Improving the efficiency of quantum annealing with controlled diagonal catalysts

TL;DR

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 and achieves an approximate quadratic speedup of the exponential scaling exponent in time to solution compared to the conventional quantum annealing.

Abstract

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.

Figures (15)

• Figure 1: Conceptual diagram of the Ising model for MWIS on $K_{4,3}$. The circles represent the vertices of $K_{4,3}$, and the black lines indicate the edges. The graph consists of two disjoint subsets, shown as dark blue and red circles. Each spin on the nodes of the graph has an associated weight denoted by $w_i$. The white arrows represent the spin directions in the ground state of the Ising model. All edge interactions in $K_{4,3}$ have the same value $J_{zz}$.
• Figure 2: (a) Energy gap $\Delta(t)$ between the ground state and the first excited state as a function of $t$ for MWIS instances with an $N$-Hamming-distance configuration. Since the energy difference between the ground state and the first excited state in the problem Hamiltonian is extremely small, the energy gap $\Delta(t)$ closes near the end of the annealing process. (b) Enlarged view of $\Delta(t)$ in the range $0.99 \leq t/\tau \leq 1$ on a logarithmic scale, showing the rapid closure of the gap near the final annealing time.
• Figure 3: Size scaling of TTS for hard MWIS instances with $\tau=512$. The vertical axis is logarithmic. Points indicate mean TTS values, error bars denote standard deviations, and dashed lines show least-squares fits (see Table \ref{['tab: TTS_sizescaling']}).
• Figure 4: Populations of the ground and first excited states during the time evolution for a hard MWIS instance with $N=11$. The annealing time is set to $\tau=512$. The blue and red solid lines show the ground-state populations for the proposed method and the linear-schedule QA, respectively, while the dashed lines indicate the corresponding first-excited-state populations. The color distinctions are the same as before.
• Figure 5: Optimized schedules for hard MWIS instances with $N=11$ at annealing time $\tau=512$. The black dashed line represents the zero reference, and different colors correspond to different MWIS instances.