On the role of overparametrization in Quantum Approximate Optimization

TL;DR

This work analyzes how overparametrization, quantified via effective quantum dimension (EQD), influences the performance of Quantum Approximate Optimization Algorithm (QAOA) for combinatorial problems. By examining MAX-CUT and MAX-2-SAT on 2-regular and random graphs, the authors show that overparametrization is sufficient to achieve exact solutions with high probability, and is also necessary for MAX-CUT on 2-regular graphs where the optimal depth scales as floor(n/2). In contrast, many MAX-2-SAT instances are solvable in the underparametrized regime, with the overparametrization depth pc growing exponentially in n while the optimal depth p* grows more slowly, implying p*/pc decreases with n. These findings suggest QAOA may be effective on NISQ devices without resorting to overly deep circuits, and motivate studying problem-dependent ansatzes and alternative QAOA variants to further mitigate trainability and noise issues.

Abstract

Variational quantum algorithms have emerged as a cornerstone of contemporary quantum algorithms research. While they have demonstrated considerable promise in solving problems of practical interest, efficiently determining the minimal quantum resources necessary to obtain such a solution remains an open question. In this work, inspired by concepts from classical machine learning, we investigate the impact of overparameterization on the performance of variational algorithms. Our study focuses on the quantum approximate optimization algorithm (QAOA) -- a prominent variational quantum algorithm designed to solve combinatorial optimization problems. We investigate if circuit overparametrization is necessary and sufficient to solve such problems in QAOA, considering two representative problems -- MAX-CUT and MAX-2-SAT. For MAX-CUT we observe that overparametriation is both sufficient and (statistically) necessary for attaining exact solutions, as confirmed numerically for up to $20$ qubits. In fact, for MAX-CUT on 2-regular graphs we show the necessity to be exact, based on the analytically found optimal depth. In sharp contrast, for MAX-2-SAT, underparametrized circuits suffice to solve most instances. This result highlights the potential of QAOA in the underparametrized regime, supporting its utility for current noisy devices.

Figures (5)

• Figure 1: The energy error $E_p^*(H) - E_g$ (blue, left ordinate) and the effective quantum dimension $\mathcal{Q}(U_p)$ (green, right ordinate) with respect to QAOA circuit depth. The problem Hamiltonian $H$ is the $20$ qubit Ising Hamiltonian of type \ref{['eq:ising_MC']}.
• Figure 2: Typical examples of energy errors for $n=7$ vertex random graphs for $q=0.6$. The best energy, found in 20 optimization runs, is depicted as a function of circuit depth. The solid vertical lines represent the optimal depth $p^*$, while the dashed lines represent the overparametrization depth $p_c$.
• Figure 3: Optimization results for MAX-CUT (in blue) and MAX-2-SAT (in red) on $n=7$ qubit instances. Solid and dashed lines depict the convergence for $\epsilon = 10^{-8}$ and $\epsilon=10^{-1}$ energy error thresholds, respectively. (a) Fraction of solved instances with respect to the normalized circuit depth $p/p_c$. The fraction is computed over all the considered instances (50 random instances for each edge probability $q$, and 50 random instances for each number of clauses $m$, for MAX-CUT and MAX-2-SAT, respectively). (b) Fraction of converged optimization runs for the instances considered in (a) with respect to the normalized circuit depth.
• Figure 4: Comparison of the overparametrization and optimal QAOA circuit depth scaling for MAX-2-SAT problems of different size $n$ and clause density $\alpha = 2$. The statistics is accumulated over 50 random instances for each problem size $n$. The color intensity represents the density of data points with the respective depth. The crosses depict the geometric means of the corresponding depths. The data is offset horizontally for better visualization.
• Figure 5: Energy errors for a MAX-CUT instance on a random graph of $n=7$ vertices with $q=0.5$ as function of depth $p$. (a) The smallest error found in 20 optimization runs at each $p$. Due to the optimization strategy used, the error achieved at $p + 1$ layers may be large compared to $p$ layers. (b) When this is the case, we set $E_{p+1}^*(H) = E_{p}^*(H)$, which ensures monotonic behaviour of the error. In both panels, the green line indicates $p_c$ (which coincides with $p^*$ in this case), and the red line stands for the error threshold $\epsilon = 10^{-8}$.

Theorems & Definitions (4)

• Definition 1: Optimal depth
• Definition 2: Effective quantum dimension
• Definition 3: overparametrization depth
• Theorem 1: Optimal QAOA depth