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arXiv:2607.13150

Emulating XX catalysts for quantum annealing via self-consistent transverse fields

Mohammadhossein Dadgar, Christopher L. Baldwin

quant-phcond-mat.stat-mechSubmitted Jul 14, 2026• Announced Jul 16, 2026• 0 citations
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Abstract

Fully-connected transverse interactions have been considered as catalysts for quantum annealing that could mitigate exponentially small gaps and circumvent first-order phase transitions, but their experimental implementation remains challenging. In this work, we introduce a procedure for emulating their effects via a self-consistent transverse field. All that is required beyond conventional transverse-field annealing is the ability to make measurements in the transverse ($\hatσ^x$) basis. We show that this protocol yields identical dynamics in the large-system limit, and study the approach to that limit in numerical simulations of the (uniform) $p$-spin model. However, realizing the protocol in practice requires us to consider a series of approximate variants, each of whose errors we quantify and demonstrate can be made sufficiently small. Lastly, we show how to map the protocol onto annealing platforms that vary only a single control parameter. Even after the multiple stages of approximation, our procedure can generate dynamics that agree well with the original transverse-interaction catalyst, establishing self-consistent transverse fields as a viable alternative on near-term quantum annealers.

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Fully-connected transverse interactions have been considered as catalysts for quantum annealing that could mitigate exponentially small gaps and circumvent first-order phase transitions, but their experimental implementation remains challenging. In this work, we introduce a procedure for emulating their effects via a self-consistent transverse field. All that is required beyond conventional transverse-field annealing is the ability to make measurements in the transverse (σ^x\hatσ^xσ^x) basis. We show that this protocol yields identical dynamics in the large-system limit, and study the approach to that limit in numerical simulations of the (uniform) ppp-spin model. However, realizing the protocol in practice requires us to consider a series of approximate variants, each of whose errors we quantify and demonstrate can be made sufficiently small. Lastly, we show how to map the protocol onto annealing platforms that vary only a single control parameter. Even after the multiple stages of approximation, our procedure can generate dynamics that agree well with the original transverse-interaction catalyst, establishing self-consistent transverse fields as a viable alternative on near-term quantum annealers.
We introduce a two-parameter approximate counter-diabatic term into the Hamiltonian of the transverse-field Ising model for quantum annealing to accelerate convergence to the solution, generalizing an existing single-parameter approach. The protocol is equivalent to unconventional diabatic control of the longitudinal and transverse fields in the transverse-field Ising model and thus makes it more feasible for experimental realization than an introduction of new terms such as non-stoquastic catalysts toward the same goal of performance enhancement. We test the idea for the ppp-spin model with p=3p=3p=3, which has a first-order quantum phase transition, and show that our two-parameter approach leads to significantly larger ground-state fidelity and lower residual energy than those by traditional quantum annealing as well as by the single-parameter method. We also find a scaling advantage in terms of the time to solution as a function of the system size in a certain range of parameters as compared to the traditional methods.
We introduce a two-parameter approximate counter-diabatic term into the Hamiltonian of the transversefield Ising model for quantum annealing to accelerate convergence to the solution, generalizing an existing single-parameter approach.The protocol is equivalent to unconventional diabatic control of the longitudinal and transverse fields in the transverse-field Ising model and thus makes it more feasible for experimental realization than an introduction of new terms such as nonstoquastic catalysts toward the same goal of performance enhancement.We test the idea for the p-spin model with p = 3, which has a first-order quantum phase transition, and show that our two-parameter approach leads to significantly larger ground-state fidelity and lower residual energy than those by traditional quantum annealing and by the single-parameter method.We also find a scaling advantage in terms of the time-to-solution as a function of the system size in a certain range of parameters as compared to the traditional methods in the sense that an exponential time complexity is reduced to another exponential complexity with a smaller coefficient.Although the present method may not always lead to a drastic exponential speedup in difficult optimization problems, it is useful because of its versatility and applicability for any problem after a simple algebraic manipulation, in contrast to some other powerful prescriptions for acceleration such as nonstoquastic catalysts in which one should carefully study in advance if it works in a given problem and should identify a proper way to meticulously control the system parameters to achieve the goal, which is generally highly nontrivial.
This paper deals with fully-connected mean-field models of quantum spins with p-body ferromagnetic interactions and a transverse field. For p=2 this corresponds to the quantum Curie-Weiss model (a special case of the Lipkin-Meshkov-Glick model) which exhibits a second-order phase transition, while for p>2 the transition is first order. We provide a refined analytical description both of the static and of the dynamic properties of these models. In particular we obtain analytically the exponential rate of decay of the gap at the first-order transition. We also study the slow annealing from the pure transverse field to the pure ferromagnet (and vice versa) and discuss the effect of the first-order transition and of the spinodal limit of metastability on the residual excitation energy, both for finite and exponentially divergent annealing times. In the quantum computation perspective this quantity would assess the efficiency of the quantum adiabatic procedure as an approximation algorithm.
Introducing a nonstoquastic catalyst is a promising avenue to improve quantum annealing with the transverse field. In the present paper, we propose a nonstoquastic catalyst for bifurcation-based quantum annealing described by the spin-1 operators to improve the efficiency of a ground-state search. To investigate the effect of the nonstoquastic catalyst, we study the ferromagnetic ppp-spin model, which has difficulty with finding the ground state due to the first-order phase transition for quantum annealing. A semiclassical analysis shows that the problematic first-order phase transition can be eliminated by introducing the proposed nonstoquastic catalyst with the appropriate amplitude. We also numerically calculate the minimum energy gap for a finite-size system by diagonalizing the Hamiltonian. We find that while the energy gap decreases exponentially with increasing system size for the original Hamiltonian, it decreases polynomially against the system size for the Hamiltonian with the nonstoquastic catalyst. This result implies that the proposed nonstoquastic catalyst has the potential to improve the performance of bifurcation-based quantum annealing.
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.
This paper deals with fully-connected mean-field models of quantum spins with p-body ferromagnetic interactions and a transverse field. For p=2 this corresponds to the quantum Curie-Weiss model (a special case of the Lipkin-Meshkov-Glick model) which exhibits a second-order phase transition, while for p>2 the transition is first order. We provide a refined analytical description both of the static and of the dynamic properties of these models. In particular we obtain analytically the exponential rate of decay of the gap at the first-order transition. We also study the slow annealing from the pure transverse field to the pure ferromagnet (and vice versa) and discuss the effect of the first-order transition and of the spinodal limit of metastability on the residual excitation energy, both for finite and exponentially divergent annealing times. In the quantum computation perspective this quantity would assess the efficiency of the quantum adiabatic procedure as an approximation algorithm.