Quantum Optimization with Classical Chaos

TL;DR

This work introduces QACOA, a chaotic-parameterization framework for QAOA that drastically reduces the number of trainable parameters by using classical chaotic maps to generate circuit angles. Through MAX-$K$-SAT benchmarks, the authors show that pure chaotic parameterization can match standard QAOA at shallow depths but suffers trainability issues at deeper circuits due to ergodic chaos, characterized by Lyapunov exponents and cost-landscape dynamics. They develop a rigorous dynamical-systems analysis, connecting phase-space chaos to gradient behavior and the ability to resolve extrema, and propose hybrid QACOA schemes that blend chaotic and standard parameterizations to recover and surpass standard QAOA performance for deeper circuits. The study demonstrates that a dynamical-map perspective can guide the design of scalable, parameter-efficient variational quantum algorithms with practical impact on near-term quantum devices.

Abstract

The Quantum Approximate Optimization Algorithm (QAOA) is a powerful tool in solving various combinatorial problems such as Maximum Satisfiability and Maximum Cut. Hard computational problems, however, require deep circuits that place high demands on classical variational parameter optimization. Ultimately, this has necessitated investigations into alternative methods for effective QAOA parameterizations. Here, we study a parameterization scheme based on classical chaotic recursive mapping, which enables significant reductions in the scaling of the variational parameter space. Through numerical investigations of hard Maximum Satisfiability problems, we demonstrate that the chaotic mapping can effectively match the performance of standard QAOA when subject to a limited number of classical optimization iterations and short-depth circuits. Insight into this behavior is elucidated through the lens of classical dynamical systems and used to inform hybridized schemes that leverage both standard and chaotic parameterizations. It is shown that these hybridized approaches can boost QAOA performance beyond that of the standard approach alone, especially for deep circuits. Through this study, we provide a new perspective that introduces a generalized framework for specifying performant, dynamical-map-based QAOA parameterizations.

Figures (16)

• Figure 1: The general procedure for the parameterization $(\mathcal{X}, \mathfrak T, \Phi=\{\phi_m\}_{m=1}^p)$ of a depth $p$ QACOA/QAOA circuit. Circuit angles are parameterized by trajectories on a phase space $\mathcal{X}$ of arbitrary dimension transforming under $\mathfrak T$. General 'pure' QACOA is given by $\mathfrak T$ ergodic (see Secs. \ref{['subsec:chaotic_parameterization']}, \ref{['sec:discussion_of_ergodicity']}), and standard QAOA (Sec. \ref{['subsec:standard_parameterization']}) corresponds to the case where $\mathcal{X}$ is the unit cube in $2p$ dimensions, $\mathcal{T}$ is the identity, and $\phi_m$ is a function simply selecting $(\gamma_m,\beta_m)$ from $\mathfrak T^{m-1}(\bm\theta)$. In the version of QACOA implemented here, we take $\mathcal{X}$ as the unit square, $\mathfrak T=l^c\times l^c$ the decoupled logistic map iterated $c$ times per layer, and $\phi_m$ the identity.
• Figure 2: The distribution of ARs $\epsilon^{(p)}_\mu$ achieved by the QACOA algorithm over the optimization iteration $j$, compared to those yielded by standard QAOA. Results are shown for a fixed $N=5$ MAX $3$-SAT problem near $\alpha=4.2$ for circuit depths $p=4,12,20$ and map speeds $c=1,5,100$. The blue curve corresponds to standard QAOA, while the others convey QACOA data. Each curve takes into account 20 optimizations, with the shaded regions representing IQRs and the solid lines are the medians. The comparison indicates that the larger map speeds yield the best performance.
• Figure 3: Pure QACOA performance results after various optimization iterations as a function of the circuit depth $p$ and map speed $c$. Results are shown for a fixed $N=5$ MAX $3$-SAT problem at $\alpha=4.2$. Each data point is averaged over 20 optimizations, and the error bars represent a $68\%$ confidence interval estimate for the mean $\overline\epsilon$. QACOA performance improves with the optimization iteration $j$ most significantly at small $p$. However, at large $p$, we see evidence of a trainability deficit as the mean AR results improve slowly with $j$. This behavior is seemingly insensitive to the choice of the map speed $c$ at large depths.
• Figure 4: The misassignment rates $h(\bm\theta^*,\Omega)/N$ corresponding to the data shown in Fig. \ref{['fig:approximation_ratios_over_j_c_fixed_problem']}. As was done in said figure, we have the misassignment rates for a fixed $N=5$ MAX $3$-SAT problem under standard QAOA and $c=1,5,100$ QACOAs for three circuit depths $p=4,12,20$. This figure supports the conclusions drawn from Fig. \ref{['fig:approximation_ratios_over_j_c_fixed_problem']}: $c=5,100$ QACOAs can be said to outperform $c=1$ at short depth, but performance at high depth is more or less indistinguishable in our results.
• Figure 5: The average ARs achieved under chaotic parameterization as a function of the circuit depth $p$, for various MAX $2$-SAT (top) and MAX $3$-SAT (bottom) problems, colored by the clause density $\alpha$, for $N=8$ qubits after 1000 optimization iterations. Down-triangle ($\triangledown$) markers indicate a problem for which $\alpha<\alpha_c$, whereas up-triangle ($\triangle$) markers indicate $\alpha\geq\alpha_c$.