Near-Optimal Parameter Tuning of Level-1 QAOA for Ising Models

TL;DR

This paper addresses the practical challenge of tuning QAOA$_1$ for QUBO problems cast as Ising models on NISQ devices. By leveraging a Fourier-series perspective, it shows the cost landscape along $\gamma$ is highly oscillatory and derives a univariate reduction where $\beta^*$ is computed analytically, together with a polynomial-time sampling bound for reliable landscape reconstruction. It then proves that for large regular graphs the globally optimal $\gamma^*$ concentrates near zero and coincides with the first local optimum, enabling efficient gradient-descent-based optimisation. The approach is validated through Recursive QAOA (RQAOA) benchmarks on dense, weighted instances, where near-optimal QAOA$_1$ parameter tuning consistently outperforms SDP and coarsely optimised baselines, and a robust Iter-QAOA variant further strengthens performance in Ising models with fields.

Abstract

The Quantum Approximate Optimisation Algorithm (QAOA) is a hybrid quantum-classical algorithm for solving combinatorial optimisation problems. QAOA encodes solutions into the ground state of a Hamiltonian, approximated by a $p$-level parameterised quantum circuit composed of problem and mixer Hamiltonians, with parameters optimised classically. While deeper QAOA circuits can offer greater accuracy, practical applications are constrained by complex parameter optimisation and physical limitations such as gate noise, restricted qubit connectivity, and state-preparation-and-measurement errors, limiting implementations to shallow depths. This work focuses on QAOA$_1$ (QAOA at $p=1$) for QUBO problems, represented as Ising models. Despite QAOA$_1$ having only two parameters, $(γ, β)$, we show that their optimisation is challenging due to a highly oscillatory landscape, with oscillation rates increasing with the problem size, density, and weight. This behaviour necessitates high-resolution grid searches to avoid distortion of cost landscapes that may result in inaccurate minima. We propose an efficient optimisation strategy that reduces the two-dimensional $(γ, β)$ search to a one-dimensional search over $γ$, with $β^*$ computed analytically. We establish the maximum permissible sampling period required to accurately map the $γ$ landscape and provide an algorithm to estimate the optimal parameters in polynomial time. Furthermore, we rigorously prove that for regular graphs on average, the globally optimal $γ^* \in \mathbb{R}^+$ values are concentrated very close to zero and coincide with the first local optimum, enabling gradient descent to replace exhaustive line searches. This approach is validated using Recursive QAOA (RQAOA), where it consistently outperforms both coarsely optimised RQAOA and semidefinite programs across all tested QUBO instances.

Figures (7)

• Figure 1: Illustration of the Main Results. This paper focuses on optimising the variational parameters of QAOA$_1$ for solving QUBO problems formulated as Ising models. (a) The Ising model is represented by the problem Hamiltonian $H_P$, which is diagonal in the Pauli-$Z$ basis. Its eigenvalues define (b) the spectrum of frequencies $\omega$ for the given problem instance, where each $\omega_i$ corresponds to the difference between eigenvalues. To solve the problem exactly, the quantum circuit must express the full spectrum of frequencies. However, (c) QAOA$_1$ is limited to representing only a small subset of the spectrum. The expectation value of the cost function in QAOA$_1$ can be expressed as (d) a truncated Fourier series, where the frequencies are determined by the problem unitary, and the coefficients depend on both problem and mixing unitaries. At $p=1$, most coefficients vanish, significantly constraining the circuit’s expressivity. We provide an analytical method to compute the maximum frequency of the cost function along $\gamma$, enabling an appropriate sampling rate for optimisation. Previous studies derived analytical expressions for classically computing the expectation value $\langle H_P(\gamma, \beta) \rangle$, yielding (e) the cost landscape. In this work, we show how the two-dimensional optimisation over $(\gamma, \beta)$ can be reduced to (f) a univariate optimisation over $\gamma$, with $\beta^*$ determined analytically as a function of $\gamma$. By leveraging the maximum frequency, the univariate cost function can be efficiently sampled, enabling accurate estimation of the optimal $\gamma^*$ without overshooting the global optimum. Finally, we prove that, on average, the global optimum $\gamma^* \in \mathbb{R}^{+}$ is concentrated near 0 and that this global optimum coincides with the first local optimum.
• Figure 2: Optimal $(\gamma, \beta)$ Paths for QAOA$_1$ on a 4-Regular Graph. These plots depict the optimal $(\gamma, \beta)$ paths for QAOA$_1$ on a 4-regular graph with 128 vertices, calculated using \ref{['qaoa_2_local_opt_params_thm']}. The top contour plots show the cost landscape, with colours indicating approximation ratios (higher is better). The line plots below display the approximation ratios computed along $\gamma$ using \ref{['QAOA_Gamma_Linear_Eqn']}, where the approximation ratio measures the quality of the obtained solution relative to the optimal solution. Solid blue lines trace the optimal trajectory, $(\gamma, \beta^*_{\gamma})$, where $\beta^*_{\gamma}$ is the optimal $\beta$ for each fixed $\gamma$. Orange 'x' markers indicate sampled points, spaced according to \ref{['qaoa_min_samples_thm']}. The left plot (unweighted graph) reveals symmetric extrema in the cost landscape, while the right plot (weighted graph with Gaussian-distributed integer edge weights, mean 5, and variance 1) shows extrema clustered near $\gamma \approx 0$ and $\pi$. Discontinuities in the optimal path, evident in both cases, result from abrupt shifts in $\beta$ for specific $\gamma$ values---an effect more pronounced in weighted graphs.
• Figure 3: QAOA$_1$ Cost Landscape for a Weighted Regular Graph with Varying $\gamma$ Resolutions. These plots illustrate the cost landscape of QAOA$_1$ for a weighted $4$-regular graph with $64$ vertices, where edge weights are sampled from a Gaussian distribution with a mean and variance of $25$ and rounded to the nearest integer. The $\gamma$ dimension is sampled at resolutions of $20$, $40$, $60$, and $560$, while the $\beta$ dimension is fixed at 20 samples. The final plot's resolution of $560$ was determined using \ref{['qaoa_min_samples_thm']}. Below each mesh grid, a line plot displays the cost landscape along $\gamma$, calculated using the expression in \ref{['QAOA_Gamma_Linear_Eqn']}. In plots (a) and (b), the maximum approximation ratio is $0.04$, with the first optimum around $\beta < \pi/4$ and $\gamma \approx 0.15$. Plot (c) shows a higher ratio of $0.06$ with $\gamma$ near $1.4$. In plot (d), sampling $\gamma$ at the appropriate resolution yields a ratio of $0.48$, with the optimum at $\beta > \pi/4$ and $\gamma \approx 0.01$. These results highlight the critical importance of appropriately sampling the $\gamma$ dimension to avoid aliasing and distortion, which can lead to misleading local optima.
• Figure 4: Probability of Adversarial Graphs in QAOA$_1$. These plots illustrate the probability of encountering adversarial instances in Ising models for $D$-regular graphs ($2 \leq D \leq 10$) and Erdős-Rényi (ER) graphs with edge probability $1/2$. (a) presents results for Ising models without external fields, whereas (b) illustrates results for models with external fields. Graph sizes range from 4 to 128 nodes, with edge weights sampled as integers from a Gaussian distribution (mean 0, variance 100). For each configuration, 100 random graph instances were evaluated using two methods: (1) a full line search over $\gamma \in [0, \pi]$ using algorithm \ref{['subdivision_algorithm']}, and (2) gradient descent initialised near $\gamma = \Delta_{\gamma}/2$. Adversarial instances are defined as those where the optima obtained from the two methods differ. The probability of adversarial instances stabilises around 0.01 for graphs with $n > 14$, implying that only 1 in 100 graphs exhibits adversarial behaviour at larger sizes. Smaller and sparser graphs show a higher occurrence of adversarial instances, reflecting the increased likelihood of misalignment between local and global optima in these cases.
• Figure 5: Benchmark Results for Ising Models Without External Fields. These plots compare four algorithms---QAOA1, RQAOA1, RQAOA1(20), and a classical SDP---for approximating the ground state of the Ising model without external fields. We generate weighted Erdős--Rényi graphs with varying edge probabilities $p$ for (a) 128-vertex and (b) 256-vertex instances. Each bar indicates the mean approximation ratio over 20 random graphs at a given $p$, while the whiskers span the best and worst ratios. QAOA1 is optimised via gradient descent initialised near $\gamma \approx 0$; RQAOA1(20) employs a coarse line search of 20 samples in $[0,\pi]$ followed by gradient descent on the best found point; RQAOA1 uses gradient descent from $\gamma \approx 0$ with $\eta=120$ recursive steps for 128-vertex graphs and $\eta=248$ for 256-vertex graphs. The SDP adapts the Goemans--Williamson relaxation by replacing the MaxCut cost function with the Ising Hamiltonian, and each solution is rounded from 1024 random hyperplanes.

Theorems & Definitions (44)

• Theorem 1
• Corollary 1
• Definition 1
• Theorem 2
• Corollary 2
• proof
• Theorem 3: Nyquist-Shannon Sampling Theorem
• Theorem 4
• proof
• Corollary 3