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arXiv:2512.03526

Stretched Exponential Scaling of Parity-Restricted Energy Gaps in a Random Transverse-Field Ising Model

G. -X. Tang, J. -Z. Zhuang, L. -M. Duan, Y. -K. Wu

cond-mat.dis-nnquant-phSubmitted Dec 3, 2025• Announced Dec 4, 2025• 0 citations
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Abstract

The success of a quantum annealing algorithm requires a polynomial scaling of the energy gap. Recently it was shown that a two-dimensional transverse-field Ising model on a square lattice with nearest-neighbor $\pm J$ random coupling has a polynomial energy gap in the symmetric subspace of the parity operator [Nature 631, 749-754 (2024)], indicating the efficient preparation of its ground states by quantum annealing. However, it is not clear if this result can be generalized to other spin glass models with continuous or biased randomness. Here we prove that under general independent and identical distributions (i.i.d.) of the exchange energies, the energy gap of a one-dimensional random transverse-field Ising model follows a stretched exponential scaling even in the parity-restricted subspace. We discuss the implication of this result to quantum annealing problems.

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14 pages, 3 figures

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The success of a quantum annealing algorithm requires a polynomial scaling of the energy gap. Recently it was shown that a two-dimensional transverse-field Ising model on a square lattice with nearest-neighbor ±J\pm J±J random coupling has a polynomial energy gap in the symmetric subspace of the parity operator [Nature 631, 749-754 (2024)], indicating the efficient preparation of its ground states by quantum annealing. However, it is not clear if this result can be generalized to other spin glass models with continuous or biased randomness. Here we prove that under general independent and identical distributions (i.i.d.) of the exchange energies, the energy gap of a one-dimensional random transverse-field Ising model follows a stretched exponential scaling even in the parity-restricted subspace. We discuss the implication of this result to quantum annealing problems.
A promising approach to solving hard binary optimisation problems is quantum adiabatic annealing (QA) in a transverse magnetic field. An instantaneous ground state --- initially a symmetric superposition of all possible assignments of NNN qubits --- is closely tracked as it becomes more and more localised near the global minimum of the classical energy. Regions where the energy gap to excited states is small (e.g. at the phase transition) are the algorithm's bottlenecks. Here I show how for large problems the complexity becomes dominated by O(log⁡N)O(\log N)O(logN) bottlenecks inside the spin glass phase, where the gap scales as a stretched exponential. For smaller NNN, only the gap at the critical point is relevant, where it scales polynomially, as long as the phase transition is second order. This phenomenon is demonstrated rigorously for the two-pattern Gaussian Hopfield Model. Qualitative comparison with the Sherrington-Kirkpatrick Model leads to similar conclusions.
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A promising approach to solving hard binary optimization problems is quantum adiabatic annealing in a transverse magnetic field. An instantaneous ground state-initially a symmetric superposition of all possible assignments of N qubits-is closely tracked as it becomes more and more localized near the global minimum of the classical energy. Regions where the energy gap to excited states is small (for instance at the phase transition) are the algorithm's bottlenecks. Here I show how for large problems the complexity becomes dominated by O(log N) bottlenecks inside the spin-glass phase, where the gap scales as a stretched exponential. For smaller N, only the gap at the critical point is relevant, where it scales polynomially, as long as the phase transition is second order. This phenomenon is demonstrated rigorously for the two-pattern Gaussian Hopfield model. Qualitative comparison with the Sherrington-Kirkpatrick model leads to similar conclusions.
Quantum Annealing (QA) encounters limitations when the energy gap along the annealing path becomes exponentially small, leading to impractically long runtimes. In contrast, the success of hybrid digital methods like the Quantum Approximate Optimization Algorithm (QAOA), which operate via discrete unitary operations, relies on the optimization of the variational parameters appearing in the state. We analyze a class of transverse-field Ising models which includes problems with exponentially small spectral gaps, but whose dynamics is described in terms of fermionic Gaussian states after Jordan-Wigner mapping. We show that, for digital alternating QAOA-like states, the number of unitaries required to reach the exact ground state scales quadratically with system size and is independent of the annealing gap. This number can be exactly computed from the algebraic properties of the Ansatz, revealing a fundamental distinction between digital methods and their analog counterpart.
The performance of quantum annealing for combinatorial optimization is fundamentally limited by the minimum energy gap ΔΔΔ encountered at quantum phase transitions. We investigate the scaling of ΔΔΔ with system size NNN for two paradigmatic quantum spin-glass models: the two-dimensional Edwards-Anderson (2D-EA) and the all-to-all Sherrington-Kirkpatrick (SK) models. Utilizing a newly proposed unbiased energy-gap estimator for continuous-time projection quantum Monte Carlo simulations, complemented by high-performance sparse eigenvalue solvers, we characterize the gap distributions across disorder realizations. It is found that, in the 2D-EA case, the inverse-gap distribution develops a fat tail with infinite variance as NNN increases. This indicates that the unfavorable super-algebraic scaling of ΔΔΔ, recently reported for binary couplings [Nature 631, 749 (2024)], persists for the Gaussian disorder considered here, pointing to a universal feature of 2D spin glasses. Conversely, the SK model retains a finite-variance distribution, with the disorder-averaged gap following a rather slow power law, close to Δ∝N−1/3Δ\propto N^{-1/3}Δ∝N−1/3. This finding provides a promising outlook for the potential efficiency of quantum annealers for optimization problems with dense connectivity.
We show that the quantum approximate optimization algorithm (QAOA) can construct, with polynomially scaling resources, the ground state of the fully connected ppp-spin Ising ferromagnet, a problem that notoriously poses severe difficulties to a vanilla quantum annealing (QA) approach due to the exponentially small gaps encountered at first-order phase transition for $p\ensuremath{\ge}3$. For a target ground state at arbitrary transverse field, we find that an appropriate QAOA parameter initialization is necessary to achieve good performance of the algorithm when the number of variational parameters 2P2P2P is much smaller than the system size NNN because of the large number of suboptimal local minima. Instead, when PPP exceeds a critical value ${P}_{N}^{*}\ensuremath{\propto}N$, the structure of the parameter space simplifies, as all minima become degenerate. This allows achieving the ground state with perfect fidelity with a number of parameters scaling extensively with NNN and with resources scaling polynomially with NNN.