Improving the efficiency of quantum annealing with controlled diagonal catalysts

Quantum annealing is a promising algorithm for solving combinatorial optimization problems. It searches for the ground state of the Ising model, which corresponds to the optimal solution of a given combinatorial optimization problem. The guiding principle of quantum annealing is the adiabatic theorem in quantum mechanics, which guarantees that a system remains in the ground state of its Hamiltonian if the time evolution is sufficiently slow. According to the adiabatic theorem, the annealing time required for quantum annealing to satisfy the adiabaticity scales inversely proportional to the square of the minimum energy gap between the ground state and the first excited state during time evolution. As a result, finding the ground state becomes significantly more difficult when the energy gap is small, creating a major bottleneck in quantum annealing. Expanding the energy gap is one strategy for improving the performance of quantum annealing; however, its implementation in actual hardware remains difficult. This study proposes a method for efficiently solving instances with small energy gaps by introducing additional local terms to the Hamiltonian and exploiting the diabatic transition remaining in the small energy gap. The proposed method achieves an approximate quadratic speedup of the exponential scaling exponent in time to solution compared to the conventional quantum annealing. In addition, we investigate the transferability of the parameters obtained with the proposed method.

💡 Research Summary

This paper addresses one of the most critical bottlenecks in quantum annealing (QA): the drastic closing of the minimum energy gap, especially in instances that exhibit so‑called perturbative crossings. In such cases the adiabatic condition demands an annealing time that scales inversely with the square of the gap, making the required runtime exponentially large for realistic problem sizes. While previous works have proposed “catalysts” that add non‑diagonal (XX, ZZ) or higher‑order interaction terms to enlarge the gap, these terms are difficult to implement on current commercial QA hardware because they require precise control of multi‑qubit couplers.

The authors propose a hardware‑friendly alternative: a purely diagonal catalyst consisting only of linear z‑field terms. The catalyst Hamiltonian is defined as
(H_{\text{catalyst}} = -N\sum_{i=1}^{N}\sigma_i^{z}),
where (N) is the number of qubits. Because it contains only single‑qubit z operators, the required schedule can be realized by locally modulating the longitudinal magnetic fields, a capability already available on existing annealers (e.g., via the h_gain_schedule feature).

The central idea is to treat the catalyst schedule (C(t)) as a variational control function. The total Hamiltonian becomes
(H(t)=A(t)H_q + B(t)H_p + C(t)H_{\text{catalyst}}),
with (C(0)=C(\tau)=0). Using optimal‑control theory, the authors minimize the final‑time expected problem energy
(J = \langle\psi(\tau)|H_p|\psi(\tau)\rangle).
They derive the functional derivative
(\frac{\partial J}{\partial C(t)} = 2,\text{Im}\bigl\langle k(t)\big| H_{\text{catalyst}}\big|\psi(t)\bigr\rangle),
where (|k(t)\rangle) is obtained by backward propagation of (|k(\tau)\rangle = H_p|\psi(\tau)\rangle). This gradient enables a gradient‑based update of (C(t)) until convergence.

For benchmarking, the authors focus on the Maximum Weighted Independent Set (MWIS) problem on a complete bipartite graph (K_{m,n}) with (m=n+1). They generate 500 instances where vertex weights (w_i) are drawn from a tiny uniform interval (