Worldline-Susceptibility Scheduling for Quantum Annealing Beyond Local-Adiabatic Evolution

INTRODUCTION

Quantum annealing solves combinatorial optimization problems by encoding solutions into the ground state of an Ising Hamiltonian and continuously evolving from a simple driver Hamiltonian to the target problem Hamiltonian. For a fixed total annealing time, traversing the most difficult part of the evolution too rapidly increases diabatic transitions, while evolving slowly throughout wastes runtime — so how the annealing parameter is varied in time plays a central role in the final success probability. Roland and Cerf showed that the annealing velocity should satisfy a local-adiabatic condition, slowing down in inverse proportion to the square of the instantaneous spectral gap. This schedule minimizes total annealing time while maintaining adiabatic accuracy, but implementing it requires complete knowledge of the instantaneous spectral gap throughout the trajectory — generally requiring repeated diagonalization of an exponentially large Hamiltonian, which becomes infeasible beyond small system sizes. This motivates the central question of this work: can an inexpensive observable, measured directly during simulated quantum annealing (SQA), reliably locate the critical region of the anneal without spectral information?

METHODOLOGY

We construct and validate the surrogate schedule in four stages: 1. WORLDLINE SUSCEPTIBILITY VIA SQA: Using the Suzuki–Trotter mapping, the quantum annealing process is represented as an equivalent (d+1)-dimensional classical worldline system of M coupled Trotter replicas, sampled by classical Monte Carlo. The equilibrium fluctuations of the average worldline magnetization define the worldline magnetization susceptibility χm(s), which — via the fluctuation–dissipation theorem — scales near criticality as χm(s) ∝ Δ(s)⁻², providing a spectral-information-free surrogate for the inverse-square spectral gap central to the Roland–Cerf condition. 2. SURROGATE SCHEDULE CONSTRUCTION: The measured susceptibility is converted into a positive time-allocation weight w(s) = χm(s) + χ0, whose cumulative normalized integral τ(s) is inverted to yield the surrogate annealing velocity ds/dt = 1/[T·w(s)] — slowing the evolution where susceptibility is large and accelerating it where it is small, using only equilibrium observables measured during the SQA simulation. 3. BENCHMARKING AGAINST EXACT SPECTRAL METHODS: For small Sherrington–Kirkpatrick spin-glass benchmark instances, exact diagonalization provides the ground-truth instantaneous spectral gap and the exact Roland–Cerf schedule, against which the linear, Roland–Cerf, and surrogate schedules are compared by propagating the time-dependent Schrödinger equation to obtain the final ground-state probability PGS. 4. QANNEAL IMPLEMENTATION: The complete workflow — SQA simulation, susceptibility measurement, cumulative time allocation, surrogate schedule generation, exact spectral benchmarking, and quantum-dynamical simulation — is implemented in the open-source Qanneal (C++17/Python) framework, used throughout for reproducibility.

RESULTS

For a representative n = 10 instance, the exact minimum gap occurs at sED = 0.883 while the susceptibility peak occurs at sSQ = 0.920 — closely co-locating the critical region without spectral information — yet the resulting schedules differ sharply: the Roland–Cerf schedule concentrates an ~4-orders-of-magnitude slowdown in a narrow interval, while the surrogate spreads the slowdown broadly and smoothly. At finite annealing time, the surrogate schedule reaches PGS = 0.295 versus 0.207 (linear) and only 0.078 (Roland–Cerf) for the n = 10 instance, and 0.517 versus 0.431 and 0.150 for a representative n = 12 instance. Two distinct finite-time failure modes explain this: a BOUNDARY-GAP TRAP (n = 12, minimum gap at sED = 1.000), where ~90% of Roland–Cerf's runtime is spent after the transverse-field driver has essentially vanished; and an OSCILLATORY INSTABILITY (n = 10), where extreme localization of runtime around an interior minimum gap produces a reproducible, non-monotonic oscillation in PGS(T) with ~5 extrema — ruled out as conventional two-level Landau–Zener–Stückelberg interference (predicted period ≈312 vs. observed ≈20) and attributed instead to genuine multilevel finite-time interference. Disorder-averaged results across n ∈ {10,...,20} confirm the surrogate schedule achieves the highest mean ground-state probability at every system size, while the fraction of boundary-gap instances grows with n, progressively degrading Roland–Cerf's disorder-averaged performance. Supplementary robustness checks over a 370-instance ensemble further show: the oscillatory instability survives two orders of magnitude of Lindblad dephasing before washing out at hardware-scale noise, confirming it is a genuine coherent effect rather than a numerical artifact; the residual sED–sSQ offset is an irreducible finite-size/disorder floor rather than a Trotter-discretization bias; and a held-out logistic-regression validation recovers a Roland–Cerf failure threshold of Δ* = 0.0159 (95% CI [0.0089, 0.0240], test accuracy 81.1%, AUC 0.890), with outcome rates remaining statistically stable from n = 10 to n = 20.

IMPLICATIONS

These results show that, under realistic finite-time conditions, the quality of an annealing schedule depends not only on how accurately it tracks the instantaneous spectral gap but on how the resulting runtime is distributed — a schedule built from an inexpensive equilibrium observable measured in a classical worldline simulation can outperform one built from exact spectral information. The boundary-gap trap and oscillatory instability identified here represent two general finite-time limitations of exact local-adiabatic scheduling that are not artifacts of small system size or numerical noise. More broadly, this suggests a shift in perspective for annealing schedule design: rather than pursuing increasingly accurate reconstructions of the instantaneous spectral gap, robust schedules can be built from inexpensive equilibrium observables that reliably identify the physically relevant crossover region — opening a scalable route to schedule optimization, implemented here in the open-source Qanneal framework, that avoids the exponential cost of exact spectral methods.

REFERENCES

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