations to find quasioptimal p-level QAOA parameters in O½polyðpÞ time, whereas the standard strategy of random initialization requires 2 OðpÞ optimization runs to achieve similar performance.We then benchmark the QAOA and compare it with quantum annealing, especially on difficult instances where adiabatic quantum annealing fails due to small spectral gaps.The comparison reveals that the QAOA can learn via optimization to utilize nonadiabatic mechanisms to circumvent the challenges associated with vanishing spectral gaps.Finally, we provide a realistic resource analysis on the experimental implementation of the QAOA.When quantum fluctuations in measurements are accounted for, we illustrate that optimization is important only for problem sizes beyond numerical simulations but accessible on near-term devices.We propose a feasible implementation of large MaxCut problems with a few hundred vertices in a system of 2D neutral atoms, reaching the regime to challenge the best classical algorithms.
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.
The quantum approximate optimization algorithm (QAOA) is a prospective near-term quantum algorithm due to its modest circuit depth and promising benchmarks. However, an external parameter optimization required in QAOA could become a performance bottleneck. This motivates studies of the optimization landscape and search for heuristic ways of parameter initialization. In this work we visualize the optimization landscape of the QAOA applied to the MaxCut problem on random graphs, demonstrating that random initialization of the QAOA is prone to converging to local minima with sub-optimal performance. We introduce the initialization of QAOA parameters based on the Trotterized quantum annealing (TQA) protocol, parameterized by the Trotter time step. We find that the TQA initialization allows to circumvent the issue of false minima for a broad range of time steps, yielding the same performance as the best result out of an exponentially scaling number of random initializations. Moreover, we demonstrate that the optimal value of the time step coincides with the point of proliferation of Trotter errors in quantum annealing. Our results suggest practical ways of initializing QAOA protocols on near-term quantum devices and reveals new connections between QAOA and quantum annealing.
Quantum Approximate Optimization Algorithm (QAOA) is a promising quantum heuristic with empirical evidence of speedup over classical state-of-the-art for some problems. QAOA uses a parameterized circuit with ppp layers, where higher ppp yields better solutions, but requires optimizing 2p2p2p independent parameters, which is challenging at large ppp. We present an iterative interpolation method that exploits the smoothness of optimal parameter schedules by expressing them in a basis of orthogonal functions, generalizing the work of Zhou et al. By optimizing a small number of basis coefficients and iteratively increasing both circuit depth and coefficient count until convergence, our method constructs high-quality schedules for large ppp. We provide theoretical justification using Jackson's theorem and Lipschitz continuity to bound the required number of basis coefficients for a given accuracy. Our approach achieves better performance with fewer optimization steps than existing methods across three benchmark problems: the Sherrington-Kirkpatrick (SK) model, portfolio optimization, and Low Autocorrelation Binary Sequences (LABS). For the largest LABS instance, we achieve near-optimal merit factors with schedules exceeding 1000 layers, an order of magnitude beyond previous methods. Additionally, we observe that a mild growth in QAOA depth suffices to solve the SK model exactly, a result of independent theoretical interest.
The quantum approximate optimization algorithm, also known in its generalization as the quantum alternating operator ansatz, (QAOA) is a heuristic hybrid quantum-classical algorithm for finding high-quality approximate solutions to combinatorial optimization problems, such as maximum satisfiability. While the QAOA is well studied, theoretical results as to its runtime or approximation ratio guarantees are still relatively sparse. We provide some of the first lower bounds for the number of rounds (the dominant component of QAOA runtimes) required for the QAOA. For our main result, we (i) leverage a connection between quantum annealing times and the angles of the QAOA to derive a lower bound on the number of rounds of the QAOA with respect to the guaranteed approximation ratio. We apply and calculate this bound with Grover-style mixing unitaries and (ii) show that this type of QAOA requires at least a polynomial number of rounds to guarantee any constant approximation ratios for most problems. We also (iii) show that the bound depends only on the statistical values of the objective functions and, when the problem can be modeled as a kkk-local Hamiltonian, can be easily estimated from the coefficients of the Hamiltonians. For the conventional transverse-field mixer, (iv) our framework gives a trivial lower bound to all bounded-occurrence local cost problems and for all strictly kkk-local cost Hamiltonians matching known results that constant approximation ratio is obtainable with a constant-round QAOA for a few optimization problems from these classes. Using our proof framework, (v) we recover the Grover lower bound for unstructured search and, with small modification, show that our bound applies to any QAOA-style search protocol that starts in the ground state of the mixing unitaries.