The efficiency of adiabatic quantum evolution is governed by the evolution time T , which typically scales as O ( Δ − 2 ) with the minimum energy gap Δ . However, the rigorous lower bound is O ( L Δ − 1 ) , where L is the adiabatic path length. Although L is formally upper bounded by O ( Δ − 1 ) , such a bound is often too loose in practice, and L can be bounded independently of Δ . This indicates the potential for a quadratic speedup through adiabatic schedule construction. Here, we introduce the constant geometric speed (CGS) schedule, which traverses the adiabatic path at a uniform rate. We show that this approach reduces the scaling of the evolution time by a factor of Δ − 1 , provided L remains bounded independently of Δ . We propose a segmented CGS protocol where path segment lengths are computed from eigenstate overlaps on the fly, reducing the prior spectral-knowledge requirement from the full gap function Δ ( s ) to just a global lower bound on the energy gap. Numerical tests on adiabatic unstructured search, N 2 , and a [2Fe-2S] cluster demonstrate the optimal Δ − 1 scaling, confirming a quadratic speedup over the standard linear schedule.
The efficiency of adiabatic quantum evolution is governed by the evolution time T , which typically scales as O ( Δ − 2 ) with the minimum energy gap Δ . However, the rigorous lower bound is O ( L Δ − 1 ) , where L is the adiabatic path length. Although L is formally upper bounded by O ( Δ − 1 ) , such a bound is often too loose in practice, and L can be bounded independently of Δ . This indicates the potential for a quadratic speedup through adiabatic schedule construction. Here, we introduce the constant geometric speed (CGS) schedule, which traverses the adiabatic path at a uniform rate. We show that this approach reduces the scaling of the evolution time by a factor of Δ − 1 , provided L remains bounded independently of Δ . We propose a segmented CGS protocol where path segment lengths are computed from eigenstate overlaps on the fly, reducing the prior spectral-knowledge requirement from the full gap function Δ ( s ) to just a global lower bound on the energy gap. Numerical tests on adiabatic unstructured search, N 2 , and a [2Fe-2S] cluster demonstrate the optimal Δ − 1 scaling, confirming a quadratic speedup over the standard linear schedule.
In an adiabatic evolution, the wave function follows the path of ground states ∣Φ0(s)⟩ of a Hamiltonian H(s) as s is tuned from 0 to 1. The geometry of this adiabatic path is described by quantities such as its length L=∫01∥∂s∣Φ0(s)⟩∥ds and its total curvature K. This geometry affects the evolution time T. For instance, T is at least of order L/Δ∗, where Δ∗ is the minimum energy gap over s∈[0,1]. The traditional approach gives L=O(Δ∗−1/2) and K=O(Δ∗−1), which are loose in many cases. We improve these bounds by using μ<(γ), the width of the region in s on which the gap is below γ. The gap profile is in width class p when μ<(γ) falls at least as fast as γ1/p. A typical avoided crossing is in width class p=1, which gives L=O(logΔ∗−1) and K=O(logΔ∗−1). A profile in width class p>1 gives power laws with exponents (p−1)/(2p) for L and (p−1)/p for K. The width class also bounds the evolution time. At p=1 we have T=O(Δ∗−2) with a schedule that advances s at a constant rate, and T=O(Δ∗−1) up to a polylogarithmic correction with a schedule that traverses the path at a constant geometric speed. The bounds on L hold for any twice continuously differentiable H(s), and the bounds on K for affine H(s). We also prove the tightness of scaling of L in Δ∗ for every integer p≥1, and the scaling of K at p=1. We demonstrate the bounds on the adiabatic Grover search, the XXZ spin chain, and molecular electronic Hamiltonians.
We develop a discrete adiabatic framework for Gibbs-state preparation in classical and quantum optimisation algorithms. For finite time-inhomogeneous reversible Markov chains, we derive a tracking theorem that bounds the distance between the evolving distribution and the instantaneous stationary state. The bound combines contraction of accumulated error with the new error generated as the stationary distribution changes. For Gibbs paths, it yields explicit step-complexity bounds and an effective tracking metric determined by the spectral gap and equilibrium energy fluctuations. The associated constant-speed schedule minimizes the metric-energy functional and, in the small-gap regime, takes the form β˙∝δ(β)/V(β), where V(β)=Varπβ(E). We then apply the discrete quantum adiabatic theorem to temperature-dependent Szegedy quantum walks, whose eigenphase gap satisfies Δw=Θ(δ+), with δ+ the underlying Markov kernel's right spectral gap. Under uniform bounds on the discrete operator variation and suitable spectral and path-integral assumptions, gap-adapted schedules give a sufficient fixed-accuracy cost of O(δ+,min−1/2). This gives a quadratic improvement in gap dependence relative to the classical inverse-gap relaxation scale under the stated assumptions. Numerical tests on a solvable two-state model and Sherrington-Kirkpatrick instances compare tracking and final preparation errors and investigate the transfer of schedules from smaller to larger systems.
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.
We present an optimized adiabatic quantum schedule for unstructured search building on the original approach of Roland and Cerf [Phys. Rev. A 65, 042308 (2002)]. Our schedule adiabatically varies the Hamiltonian even more rapidly at the endpoints of its evolution, preserving Grover's well-known quadratic quantum speedup. In the errorless adiabatic limit, the probability of successfully obtaining the marked state from a measurement increases directly proportional to time, suggesting efficient parallelization. Numerical simulations of an appropriate reduced two-dimensional Schrödinger system confirm adiabaticity while demonstrating superior performance in terms of probability compared to existing adiabatic algorithms and Grover's algorithm, benefiting applications with possible premature termination. We introduce a protocol that ensures a marked-state probability at least p in time of order N(1+p/ε), and analyze its implications for realistic bounded-resource scenarios. Our findings suggest that quantum advantage may still be achievable under constrained coherence times (where other algorithms fail), provided the hardware allows for them to be sufficiently long.
We analyze state preparation within a restricted space of local control parameters between adiabatically connected states of control Hamiltonians. We formulate a conjecture that the time integral of energy fluctuations over the protocol duration is bounded from below by the geodesic length set by the quantum geometric tensor. The conjecture implies a geometric lower bound for the quantum speed limit (QSL). We prove the conjecture for arbitrary sufficiently slow protocols using adiabatic perturbation theory, and show that the bound is saturated by geodesic protocols, which keep the energy variance constant along the trajectory. Our conjecture implies that any optimal unit-fidelity protocol, even those which drive the system far from equilibrium, are fundamentally constrained by the quantum geometry of adiabatic evolution. When the control space includes all possible couplings, spanning the full Hilbert space, we recover the well-known Mandelstam-Tamm bound. However, using only accessible local controls to anneal in complex models such as glasses, or target individual excited states in quantum chaotic systems, the geometric bound for the QSL can be exponentially large in the system size due to a diverging geodesic length. We validate our conjecture both analytically by constructing counter-diabatic and fast-forward protocols for a three-level system, and numerically in non-integrable spin chains using optimal control.
We analyze state preparation within a restricted space of local control parameters between adiabatically connected states of control Hamiltonians.We formulate a conjecture that the time integral of energy fluctuations over the protocol duration is bounded from below by the geodesic length set by the quantum geometric tensor.The conjecture implies a geometric lower bound for the quantum speed limit (QSL).We prove the conjecture for arbitrary, sufficiently slow protocols using adiabatic perturbation theory and show that the bound is saturated by geodesic protocols, which keep the energy variance constant along the trajectory.Our conjecture implies that any optimal unit-fidelity protocol, even those that drive the system far from equilibrium, are fundamentally constrained by the quantum geometry of adiabatic evolution.When the control space includes all possible couplings, spanning the full Hilbert space, we recover the well-known Mandelstam-Tamm bound.However, using only accessible local controls to anneal in complex models such as glasses or to target individual excited states in quantum chaotic systems, the geometric bound for the quantum speed limit can be exponentially large in the system size due to a diverging geodesic length.We validate our conjecture both analytically by constructing counter-diabatic and fast-forward protocols for a three-level system, and numerically in nonintegrable spin chains and a nonlocal SYK model.