Iterative quantum optimisation with a warm-started quantum state

TL;DR

This work tackles the known stuckness of standard QAOA by introducing an iterative warm-started approach that builds the initial quantum state from a superposition of the best-measured classical strings and refines it through multiple optimisation cycles. The method combines a $t$-order statistic or percentile state construction with a Permutation Grover-Rudolph circuit to prepare the warm-start, and applies it within QAOA to both MaxCut on $3$-regular graphs and the DGMVP portfolio model. Across extensive simulations, the approach yields improved approximation ratios for MaxCut at $p=1$, and achieves better mean/minimum value metrics and sampling probabilities for DGMVP, with favorable scaling in problem size and resilience to noise; however, gains tend to plateau at larger depths likely due to classical optimiser limitations. The results suggest iterative warm-starting as a viable strategy to surpass single-shot warm-start limitations and motivate extensions to other VQAs and warm-start schemes with theoretical performance bounds.

Abstract

We provide a method to prepare a warm-started quantum state from measurements with an iterative framework to enhance the quantum approximate optimisation algorithm (QAOA). The numerical simulations show the method can effectively address the "stuck issue" of the standard QAOA using a single-string warm-started initial state described in [Cain et al., 2023]. When applied to the $3$-regular MaxCut problem, our approach achieves an improved approximation ratio, with a lower bound that iteratively converges toward the best classical algorithms for $p=1$ standard QAOA. Additionally, in the context of the discrete global minimal variance portfolio (DGMVP) model, simulations reveal a more favourable scaling of identifying the global minimal compared to the QAOA standalone, the single-string warm-started QAOA and a classical constrained sampling approach.

Figures (12)

• Figure 1: A diagram for iterative warm-started quantum approximation optimisation.
• Figure 2: The $r$(a), $R$(b) and $P$(c) for the iterative warm-started quantum approximate optimisation applied over four iterations to the $100$ randomly generated $3$-regular graphs, as a function of graph size $N$. The quantum ansatz employs a $p=1$ standard QAOA with a $20$-th ordered statistic state. The classical optimiser is DA with $\mathcal{I} = 5000$, $m = 8000$, and $M = 8000$.
• Figure 3: The worst-cases(a)(c) and best-cases(b)(d) of approximation ratios $r$ for the iterative warm-started quantum approximate optimisation in each iteration applied over four iterations to $3$-regular graphs, as a function of graph size $N$. The hidden dots is (b)(d) mean the instances with $r=0$. The quantum ansatz employs a $p=1$ standard QAOA with a $20$-th ordered statistic state. The classical optimiser is DA with $\mathcal{I} = 5000$, $m = 8000$, and $M = 8000$.
• Figure 4: The $r$(a) and $R$(b) for the iterative warm-started quantum approximate optimisation applied over four iterations to the non-planar worst-case of the $3$-regular graphs for a $p=1$ standard QAOA, as a function of graph size $N$. The quantum ansatz employs a $p=1$ standard QAOA with a $20$-th ordered statistic state. The worst-case graphs are displayed next to their corresponding data points. The classical optimiser is DA with $\mathcal{I} = 5000$, $m = 8000$, and $M = 8000$.
• Figure 5: The $r$(a), $R$(b) and $P$(c) for the iterative warm-started quantum approximate optimisation applied over four iterations to $100$ randomly generated $3$-regular graphs with $12$ vertices each, as a function of the prepared initial statistic state order $k$. The quantum ansatz employs a $p=4$ standard QAOA. The classical optimiser is DA with $\mathcal{I} = 5000$, $m = 8000$, and $M = 8000$.