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arXiv:2503.15244 · Physical Review A

Improving the efficiency of quantum annealing with controlled diagonal catalysts

T Hattori, Shu Tanaka

Published Feb 3, 2026• 3 citations
arXivHTMLPDF

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.

DOI
10.1103/vff6-cf7c

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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.
Quantum annealing (QA) is a method for solving combinatorial optimization problems. We can estimate the computational time for QA using what is referred to as the adiabatic condition derived from the adiabatic theorem. The adiabatic condition consists of two parts: an energy gap and a transition matrix. Most past studies have focused on the relationship between the energy gap and computational time. The success probability of QA is considered to decrease exponentially owing to the exponentially decreasing energy gap at the first-order phase-transition point. In this study, through a detailed analysis of the relationship between the energy gap, transition matrix, and computational cost during QA, we propose a general method for constructing counterintuitive models wherein QA with a constant annealing time fails despite a constant energy gap, based on polynomial energy. We assume that the energy of the total Hamiltonian is at most $\mathrm{\ensuremath{\Theta}}(L)$, where LLL is the number of qubits. In our formalism, we choose a known model that exhibits an exponentially small energy gap during QA, and modify the model by adding a specific penalty term to the Hamiltonian. In the modified model, the transition matrix in the adiabatic condition becomes exponentially large as the number of qubits increases, while the energy gap remains constant. Moreover, we achieve a quadratic speedup, for which the upper bound for improvement in the adiabatic condition is determined by the polynomial energy. For concrete examples, we consider the adiabatic Grover search and the ferromagnetic ppp-spin model. In these cases, with the addition of the penalty term, although the success probability of QA on the modified models becomes exponentially small despite a constant energy gap, we are able to achieve a success probability considerably higher than that of conventional QA. Moreover, in concrete examples, we numerically show the scaling of the computational cost is quadratically improved compared to the conventional QA. Our findings pave the way for a better understanding of QA performance.
Abstract Quantum annealing (QA) is a continuous-time heuristic quantum algorithm for solving or approximately solving classical optimization problems. The algorithm uses a schedule to interpolate between a driver Hamiltonian with an easy-to-prepare ground state and a problem Hamiltonian whose ground state encodes solutions to an optimization problem. The standard implementation relies on the evolution being adiabatic: keeping the system in the instantaneous ground state with high probability and requiring a time scale inversely related to the minimum energy gap between the instantaneous ground and excited states. However, adiabatic evolution can lead to evolution times that scale exponentially with the system size, even for computationally simple problems. Here, we study whether non-adiabatic evolutions with optimized annealing schedules can bypass this exponential slowdown for one such class of problems called the frustrated ring model. For sufficiently optimized annealing schedules and system sizes of up to 39 qubits, we provide numerical evidence that we can avoid the exponential slowdown. Our work highlights the potential of highly-controllable QA to circumvent bottlenecks associated with the standard implementation of QA.
Abstract In adiabatic quantum annealing, the speed with which an anneal can be run, while still achieving a high final ground state (GS) fidelity, is dictated by the size of the minimum gap that appears between the ground and first excited state in the annealing spectrum. To avoid the exponential slowdown associated with exponentially closing gaps, diabatic transitions to higher energy levels may be exploited in such a way that the system returns to the GS before the end of the anneal. In certain cases, this is facilitated by the original annealing spectrum. However, there are also examples where careful manipulation of the annealing Hamiltonian has been used to alter the spectrum to create a diabatic path to the GS. Since diabatic transitions depend on the evolution rate and the gap sizes in the spectrum, it is important to consider the sensitivity of any potential enhancement to changes in the anneal time as well as any parameters involved in the manipulation of the spectrum. We explore this sensitivity using annealing spectra containing an exponentially closing gap and an additional, tuneable, small gap created by a catalyst. We find that there is a trade-off between the precision needed in the catalyst strength and the anneal time in order to maintain the enhancement to the final GS fidelity.
In the computational model of quantum annealing, the size of the minimum gap between the ground state and the first excited state of the system is of particular importance, since it is inversely proportional to the running time of the algorithm. Thus, it is desirable to keep the gap as large as possible during the annealing process, since it allows the computation to remain under the protection of the adiabatic theorem while staying efficient. We propose steered quantum annealing as a new method to enlarge the gap throughout the process, in the case of diagonal final Hamiltonians, based on the exploitation of some assumptions we can make about the particular problem instance. In order to introduce this information, we propose beginning the anneal from a biased Hamiltonian that incorporates reliable assumptions about the final ground state. Our simulations show that this method yields a larger average gap throughout the whole computation, which results in an increased robustness of the overall annealing process.
The potential quantum speedup in solving optimization problems via adiabatic quantum annealing is often hindered by the closing of the energy gap during the anneal, especially when this gap scales exponentially with system size. In this work, we alleviate this bottleneck by demonstrating that for the NP-complete Maximum Weighted Independent Set (MWIS) problem, an informed choice of n -local catalysts (operators involving n qubits) can reopen the gap during the annealing process. By analyzing first-order phase transitions in toy instances of the MWIS problem, we first identify direct-tunneling catalysts that effectively eliminate the transition and provide an analytical discussion on when the sign of the catalyst influences its impact. We then reveal that n -local catalysts exponentially improve gap scaling and in certain scenarios are as effective as direct-tunnel coupling between two minima. Utilizing this understanding, we show that they also increase the efficiency of ground-state preparation via adiabatic quantum annealing in random graphs and analytically demonstrate the necessity of their placement across unfrustrated loops in the graph for effective performance in MWIS problems. Additionally, using a circuit implementation of the n -local catalyst (requiring 2 n nearest-neighbor gates), we demonstrate that both the circuit depth and the total number of gates required to solve the problem are reduced by several orders of magnitude when compared to the discrete-time version of traditional quantum annealing using local drivers. Our analysis suggests that nonlocal quantum fluctuations entangling multiple qubits as a catalyst are key to achieving the desired quantum advantage.
Abstract One of the main challenges in solving combinatorial optimisation problems with quantum annealers is the emergence of extremely small energy gaps between the ground state and the first excited state of the annealing Hamiltonian. These small gaps may be symptoms of an underlying first-order phase transition, which, according to the adiabatic theorem, can significantly extend the required anneal time, making practical implementation effectively infeasible. In this paper we demonstrate that attaching an XX-catalyst on all the edges of a graph upon which a MWIS (Maximum Weighted Independent Set) problem is defined, significantly enhances the minimum energy gap. Remarkably, our analysis shows that the smaller the energy gap, the more effective the catalyst is in opening it. This result is based on a detailed statistical analysis performed on a large number of randomly generated MWIS problem instances on both Erdõs–Rényi and Barabáasi–Albert graphs. We perform the analysis using both stoquastic and non-stoquastic catalysts.