Depth One Quantum Alternating Operator Ansatz as an Approximate Gibbs Distribution Sampler

TL;DR

The paper investigates whether depth‑one QAOA can sample from a Gibbs distribution p(z) ∝ e^{−β E(z)} for a 15‑spin SK Ising model, comparing the X and Grover mixers. Using noiseless Hamiltonian simulations with JuliQAOA, it performs a high‑resolution 200×200 search over QAOA angles and β to fit Boltzmann distributions, assessing closeness via TVD and characterizing sampling via Shannon entropy. The main finding is that depth‑one Grover mixer QAOA yields more accurate Boltzmann sampling than the X mixer at high temperatures, though neither reaches ground‑state Boltzmann sampling at p=1; both show that accuracy strongly depends on angle choices. The study demonstrates the feasibility and limitations of using short‑depth quantum circuits for thermal sampling of complex Ising models, suggesting that deeper GM‑QAOA or alternative mixers may improve low‑temperature performance and that near‑term quantum devices could explore Boltzmann sampling tasks in frustrated systems.

Abstract

This study numerically investigates the thermal sampling properties of QAOA, the Quantum Alternating Operator Ansatz which was generalized from the original Quantum Approximate Optimization Algorithm. Specifically, the ability of QAOA to sample from the Gibbs distribution, equivalently the Boltzmann distribution, defined by a classical Ising model, specifically a fully connected disordered spin glass (Sherrington-Kirkpatrick) model. We focus on two different QAOA mixers; the standard transverse field X mixer, and the Grover mixer. At a QAOA depth of one we examine, for a single full QAOA parameter search space period, the energy landscape, the Shannon entropy landscape of the QAOA probability distribution, and the tradeoff between Boltzmann distribution sampling temperature and error rate (how close to the true Boltzmann distribution is the QAOA distribution). We find that at very high temperatures one-round Grover mixer QAOA can sample from the Boltzmann distribution more accurately than the standard X mixer QAOA at one round. Both X mixer and Grover mixer depth one QAOA can serve as approximate Boltzmann distribution samplers, and how good this approximation is depends heavily on the QAOA angle choice.

Figures (6)

• Figure 1: Energy $p=1$ QAOA angle landscape for the X mixer (left) and the Grover mixer (right).
• Figure 2: Full QAOA probability distribution Shannon entropy (normalized) $p=1$ angle landscape, for the X mixer (left) and Grover mixer (right). A Shannon entropy of $1$ is the maximum possible value, corresponding to a uniform distribution, and a Shannon entropy of $0$ corresponds to a maximally biased distribution.
• Figure 3: Boltzmann distribution sampling properties of QAOA using the X mixer (left column) and the Grover mixer (right column). Top row contains heatmaps of the best effective inverse temperature $\beta$ that was fitted to the full noiseless QAOA probability distribution, with a logarithmic scale heatmap. The bottom row contains heatmaps of the error measure, TVD, between the ideal Boltzmann distribution and the QAOA probability distribution, at the best fitted $\beta$. The TVD values, and the corresponding $\beta$ value, is the absolute best-fit that was found -- in other words, the lowest TVD that was found during the numerical fitting process. The heatmap landscape coordinates are given by the depth one QAOA parameters of $\vec{\gamma}$ (y-axis) and $\vec{\beta}$ (x-axis). The type of sampling that would be most interesting computationally is low temperature sampling, so maximize $\beta$. However, we also want accurate sampling, so small TVD measures -- most of the low temperature sampling regimes we see in these heatmaps also have high error rates.
• Figure 4: Scatterplot of the TVD error (y-axis), with respect to the ideal Boltzmann distribution, of the best-fitted temperature ($1/\beta_{\text{eff}}$) found for the given QAOA sample distribution, as a function of that effective temperature (x-axis) that was fitted. X mixer QAOA distribution is shown on the left, and the Grover mixer QAOA distribution is shown on the right. TVD closer to $0$ means a better approximation of the Boltzmann distribution at that effective temperature ($T_\text{eff}$). For these tradeoff plots, the aim is to focus on low temperature sampling, and therefore effective temperatures greater than $100$ are not plotted. Specifically, this data is a different representation of the high resolution $200\times 200$ grid-search plots in Figure \ref{['fig:TVD_Boltzmann_distribution']}, but only the points with an estimated effective temperature less than $100$ are shown. We see that there are many $\vec{\beta},\vec{\gamma}$ configurations which result in extremely high error rates of $\approx 0.5$, and for low error rates the effective temperature becomes higher.
• Figure 5: Comparing the two mixers by a TVD threshold analysis where the largest effective fitted $\beta$ (y-axis) that has TVD below the threshold (x-axis). logarithmic scale on both axes. We see a sharp dropoff of the X mixer sampling capability in the effective $\beta$ as the TVD threshold is made smaller, whereas at very small TVD thresholds Grover mixer QAOA is able to sample at lower temperatures compared to X mixer QAOA.