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arXiv:2409.16350 · Quantum Science and Technology

Enhancing the energy gap of random graph problems via XX-catalysts in quantum annealing

Luca A. Nutricati, Roopayan Ghosh, Natasha Feinstein, Sougato Bose, Paul A. Warburton

Published Jul 22, 2025• 1 citation
arXivHTMLPDF

Abstract

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.

DOI
10.1088/2058-9565/adf2d5

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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.
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.
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 promises to solve combinatorial optimization problems by preparing the ground state of a target Hamiltonian. Standard annealing protocols are, however, stoquastic and can thus be simulated by sign-problem-free quantum Monte-Carlo methods. To obtain a true quantum advantage, it has been proposed to use non-stoquastic catalyst Hamiltonians. Active only at intermediate stages of the protocol, these can, for certain problems, convert first-order into second-order quantum phase transitions and thus permit an exponential speedup over the stoquastic protocol. At the same time, the non-stoquastic catalyst renders quantum Monte-Carlo methods inefficient. It remains, however, an open question how other classical computation methods are affected by the non-stoquastic terms. We address this question by computing quantum resources -- entanglement entropy and stabilizer Rényi entropy -- whose presence makes classical computations based on tensor networks and stabilizer-tableau methods exponentially hard. We compare these with the spectral gap along the annealing path for two paradigmatic benchmark models, the fully connected ppp-spin model and a geometrically local Ising model. While the exact behavior shows a subtle dependency on the underlying model and the annealing path, our numerics suggest consistently that the scaling of entanglement and non-stabilizerness is at least maintained in the deeply non-stoquastic regime and in some cases even significantly enhanced. Our results thus suggest that improvements of quantum performance in non-stoquastic annealing coincide with significant presence of quantum computational resources.
In adiabatic quantum computation the required run-time to reach a given ground-state fidelity is dictated by the size of the minimum gap that appears between the ground and first-excited state in the annealing spectrum. In general the presence of avoided level crossings demands an exponential increase in the annealing time with the system size which has consequences both for the efficiency of the algorithm and the required qubit coherence times. One promising avenue being explored to produce more favorable gap scaling is the introduction of nonstoquastic X X couplings in the form of a catalyst—of particular interest are catalysts which utilize accessible information about the optimization problem in their construction. Here we show extreme sensitivity of the effect of an X X catalyst to subtle changes in the encoding of the optimization problem. In particular, we observe that a targeted catalyst containing a single coupling can significantly reduce the gap closing with system size at an avoided level crossing. For slightly different encodings of the same problems however, these same catalysts result in closing gaps in the annealing spectrum. To understand the origin of these closing gaps, we study how the evolution of the ground-state vector is altered by the presence of the catalyst and find that the negative components of the ground-state vector are key to understanding the response of the gap spectrum. We also consider how and when these closing gaps could be utilized in diabatic quantum annealing protocols—a promising alternative to adiabatic quantum annealing in which transitions to higher energy levels are exploited to reduce the run time of the algorithm. Published by the American Physical Society 2024
Non-stoquastic Hamiltonians have both positive and negative signs in off-diagonal elements in their matrix representation in the standard computational basis and thus cannot be simulated efficiently by the standard quantum Monte Carlo method due to the sign problem. We describe our analytical studies of this type of Hamiltonians with infinite-range non-random as well as random interactions from the perspective of possible enhancement of the efficiency of quantum annealing or adiabatic quantum computing. It is shown that multi-body transverse interactions like XX and XXXXX with positive coefficients appended to a stoquastic transverse-field Ising model render the Hamiltonian non-stoquastic and reduce a first-order quantum phase transition in the simple transverse-field case to a second-order transition. This implies that the efficiency of quantum annealing is exponentially enhanced, because a first-order transition has an exponentially small energy gap (and therefore exponentially long computation time) whereas a second-order transition has a polynomially decaying gap (polynomial computation time). The examples presented here represent rare instances where strong quantum effects, in the sense that they cannot be efficiently simulated in the standard quantum Monte Carlo, have analytically been shown to exponentially enhance the efficiency of quantum annealing for combinatorial optimization problems.
Quantum Annealing (QA) has already put into practical use and considered useful for solving many social issues, such as reduction of traffic congestion and delivery optimization. But in QA, when the energy difference between the ground state and the first excited state is small, the transition probability between them increases. Therefore, in order to decrease the transition, QA has to be performed at extremely low temperatures. To simulate the situation, it has the problem in that it takes much time. Many studies are underway to accelerate QA theoretically, one of which is to incorporate non-stoquastic Hamiltonian. On the other hand, I proposed Nested Simulated Annealing (NSA) inspired by Quantum Monte Carlo (QMC). I showed the computational speedup could be achieved dramatically, by considering the idea that the effect of flipping a spin preferentially influenced the spins directly interacting with it. Although NSA was based on such classical concept of causality, the hybrid computation both quantum and classical approach worked well. In this paper, in order to discuss the relationship between NSA and non-stoquastic Hamiltonian like XX-interaction, the spins are treated as continuous variables. I derive the formula to calculate the total energy in both with a problem Hamiltonian and the perturbation Hamiltonian induced by a transverse electromagnetic field. And I will show a clear relationship between local-maxima and the convergence speed when calculating it with the binary spin. More precisely, the annealing path with the smallest local-maxima converges fast, even though it consumes fewer computational resources. Therefore, it is possible to induce an optimal adiabatic phase transition by selecting NSA parameters appropriately. This paper shows an effective method to choose the parameters when simulating QMC on a classical computer.