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

Technical Summary: Worldline-Susceptibility Scheduling for Quantum Annealing

Problem Statement
The performance of quantum annealing (QA) is critically dependent on the annealing schedule, which dictates how the total runtime is distributed along the evolution path. While the Roland–Cerf local-adiabatic schedule is theoretically optimal—allocating time inversely proportional to the square of the instantaneous spectral gap ($\Delta(s)^{-2}$)—it is practically infeasible for large optimization problems. Constructing this schedule requires complete knowledge of the instantaneous low-energy spectrum, necessitating repeated diagonalization of an exponentially large Hamiltonian. Consequently, there is a need for a computationally inexpensive surrogate that can identify the critical regions of the anneal without explicit spectral information.

Methodology
The authors propose a surrogate scheduling framework based on worldline magnetization susceptibility ($\chi_m$) measured during Simulated Quantum Annealing (SQA). The methodology proceeds as follows:

1. SQA Simulation: The quantum annealing process is mapped to an equivalent classical statistical-mechanical system using the Suzuki–Trotter decomposition. This transforms the transverse-field Ising model into a $(d+1)$-dimensional classical system with $M$ coupled replicas (Trotter slices).
2. Observable Measurement: Equilibrium configurations are sampled using classical Monte Carlo techniques. From these, the worldline magnetization $m$ is calculated, and its susceptibility is derived via the fluctuation–dissipation theorem: $\chi_m = NM(\langle m^2 \rangle - \langle |m| \rangle^2)$.
3. Surrogate Construction: The measured susceptibility $\chi_m(s)$ serves as a proxy for the inverse-square spectral gap. A time-allocation weight $w(s) = \chi_m(s) + \chi_0$ (where $\chi_0$ is a regularization constant) is defined. The cumulative time allocation function $\tau(s)$ is constructed by integrating this weight, and the final schedule $s(t)$ is obtained by inverting $\tau(s)$.
4. Validation: The framework, implemented in the open-source Qanneal software, validates the surrogate schedules against exact diagonalization (ED) results for Sherrington–Kirkpatrick (SK) spin-glass instances. Performance is quantified by the final ground-state probability ($P_{GS}$) obtained by solving the time-dependent Schrödinger equation for linear, exact Roland–Cerf, and the proposed surrogate schedules.

Key Results
The study demonstrates that the worldline-susceptibility schedule consistently outperforms both conventional linear annealing and, counterintuitively, the exact Roland–Cerf schedule for a substantial fraction of SK instances.

• Identification of Critical Regions: The peak of the worldline susceptibility reliably identifies the broad region where the spectral gap is minimal, even though it does not quantitatively reproduce the detailed gap profile.
• Superior Finite-Time Performance: For representative instances ($n=10$ and $n=12$), the surrogate schedule yields higher ground-state probabilities than the exact Roland–Cerf schedule across various total annealing times.
• Discovery of Failure Modes: The authors identify two distinct finite-time failure modes of the exact local-adiabatic schedule that explain its underperformance:
  1. Boundary-Gap Trap: When the minimum gap occurs at the end of the anneal ($s \approx 1$), the Roland–Cerf schedule allocates the majority of the runtime to a region where the transverse-field driver has effectively vanished. This renders the additional time dynamically ineffective as quantum fluctuations are suppressed. The susceptibility-based schedule avoids this by concentrating time in the broader crossover region where quantum fluctuations remain active.
  2. Oscillatory Instability: When the minimum gap is interior, the exact schedule concentrates runtime so sharply around the gap that it induces coherent multilevel interference. This results in a non-monotonic, oscillatory dependence of the success probability on the total annealing time, which the smoother susceptibility-based schedule avoids.
• Statistical Robustness: Large-scale numerical experiments across disorder realizations and system sizes ($n=10$ to $n=20$) confirm that the superiority of the surrogate schedule is a generic feature, not an artifact of specific instances. As system size increases, the prevalence of boundary-gap instances rises, further degrading the performance of the exact Roland–Cerf schedule relative to the surrogate.

Significance and Claims
The paper argues that under realistic finite-time conditions, robustness in scheduling is more critical than the precise reproduction of the instantaneous spectral gap profile. The authors claim that:

• Inexpensive equilibrium observables (specifically worldline susceptibility from SQA) can serve as effective surrogates for spectral-gap-based strategies.
• The exact local-adiabatic condition, while theoretically optimal in the asymptotic limit, can be counterproductive in finite-time regimes due to the "boundary-gap trap" and "oscillatory instability."
• The proposed methodology offers a scalable path for schedule optimization that bypasses the computational intractability of exact spectral calculations, making it applicable to problem sizes beyond the reach of exact diagonalization.

The work is implemented in the open-source Qanneal framework, providing a unified platform for SQA, susceptibility measurement, and schedule construction, thereby facilitating reproducible research in quantum annealing dynamics.