Rigorous elementary bounds for the quantum adiabatic theorem — from the Ambainis–Regev elementary proof (2004) to Duan's continuous integral-representation proof (2020) — guarantee ϵ\epsilon-adiabaticity of plain Schrödinger evolution only for runtimes whose worst-case dependence on the minimal spectral gap λ\lambda is at least λ−3\lambda^{-3}, e.g. T=O(ϵ−1λ−2max⁡{∥H′∥,∥H′′∥,∥H′∥2/λ})T=O(\epsilon^{-1}\lambda^{-2}\max\{\|H'\|,\|H''\|,\|H'\|^{2}/\lambda\}), the last term degrading to ϵ−1∥H′∥2/λ3\epsilon^{-1}\|H'\|^{2}/\lambda^{3}. Inverse-square gap scaling is widely regarded as optimal and is corroborated by exactly solved two-level sweep problems and by randomized eigenpath-traversal algorithms, but the only rigorous λ−2\lambda^{-2}-type theorem for plain evolution (Elgart–Hagedorn 2012) assumes Gevrey-class smoothness and uses involved superadiabatic machinery. This record states the problem left open by Duan: prove that for twice-continuously-differentiable Hamiltonian families a runtime with clean total gap dependence λ−2\lambda^{-2}, all other factors λ\lambda-independent, suffices for ϵ\epsilon-adiabaticity — ideally by an elementary argument — or refute it by an explicit counterexample family that provably requires strictly faster-than-λ−2\lambda^{-2} runtimes.

Background

The quantum adiabatic theorem describes what happens when the Hamiltonian (the Hermitian energy operator) governing a quantum system is changed slowly in time. Write the time dependence as H(s)H(s) with normalized time s=t/T∈[0,1]s=t/T\in[0,1], where TT is the total evolution time, and assume HH is twice continuously differentiable in ss. Let φ(s)\varphi(s) be a normalized eigenvector of H(s)H(s) whose eigenvalue is separated from the rest of the spectrum at time ss by a spectral gap λ(s)\lambda(s) (the energy distance to the nearest other eigenvalue), and let λ=min⁡0≤s≤1λ(s)>0\lambda=\min_{0\le s\le 1}\lambda(s)>0. If the system starts in φ(0)\varphi(0) and evolves under the Schrödinger equation i∂tψ=H(t/T)ψi\partial_t\psi=H(t/T)\psi (units with ℏ=1\hbar=1), the theorem asserts that for TT large the final state ψ(1)\psi(1) is close to φ(1)\varphi(1). A quantitative version asks how large TT must be in terms of the error tolerance ϵ\epsilon, the gap λ\lambda, and the derivative norms ∥H′∥=max⁡s∥dH/ds∥\|H'\|=\max_s\|dH/ds\| and ∥H′′∥\|H''\|, where ∥⋅∥\|\cdot\| is the operator norm (largest singular value; for Hermitian operators the largest spectral magnitude). The question matters far beyond foundations: adiabatic quantum computation, a model proposed in 2000 and equivalent in computational power to the standard circuit model, prepares target states precisely by slow Hamiltonian deformation, and the minimum gap of interesting instances typically shrinks (often exponentially) with problem size, so each extra power of λ−1\lambda^{-1} in a rigorous runtime bound can dominate the algorithm's claimed complexity.

Elementary proofs of the theorem give bounds of the following shape. Ambainis and Regev (2004) discretized the evolution into small unitary steps and obtained an elementary proof with a sufficient time of order ϵ−2λ−3\epsilon^{-2}\lambda^{-3} (in the derivative norms). Duan (2020) gave a shorter 'continuous' proof via an exact integral representation of the state error, reaching T=O(ϵ−1λ−2max⁡{∥H′∥,∥H′′∥,∥H′∥2/λ})T=O\big(\epsilon^{-1}\lambda^{-2}\max\{\|H'\|,\|H''\|,\|H'\|^{2}/\lambda\}\big) with explicit constants. Because of the ∥H′∥2/λ\|H'\|^{2}/\lambda entry, this still degrades to ϵ−1∥H′∥2/λ3\epsilon^{-1}\|H'\|^{2}/\lambda^{3} in the worst case — the same worst-case cubic gap dependence already present in the bounds of Jansen, Ruskai, and Seiler (2007), derived by resolvent methods (the resolvent of an operator HH being the shifted inverse (z−H)−1(z-H)^{-1} for complex zz away from the spectrum), who made the gap dependence explicit, noted the 'general wisdom' that T∼λ−2T\sim\lambda^{-2} should suffice, and showed that their own method (iterated integration by parts) cannot prove the clean λ−2\lambda^{-2} criterion they conjectured. Heuristically λ−2\lambda^{-2} is the right scaling: the leading error term behaves like ∥H′∥/(Tλ2)\|H'\|/(T\lambda^{2}), exactly solved two-level sweep models show inverse-square gap dependence, and randomized eigenpath-traversal algorithms (Boixo–Knill–Somma 2009; Chiang–Xu–Somma 2014), which replace plain evolution by discretized random-time evolutions, can be arranged with cost scaling as 1/λ21/\lambda^{2}. The only rigorous λ−2\lambda^{-2}-type result for plain evolution, due to Elgart and Hagedorn (2012), requires the Hamiltonian to be infinitely differentiable with Gevrey-class derivative bounds (all derivatives exist and grow at most like CRk(k!)αC R^{k}(k!)^{\alpha} for some constants) and yields TT of order λ−2∣ln⁡λ∣6α\lambda^{-2}|\ln\lambda|^{6\alpha} through involved superadiabatic (near-diagonal basis-change) machinery; as the 2018 review of Albash and Lidar summarizes, all other rigorous versions of the theorem known to date 'have a worse gap dependence (cubic or higher)'. Thus a gap remains between the conjectured optimal λ−2\lambda^{-2} scaling and what rigorous bounds — in particular elementary ones valid under mere twice-continuous differentiability — actually guarantee, and Duan (2020) closes his paper by asking whether the residual λ\lambda-dependence can be removed by a straightforward approach.

Problem Statement

In the setting of the elementary adiabatic-theorem proofs — a twice-continuously-differentiable family H:[0,1]→H:[0,1]\to bounded Hermitian operators on a Hilbert space, with a followed non-degenerate eigenvalue separated from the rest of the spectrum by a minimum gap λ>0\lambda>0 and with ∥H′∥,∥H′′∥\|H'\|,\|H''\| finite — determine whether the sufficient evolution time for ϵ\epsilon-adiabaticity can be given a clean inverse-square gap dependence. Specifically: prove that there exist an explicit constant CC and an explicit polynomial, both independent of λ\lambda and ϵ\epsilon, such that every evolution duration

T  ≥  C poly(∥H′∥,∥H′′∥)ϵ λ2T \;\ge\; C\,\frac{\mathrm{poly}(\|H'\|,\|H''\|)}{\epsilon\,\lambda^{2}}

guarantees ∥ψ(1)−φ(1)∥≤ϵ\|\psi(1)-\varphi(1)\|\le\epsilon for the plain Schrödinger evolution i∂tψ=H(t/T)ψi\partial_t\psi=H(t/T)\psi started at φ(0)\varphi(0), where the followed eigenvector is fixed in the standard gauge ⟨φ′(s)∣φ(s)⟩=0\langle\varphi'(s)|\varphi(s)\rangle=0 and ψ(1)\psi(1) is compared with φ(1)\varphi(1) after the customary removal of the dynamical phase e−iT∫01γ(s) dse^{-iT\int_0^1\gamma(s)\,ds} of the followed eigenvalue γ(s)\gamma(s) (equivalently, after the source's reduction of the general case to eigenvalue zero via H(s)↦H(s)−γ(s)1H(s)\mapsto H(s)-\gamma(s)\mathbb{1}), so that the error measure is closeness modulo phase; equivalently, remove the spurious ∥H′∥2/λ3\|H'\|^{2}/\lambda^{3} worst-case term afflicting the Ambainis–Regev, Jansen–Ruskai–Seiler, and Duan bounds, so that λ−2\lambda^{-2} is the only gap dependence anywhere in the sufficient condition; or refute this by exhibiting an explicit twice-continuously-differentiable Hamiltonian family with λ→0\lambda\to0 and bounded derivative norms for which achieving fixed error ϵ\epsilon, in the same modulo-phase sense, provably requires TT growing strictly faster than λ−2\lambda^{-2}. A resolution valid under mere twice-continuous differentiability (no Gevrey-class or analyticity assumptions, no boundary-cancellation schedule conditions, no assumptions on the shape of the gap function) is the mathematical core; a resolution by elementary means — avoiding resolvent calculus and superadiabatic machinery, with explicit constants — is the desideratum stated in the source, not a requirement of the claim.

References

  1. Runyao Duan, "Quantum Adiabatic Theorem Revisited", arXiv preprint (2020), arXiv:2003.03063, DOI: 10.48550/arXiv.2003.03063, https://arxiv.org/abs/2003.03063
  2. Andris Ambainis and Oded Regev, "An Elementary Proof of the Quantum Adiabatic Theorem", arXiv preprint (2004), arXiv:quant-ph/0411152, https://arxiv.org/abs/quant-ph/0411152
  3. Sabine Jansen, Mary-Beth Ruskai, and Ruedi Seiler, "Bounds for the adiabatic approximation with applications to quantum computation", Journal of Mathematical Physics 48, 102111 (2007), DOI: 10.1063/1.2798382, arXiv:quant-ph/0603175, https://arxiv.org/abs/quant-ph/0603175
  4. Alexander Elgart and George A. Hagedorn, "A note on the switching adiabatic theorem", Journal of Mathematical Physics 53, 102202 (2012), DOI: 10.1063/1.4748968, arXiv:1204.2318, https://arxiv.org/abs/1204.2318
  5. Tameem Albash and Daniel A. Lidar, "Adiabatic quantum computation", Reviews of Modern Physics 90, 015002 (2018), DOI: 10.1103/RevModPhys.90.015002, arXiv:1611.04471, https://arxiv.org/abs/1611.04471
  6. Sergio Boixo, Emanuel Knill, and Rolando Somma, "Eigenpath traversal by phase randomization", Quantum Information and Computation 9(9-10), 0833-0855 (2009), DOI: 10.26421/QIC9.9-10-7
  7. Hao-Tien Chiang, Guanglei Xu, and Rolando D. Somma, "Improved Bounds for Eigenpath Traversal", Physical Review A 89, 012314 (2014), DOI: 10.1103/PhysRevA.89.012314, arXiv:1311.7073, https://arxiv.org/abs/1311.7073
  8. Donny Cheung, Peter Hoyer, and Nathan Wiebe, "Improved Error Bounds for the Adiabatic Approximation", Journal of Physics A: Mathematical and Theoretical 44, 415302 (2011), DOI: 10.1088/1751-8113/44/41/415302, arXiv:1103.4174, https://arxiv.org/abs/1103.4174
  9. Xi Guo and Dong An, "Improved gap dependence in adiabatic state preparation by adaptive schedule", arXiv preprint (2025), arXiv:2512.10329, DOI: 10.48550/arXiv.2512.10329, https://arxiv.org/abs/2512.10329
  10. Zhenwei Cao and Alexander Elgart, "On the efficiency of Hamiltonian-based quantum computation for low-rank matrices", Journal of Mathematical Physics 53, 032201 (2012), DOI: 10.1063/1.3690045, arXiv:1004.4911, https://arxiv.org/abs/1004.4911