发表机构
IMPACT AI, Office of Data Science and Informatics (ODSI)/NASA MSFC; The University of Alabama in Huntsville（IMPACT AI，数据科学与信息办公室/美国国家航空航天局马歇尔太空飞行中心; 阿拉巴马大学亨茨维尔分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究集体相位门 $e^{-i\gamma K^2}$ 的LCU最小采样开销，发现其由 $\gamma/\pi$ 的连分数决定，理性角度下开销为 $q$，并给出主公式及指数增长的反例，同时证明相关分解的最优性。
AI 中文摘要
我们考虑实现二次集体相位 $e^{-i\gamma K^2}$ 的成本，其中 $K$ 是一个具有均匀间隔谱的集体可观测量，例如置换对称的汉明权重。该门是约束量子优化中的基数惩罚层、自旋压缩中的单轴扭转门以及玻色子模式的克尔相位。它可以通过单量子比特旋转层的线性组合（即LCU）以开销 $\Gamma$ 进行采样，而非使用双量子比特门。我们确定了最小开销。无辅助比特的层采样能以该因子重现目标的结果概率，且对任何输入而言，更小的因子均不可行；当 $K$ 为置换对称时，任意单量子比特门也无法做得更好。最小开销随寄存器规模 $n$ 的增长由 $\gamma/\pi$ 的连分数决定：最简分数形式的理性角度 $\pi p/q$ 对于所有 $n\ge2q-2$ 恰好花费 $\Gamma=q$；当且仅当 $\gamma/\pi$ 是坏逼近数时，开销为 $\Theta(n)$；一个主公式（在低分母有理数附近被证明是双侧紧的）可从最近的有理数读出开销。在没有对称性的情况下，单量子比特门上的开销可能呈指数增长：对于二进制加权的 $K=\sum_j2^jx_j$，在 $m$ 个量子比特上进行傅里叶算术的平方相位至少花费 $2^{0.557m-O(1)}$，这超出了每个切割界限的 $2^{m/2}$ 上限，这是通过一种新的证书提升论证实现的。Carrera Vazquez、Egger 和 Woerner（2026）提出的基于傅里叶的LCU（保证 $\Gamma\le n+1$）在固定 $n$ 的典型角度下是最优的，最多相差一个常数因子；在理性角度下，唯一的最优解是克尔光学中的分数复兴分解，该方案与 $n$ 无关。在玻色子模拟中，Upreti、Quesada 和 Chabaud（2026）提出的克尔门移相器分解是移相器上的唯一最优解，且仅当 $\gamma/\pi$ 为有理数时才存在有限成本解。
英文摘要
The minimal sampling overhead of the quadratic collective phase $e^{-iγK^2}$ over one layer of single-qubit rotations is decided by the continued fraction of $γ/π$. Here $K$ is a collective observable with spectrum $\{0,\dots,n\}$, such as the permutation-symmetric Hamming weight. This gate is the cardinality-penalty layer of constrained optimization, the one-axis-twisting gate of spin squeezing and the Kerr phase of a bosonic mode. Instead of compiling it to two-qubit gates, we sample it as an LCU over such layers at overhead $Γ$. Our results give an optimality theory for this overhead. Ancilla-free sampling reproduces the target's outcome probabilities up to the minimal $Γ$, and no smaller factor works for every input; for permutation-symmetric $K$, independent per-qubit angles or one layer of arbitrary single-qubit gates give no further reduction. Rational angles $πp/q$ in lowest terms cost exactly $q$ once $n\ge2q-2$, attained uniquely by a uniform $q$-point sampler; the cost is $Θ(n)$ if and only if $γ/π$ is badly approximable, and bounded in $n$ if and only if it is rational. A second result shows what the symmetry buys: without it, the squaring phase of Fourier arithmetic on $m$ qubits costs at least $2^{0.557m-O(1)}$ over single-qubit gates, beyond the $2^{m/2}$ of every cut bound, by a new certificate-lifting argument. For constructions known only through upper bounds, it gives the optimum: the Fourier-based LCU of Carrera Vazquez, Egger and Woerner is, at fixed $n$, optimal up to a constant at typical angles and beaten by $(n+1)/q$ at rational ones with an $n$-independent sampler; the Kerr decomposition of Upreti, Quesada and Chabaud is the unique optimum over phase shifters at rational parameters without photon-number cutoff; and it identifies the rational couplings where one-axis twisting admits optimal cat decompositions.
Commentsv2: concise rewrite, 28 pages, 2 figures
