Quantum-Inspired Optimization through Qudit-Based Imaginary Time Evolution

TL;DR

This paper tackles combinatorial optimization with integer-valued decision variables using a quantum-inspired classical approach based on imaginary-time evolution.It extends quantum imaginary time evolution (QITE) to qudits, encoding each variable into a d-level state and constraining evolution to a product state to enable efficient classical simulation, complemented by a gradient-based adaptive ansatz.The Min-$d$-Cut problem with capacity constraints serves as a testbed, showing that qudit-based QITE can outperform penalized binary formulations in certain regimes (notably at $d=7$), while underscoring the critical role of constraint encoding.Overall, the work demonstrates the potential of qudit-based quantum-inspired optimization to reduce problem size and enhance convergence, while outlining avenues for improved constraint handling and broader applicability.

Abstract

Imaginary-time evolution has been shown to be a promising framework for tackling combinatorial optimization problems on quantum hardware. In this work, we propose a classical quantum-inspired strategy for solving combinatorial optimization problems with integer-valued decision variables by encoding decision variables into multi-level quantum states known as qudits. This method results in a reduced number of decision variables compared to binary formulations while inherently incorporating single-association constraints. Efficient classical simulation is enabled by constraining the system to remain in a product state throughout optimization. The qudit states are optimized by applying a sequence of unitary operators that iteratively approximate the dynamics of imaginary time evolution. Unlike previous studies, we propose a gradient-based method of adaptively choosing the Hermitian operators used to generate the state evolution at each optimization step, as a means to improve the convergence properties of the algorithm. The proposed algorithm demonstrates promising results on Min-d-Cut problem with constraints, outperforming Gurobi on penalized constraint formulation, particularly for larger values of d.

Figures (7)

• Figure 1: (a) Imaginary time evolution, as proposed by Motta et al. aims to approximate the imaginary time evolution by applying a sequence of unitary gates motta_determining_2020. Each unitary gate approximately propagates the quantum state for some finite imaginary time interval $\Delta\tau$. (b) Alam et al. proposed an implementation of QITE relying solely on single-qubit $R_y$ gates alam_solving_2023. By operating on product states of qubits classical simulation of large problem instances is enabled, albeit with strongly limited expressibility of the quantum state.
• Figure 2: Mean approximation ratio (AR) as a function of the optimization step. The AR is calculated as the relative performance compared to Gurobi with capacity constraints encoded using unbalanced penalization. The color bands indicate the mean $\pm$ one standard deviation. Results are depicted separately for (a) 50, (b) 100, and (c) 150 vertices. The numerical results display particularly good performance for $d=7$ partitions, achieving an mean AR $< 1$ for all considered vertex counts.
• Figure 3: AR provided by QITE compared to Gurobi with capacity constraints encoded as hard constraints, for (a) 50, (b) 100, and (c) 150 vertices. QITE never acquires an AR superior to Gurobi on average when capacity constraints are encoded in Gurobi as hard constraints.
• Figure 4: (a) Example output provided by qudit-based QITE on a graph with $N=100$ vertices, demonstrating clear clustering. (b) The cost of the rounded output and expectation value of the cost Hamiltonian as function of the number of optimization steps. Interestingly, the cost appears to reduce in a step-wise fashion. (c) Evolution of partition sizes throughout optimization, with the capacity constraint indicated by the horizontal dashed line. The colors of the partitions correspond to the colors of vertices in (a).
• Figure 5: The distributions demonstrate the distribution of partition sizes, as a percentage of the total number of vertices in the graph and a resolution of one vertex. The qudit-based implementation of QITE systematically adheres to the capacity constraint of the Min-$d$-Cut problem, regardless the number of vertices and value of $d$. The maximum partition capacity $C_{\text{max}} = \frac{2N}{d}$ is indicated by the vertical dashed lines. Although the constraint is satisfied in almost all problem instances, many partitions are generated populated only by a small number of vertices. This issue appears slightly less severe for Gurobi, for which distributions are shown in the insets.