发表机构
George Mason University（乔治梅森大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一种离散化连续酉演化的方法，建立双向对应关系，将离散绝热定理归约为连续形式，并证明乘积逼近误差为O(1/T)，且无需离散绝热定理即可达到相同缩放。
AI 中文摘要
我提出了一种对连续酉演化进行离散化的简短方法。该方法在有限维中建立了连续酉演化与单位算符乘积之间的双向对应关系，并利用它将对易绝热定理归约为连续绝热定理。在精细步长下，在合适的有限划分上持有受调控哈密顿量的截断样本，可得到一个近似乘积。相反地，对时钟进行舍入，可将$T$个单位步长样本的乘积识别为连续演化，并带有两个半步相位修正。通过在每个时间步内的抵消，我将其与原始演化的算子范数距离界定为$O(1/T)$。这假设了有界的一阶和二阶导数、有界的谱宽度以及特征值差远离非零的$2\pi$倍数，且这些界对$T$一致成立。对于具有这些导数界的酉游走，一个固定的公共自由谱弧提供了满足比较假设的厄米对数路径。连续模拟保证随后转移到相位修正乘积，连续绝热界转移到乘积本身，并带有显式的加性$O(1/T)$误差，且保留每个绝热定理的假设。对于条件数为$\kappa$的正定量子线性系统，该比较和连续绝热理论给出了一个重标度的量子化游走，泄漏$\delta$所需的步数为$O(\kappa/\delta)$，与先前通过离散绝热定理获得的缩放一致。这些乘积不需要离散绝热定理。
英文摘要
I present a short approach to discretizing continuous unitary evolution. It gives a correspondence in both directions between continuous unitary evolutions and products of unitaries in finite dimensions, and I use it to reduce discrete adiabatic theorems to continuous ones. At fine steps, holding truncated samples of a regulated Hamiltonian on a suitable finite partition gives an approximating product. Conversely, rounding the clock identifies a product of $T$ unit-step samples with a continuous evolution, up to two half-step phases. I bound its operator-norm distance from the original evolution by $O(1/T)$ through cancellation within each time step. This assumes bounded first and second derivatives, bounded spectral width, and eigenvalue differences bounded away from nonzero multiples of $2π$, uniformly in $T$. For a unitary walk with these derivative bounds, a fixed common free spectral arc supplies a Hermitian logarithm path satisfying the comparison hypotheses. Continuous simulation guarantees then transfer to the phase-corrected product, and continuous adiabatic bounds to the product itself, with an explicit additive $O(1/T)$ error and each adiabatic theorem's hypotheses retained. For positive definite quantum linear systems of condition number $κ$, this comparison and continuous adiabatic theory give a rescaled qubitized walk with $O(κ/δ)$ steps for leakage $δ$, matching the scaling previously obtained through a discrete adiabatic theorem. These products need no discrete adiabatic theorem.
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