Scalable Quantum Optimisation using HADOF: Hamiltonian Auto-Decomposition Optimisation Framework

TL;DR

This paper presents HADOF, a framework that auto-decomposes a QUBO-encoded optimization into sub-Hamiltonians to enable scalable quantum and classical optimization on NISQ devices. It leverages an iterative, sampling-based procedure to solve subproblems with cost and mixer Hamiltonians and then aggregates results to form a global solution, aiming to overcome qubit limitations. Classical simulations show HADOF scalable performance and competitive quality relative to CPLEX up to $n=500$, with a hardware demonstration on an IBM device indicating practical potential. The approach is modular, compatible with multiple optimizers, and highlights opportunities for enhanced aggregation, parallelism, and multi-level decomposition as hardware evolves.

Abstract

Quantum Annealing (QA) and QAOA are promising quantum optimisation algorithms used for finding approximate solutions to combinatorial problems on near-term NISQ systems. Many NP-hard problems can be reformulated as Quadratic Unconstrained Binary Optimisation (QUBO), which maps naturally onto quantum Hamiltonians. However, the limited qubit counts of current NISQ devices restrict practical deployment of such algorithms. In this study, we present the Hamiltonian Auto-Decomposition Optimisation Framework (HADOF), which leverages an iterative strategy to automatically divide the Quadratic Unconstrained Binary Optimisation (QUBO) Hamiltonian into sub-Hamiltonians which can be optimised separately using Hamiltonian based optimisers such as QAOA, QA or Simulated Annealing (SA) and aggregated into a global solution. We compare HADOF with Simulated Annealing (SA) and the CPLEX exact solver, showing scalability to problem sizes far exceeding available qubits while maintaining competitive accuracy and runtime. Furthermore, we realise HADOF for a toy problem on an IBM quantum computer, showing promise for practical applications of quantum optimisation.

Figures (8)

• Figure 1: Standard circuit with alternating cost and mixer HamiltoniansQUBO-pennylane. The output produces a probability distribution over the solution space which can be samples with the shots parameter of the quantum simulation or backend. Higher sampled solutions are more likely to be solutions with better objective value.
• Figure 2: Trotterised parameters based on Ref QUBO-pennylane, openQAOA. We move $\beta_m$ from 1 to 0 and $\gamma_m$ from 0 to 1 allowing the system to stay close to the ground state of the mixer Hamiltonian to the Cost Hamiltonian.
• Figure 3: General overview of the framework. Here, we use as the optimiser, which is called iteratively. (1) We choose subsets of sizes 5 and 10 from the binary variables of the global problem. (2) These are used to form the sub-Hamiltonians using $P(x_i)$, approximated as the expected value of each qubit. (3) The circuit set up with $t=1$ layers and in every iteration we add a layer. In this study, we use 10 layers. (4) Once the $n/k$ sub-Hamiltonians are optimised, we sample them and use an aggregation policy to form the global solution probability distribution.
• Figure 4: Time to solution as a function of problem size for CPLEX (exact classical solver), SA, HADOF QAOA and SA with 5 variable and 10 variable sub-QUBOs for problem sizes $n=10$ to $n=80$. CPLEX exhibits exponential scaling. SA scales the best in time as problem size increases. The inset shows the other algorithms, excluding CPLEX for clarity.
• Figure 5: Time to solution as a function of problem size for SA, HADOF QAOA and SA with 5 variable and 10 variable sub-QUBOs for problem sizes $n=100$ to $n=500$.