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

TL;DR

This paper establishes a rigorous link between the Quantum Approximate Optimization Algorithm (QAOA) and quantum annealing for the Sherrington-Kirkpatrick model by proving that QAOA with angles smoothly varying in depth and bounded total angle reproduces the energy density of linear-time quantum annealing in the large-depth limit. The authors develop a two-tier analytic framework based on Quadratic Generalized Multinomial Sums (QGMS): first, a representation of SK-QAOA energy via correlation tensors, and second, a saddle-point expansion in the interaction parameter λ that yields a continuum limit independent of system size n. They show that the resulting continuum limit matches the continuous-time annealing energy, yielding a uniform convergence result in both depth p and system size n, and they demonstrate numerically that near-optimized QAOA angles still exhibit strong QAOA–QA correspondence. As a practical consequence, the work implies that linear-time annealing can be compiled with a larger, more aggressive Trotter step than standard bounds suggest, reducing gate counts and depth (e.g., a quadratic-depth savings for SK Trotterization). The findings challenge the folklore that QAOA operates via a mechanism fundamentally different from quantum annealing in large-depth regimes and open avenues for analyzing other problem classes under smooth, constant-angle schedules.

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.

Figures (30)

• Figure 1: Summary of the numerical experiments. A Two numerically evaluated regimes overlaid onto an example QAOA performance diagram for a 20-spin SK model. Points along the orange line correspond to constant total evolution time and satisfy the conditions of Theorem \ref{['th:approximation_continuous_time_annealing_qaoa']}, with example schedules for varying $p$ plotted in B. Points along the blue line correspond approximately to the conjectured optimal angle behavior, with example schedules in C. When conjectured optimal angles are used, QAOA approximation ratio approaches 1 (D), whereas for angles with constant total evolution time QAOA approximation ratio is flat. The point marked with red star corresponds exactly to infinite-size-limit optimized parameters for $p=17$ of Ref. qaoa_maxcut_high_girth. For lines in D, use the legend from A.
• Figure 2: Equivalence between QAOA and quantum annealing for constant total evolution time. A Difference between energy achieved by QAOA and quantum annealing decays with $p$. The rate of decay is $1/p$ (B) and is independent of $n$ (C), as predicted by Theorem \ref{['th:approximation_continuous_time_annealing_qaoa']}.
• Figure 3: Equivalence between QAOA and quantum annealing for constant angle magnitude. A For angle magnitude close to but smaller than those corresponding to conjectured optimal QAOA parameters ($\Delta=0.8$), relative residual approximation ratio decays rapidly with $p$. B At conjectured optimal QAOA angles ($\Delta = 1$), a correlation is still observed between QAOA and annealing approximation ratios, despite the error between the two not vanishing in the large $p$ limit (see complementary numerical results in appendix \ref{['sec:appendix_extra_numerics']}). C For large $\Delta$, QAOA and quantum annealing energies diverge.
• Figure 4: Outline of general method to take the $p \to \infty$ limit in the constant total angle regime. The central insight is that the order of correlation tensors $\bm{C}^{(d)}$ can be shown to be independent of $n$. Expanding correlation tensors in terms of noninteracting correlation tensors which have a natural continuum limit, this give uniform in $n$ convergence of SK-QAOA energy to its continuum limit.
• Figure 5: An example term in the expansion of $\bm{\theta}^*\left(\lambda\right)$ as a series in $\lambda$. Each noninteracting tensor block $\bm{\overline{C}}^{(d)}$ contributes a factor $\lambda^d$, where $\lambda = \gamma_{\mathrm{max}}p^{-1}$. Hence, the represented term is of order $\lambda^3$. Besides this explicit prefactor, $p$ also appears implicitly in the bond dimensions, since tensors are indexed by $\mathcal{A}$ and $\left|\mathcal{A}\right| = \left(2p + 2\right)^2$ (Eq. \ref{['eq:sk_qaoa_energy_qgms_A_index_set_size_main_text']}). Symbol $\supset$ signals that the right-hand-side is a single additive contribution to the left-hand side.

Theorems & Definitions (113)

• Definition 2.1: Sherrington-Kirkpatrick (SK) model
• Definition 2.2: Total $\gamma$ and $\beta$ angles
• Definition 3.1: QAOA angles derived from continuous annealing schedule
• Definition 3.2: Total evolution times for continuous schedule
• Definition 3.3: QAOA and quantum annealing energy
• Theorem 3.4: Equivalence between quantum annealing and QAOA with $n$-independent angles
• Corollary 3.5: Quadratic depth reduction in Trotterization of SK model annealing
• proof
• Definition B.1: Pseudo-Moment Generating Function of Quadratic Generalized Multinomial Sums (QGMS-MGF)
• Definition B.2: Pseudo-Moment Generating Function of Parametrized Quadratic Generalized Multinomial Sums (QGMS-MGF)