Approximate Quadratization of High-Order Hamiltonians for Combinatorial Quantum Optimization

Abstract: Combinatorial optimization problems have wide-ranging applications in industry and academia. Quantum computers may help solve them by sampling from carefully prepared Ansatz quantum circuits. However, current quantum computers are limited by their qubit count, connectivity, and noise. This is particularly restrictive when considering optimization problems beyond the quadratic order. Here, we introduce Ansatze based on an approximate quadratization of high-order Hamiltonians which do not incur a qubit overhead. The price paid is a loss in the quality of the noiseless solution. Crucially, this approximation yields shallower Ansatze which are more robust to noise than the standard QAOA one. We show this through simulations of systems of 8 to 16 qubits with variable noise strengths. Furthermore, we also propose a noise-aware Ansatz design method for quadratic optimization problems. This method implements only part of the corresponding Hamiltonian by limiting the number of layers of SWAP gates in the Ansatz. We find that for both problem types, under noise, our approximate implementation of the full problem structure can significantly enhance the solution quality. Our work opens a path to enhance the solution quality that approximate quantum optimization achieves on noisy hardware.