Variational (matrix) product states for combinatorial optimization

TL;DR

The resulting quantum-inspired ILS algorithms are benchmarked on maximum cut problems of up to 50000 variables and show that they can outperform traditional (M)PS methods, classical ILS, the quantum approximate optimization algorithm and other variational quantum-inspired solvers.

Abstract

To compute approximate solutions for combinatorial optimization problems, we describe variational methods based on the product state (PS) and matrix product state (MPS) ansatzes. We perform variational energy minimization with respect to a quantum annealing Hamiltonian and utilize randomness by embedding the approaches in the metaheuristic iterated local search (ILS). The resulting quantum-inspired ILS algorithms are benchmarked on maximum cut problems of up to 50000 variables. We show that they can outperform traditional (M)PS methods, classical ILS, the quantum approximate optimization algorithm and other variational quantum-inspired solvers.

Figures (5)

• Figure 1: Performance comparison for G12. Average relative error $1 - r$, averaged over 100 random initial states, for G12 Gset, a toroidal weighted MaxCut problem of 800 variables and weights from $\{-1, 1\}$. If not visible, error bars are smaller than their associated symbols. The offset for QiILS results from additional sweeps used to tune a hyperparameter that aims to obtain more consistent and faster improvement per sweep. Despite this offset, QiILS achieves the best performance. For each method, results correspond to the best hyperparameter choice within the explored hyperparameter search space. Each sweep of QiILS and ILS runs in a similar amount of time, whereas LQA (GCS) is roughly 7 (9000.0) times slower. Further details on the hyperparameter search as well as on the runtimes per sweep are provided in supp.
• Figure 2: Study of QiILS hyperparameters. Performance is measured by the average relative error $1 - r$, averaged over 1000.0 random graph instances. (a) For u3R graphs of size $n = 50$, we consider various values of $\lambda$, numbers of iterations $\iota$ and bond dimensions $\chi$. (b) For u3R graphs of different sizes, we study the performance as a function of the iteration index $\iota$ using PS ($\chi = 1$) and fixing $\lambda = 0.4$. The inset shows the performance for the same u3R graphs when varying $\chi$ while fixing the iteration index to $\iota = 1$. (c) For $n = 200$ and several perturbation strengths, we compare the performance for u3R graphs (main panel) and w3R graphs (inset). [Fixed hyperparameters: sweeps = 80; in (a) and (b) $p = 0.5$; in (c) $\lambda = 0.4$ for u3R and $\lambda = 0.5$ for w3R.]
• Figure 3: Comparison with QAOA. Performance is measured by the average relative error $1 - r$, averaged over 1000.0 random graph instances. We consider u3R graphs (a) and w3R graphs (b). The main panels show the performance of QiILS with $\chi = 1$ as a function of $\iota$. The insets show the performance as a function of the bond dimension $\chi$, using a single iteration ($\iota = 1$). Closed markers are results from ZhEtAl20, where the model function is $1 - r \propto \exp(-q/q_0)$ for unweighted graphs and $1 - r \propto \exp(-\sqrt{q/q_0})$ for weighted graphs, with $q$ denoting the quantum circuit depth of QAOA. The quantity on the horizontal axis ($\iota$ in the main panels or $\chi$ in the insets) plays an analogous role to the QAOA circuit depth $q$. [Fixed hyperparameters: sweeps = $80$; in (a) $\lambda = 0.55$ and $p = 0.5$; in (b) $\lambda = 0.75$ and $p = 0.3$. Further details on the selection of $\lambda$ are provided in supp.]
• Figure 4: Comparison of QiIGS with QiILS. In the main panel, we consider G81 Gset and performance is measured by the average relative error $1 - r$ as a function of wall-clock time (in seconds), computed over 10000.0 iterations and averaged over 20 random initial states. The inset shows the average time per iteration for problem sizes $n = 1000.0, 10000.0, 20000.0, 30000.0, 40000.0, 50000.0$ across 20 random instances of u3R graphs. For QiIGS, angle updates were executed in parallel on an NVIDIA L4 GPU (24 GB VRAM, Ada Lovelace architecture) via Lightning AI. The CPU computations were performed on a 2023 MacBook Pro with an Apple M2 chip (8-core CPU, 10-core GPU) and 16 GB of unified memory, running macOS 15.6.1. [Fixed hyperparameters: sweeps = 200, $p = 0.15$, $\lambda = 0.35$, $\tau = 0.1$.]
• Figure 5: Study of QiILS hyperparameters for w3R graphs. Performance is measured by the average relative error $1 - r$, averaged over 500 random w3R graph instances. (a) For $n = 50$, we consider several values of $\lambda$, numbers of iterations $\iota$ and bond dimensions $\chi$. (b) For graphs of various sizes, we investigate the performance as a function of the iteration index $\iota$ using $\chi = 1$ at $\lambda = 0.6$. The inset shows the performance for the same graphs as a function of $\chi$ and fixing $\iota = 1$. [Fixed hyperparameters: sweeps = 80, $p = 0.3$.]