Improved gap dependence in adiabatic state preparation by adaptive schedule

Abstract: Adiabatic quantum computing is a powerful framework for state preparation, while its evolution time often scales quadratically in the inverse Hamiltonian spectral gap, leading to sub-optimal computational complexity. In this work, we introduce a nonlinear adaptive strategy for finding the time scheduling function, and show that the gap dependence can be quadratically improved to be inverse linear for a wide range of systems under a mild gap measure condition. Through variational analysis, we further demonstrate the optimality of our schedule for systems with linear gap and the partial optimality for general systems, while we also rigorously show that the commonly used linear schedule is never optimal.

• The paper presents an adaptive scheduling approach that achieves quadratic improvement by reducing runtime scaling from O(1/Δ²) to O(1/Δ).
• It leverages power-law schedules that slow evolution near small gaps and accelerate in large-gap regions, thereby minimizing error accumulation.
• The analysis uses a variational framework to show that linear schedules are suboptimal except in constant-gap scenarios, influencing quantum algorithm design.

Improved Gap Dependence in Adiabatic State Preparation through Adaptive Scheduling

Introduction

Adiabatic quantum computing (AQC) furnishes a universal paradigm for quantum ground state preparation, with direct implications for quantum chemistry, many-body simulation, and quantum algorithmics. The central bottleneck in AQC is the scaling of runtime with the inverse spectral gap, $\Delta_*$, of the time-dependent Hamiltonian: conventional approaches exhibit quadratic scaling, hindering practical applicability for many relevant problem instances. The work "Improved gap dependence in adiabatic state preparation by adaptive schedule" (2512.10329) systematically analyzes and overcomes this limitation by leveraging a family of adaptive, gap-informed nonlinear scheduling functions, demonstrating a quadratic improvement in gap dependence under physically reasonable conditions.

Background and Problem Formulation

AQC evolves a system according to a time-dependent Hamiltonian interpolating between an easily prepared initial Hamiltonian $H_0$ and a target Hamiltonian $H_1$. The time-evolved state $\ket{\psi_T(s)}$ follows the rescaled Schrödinger equation: $$\frac{1}{T}i\frac{\partial}{\partial s}\ket{\psi_T(s)} = H(u(s))\ket{\psi_T(s)}$$ with $H(u(s)) = (1-u(s))H_0 + u(s)H_1$ and a scheduling function $u: [0,1]\to[0,1]$.

Quantum adiabatic theorems guarantee success in the adiabatic limit ($T\to\infty$), with finite-time error predominantly governed by the minimum gap $\Delta_* = \min_s \Delta(u(s))$. Traditional error bounds—exemplified by Jansen, Ruskai, and Seiler—suggest $T = \mathcal{O}(1/\Delta_*^2)$ suffices for fixed accuracy but precludes efficient scaling for small-gap scenarios (e.g., in Grover's problem, $T = \mathcal{O}(N)$, nullifying quadratic speedup).

Adaptive Scheduling and Gap Measure Condition

The authors propose adaptive scheduling governed by the ODE: $u'(s) \propto \Delta^p(u(s)), \quad p\in(1,2)$ This "power-law schedule" accelerates the evolution in large-gap zones and slows in small-gap regions, mitigating error accumulation near gap minima.

A crucial component is the spectral gap measure condition, constraining the Lebesgue measure of small-gap regions: $$\mu\left(\{s\colon \Delta(s)\leq x\}\right)=\mathcal{O}(x)$$ This prevents extensive small-gap intervals, holding for important classes of Hamiltonians, including quantum search and linear system solvers.

Main Technical Results

Quadratic Improvement of Gap Scaling

Under the measure condition and power-law scheduling, the adiabatic error scales as: $\eta(1)\leq \mathcal{O}(T^{-1}\Delta_*^{-1})$ This signifies a quadratic improvement over linear schedules ($T^{-1}\Delta_*^{-2}$). It suffices to choose $T=\mathcal{O}(1/\Delta_*)$ to maintain constant accuracy—restoring, for example, the quadratic complexity of Grover's adiabatic algorithm.

The derivation applies a careful integral analysis exploiting the power-law schedule; the normalization constant $c_p$ is upper-bounded using the gap measure condition, and the terms controlling the adiabatic error are shown to scale no worse than $\Delta_*^{-1}$.

Optimality Analysis via Variational Methods

The work establishes, through functional analytic methods, that schedules solving $u'(s) \propto \Delta^{3/2}(u(s))$ are (partially) optimal, minimizing the key first-order terms dominating the adiabatic error functional for linearly varying or piecewise linear gaps:

• For linear gap: the power-law schedule with $p=3/2$ globally minimizes the upper bound on adiabatic error derived from [Jansen et al].
• For general gaps: the same schedule minimizes the leading cubic-inverse-gap term, which is typically dominant in practical settings.

Crucially, the authors rigorously prove that linear schedules (i.e., $u(s)=s$) are never optimal except for constant gaps: the Euler-Lagrange equation derived from the error upper bound is not satisfied for nontrivial gap functions. Hence, deviation from linear interpolation is always beneficial under spectral variation.

Theoretical and Practical Implications

Theoretical Consequences

• Optimal Scheduling Theory: The variational framework supplies necessary conditions for schedule optimality, and the results extend to piecewise linear gap landscapes prevalent in quantum optimization.
• Error Control: The reduction in inverse-gap scaling modifies the landscape of feasible quantum algorithms for structured and unstructured search, quantum linear system solving, and quantum chemistry ground state problems.
• Rigorous Performance Guarantees: The analysis is constructive—bounding both adiabatic runtime and necessary schedule smoothness, and connecting to existing gap-uninformed and informed methods.

Practical Considerations

• Gap Knowledge Requirement: The construction presumes access to a lower bound on the gap profile. While generally QMA-hard, several physically important Hamiltonian families possess analytic or efficiently computable lower bounds enabling practical scheduling.
• Numerical Instantiation: In settings with approximate gap knowledge, the ODE for $u(s)$ can be solved numerically to high precision with cost logarithmic in the system size.
• Breadth of Applicability: The measure condition encompasses key classes (Grover, adiabatic linear system solvers), and the paper discusses how to construct piecewise linear lower bounds for more general smooth gap functions, ensuring broad relevance.

Relation to Prior Work

Power-law schedules had appeared previously in specific contexts (e.g., $p=2$ in adiabatic search, $p\in[1,2]$ in unstructured optimization), but a general, rigorous statement of quadratic gap improvement for wide Hamiltonian classes was lacking. This work offers such a generalization and clarifies the optimality structure, in contrast to both heuristic-based and gap-uninformed scheduling methods (see, e.g., (Wan et al., 2020, Shingu et al., 21 Jan 2025), [q-2025-07-11-1790]).

Conclusions

This work provides a mathematically rigorous foundation for adaptive scheduling in adiabatic state preparation, reducing the scaling of required runtime with the inverse spectral gap from quadratic to linear in broad classes of systems. The variational analysis establishes the strong optimality properties of power-law schedules, and a critical negative result proves the strict sub-optimality of linear schedules except in trivial cases. These advances have significant consequences for quantum algorithm design, providing constructive strategies to avoid excessive complexity due to small spectral gaps. Extending this framework to discrete quantum circuits, multi-schedule Hamiltonians, or systems with boundary cancellation remains open and is a promising future direction.

Reference: "Improved gap dependence in adiabatic state preparation by adaptive schedule" (2512.10329)