通过李代数泄漏界实现有限维玻色子采样的近最优模式缩放
Near-Optimal Mode Scaling for Finite-Dimensional Boson Sampling via Lie-Algebraic Leakage Bounds
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中文总结 AI 辅助
研究有限维玻色子采样的近最优模式缩放问题，通过李代数框架和戴森级数分析界定聚束泄漏，证明谱范数集中在\(\tilde{O}(\sqrt{n})\)，将所需模式数收紧，适用于多种架构，量化空间资源需求。
中文摘要 AI 辅助
玻色子采样通过不可区分粒子的干涉展示量子优势，其输出概率由矩阵积和决定。在基于物质的确定性平台上实现它需要在有限维局部希尔伯特空间中编码玻色模式，这引入了线性光学中不存在的泄漏通道：超过局部截断\(d\)的多粒子聚束。我们为紧致李群不可约表示上的非相互作用采样开发了一个统一框架，其中跃迁振幅是单粒子跃迁矩阵子矩阵的不变量，在玻色子情况下恢复积和。在此框架内，我们通过戴森级数分析来界定聚束泄漏：将相关的多体泄漏算子分解为独立随机矩阵并应用非交换集中不等式，我们证明，在跃迁矩阵的高斯模型中，其谱范数集中在\(\tilde{O}(\sqrt{n})\)，而不是先前基于自旋模拟的\(O(n)\)最坏情况；向物理哈尔系综的过渡简化为单个子矩阵比较输入，并在前导阶得到验证。跨越局部维度\(d = 2\) - \(5\)的精确数值表明该界是紧的，哈尔系综范数与封闭形式\(\sqrt{d(n - d + 1)}\)匹配到亚百分比精度。这将所需模式数从\(m=\Omega(n^4)\)收紧到近最优的\(m=\tilde{\Omega}(n^{1 + 2/(d - 1)})\)；对于自旋 - 1 表示（\(d = 3\)），开销降至\(m=\tilde{\Omega}(n^2)\)，与无碰撞阈值匹配。结果与粒子统计无关，并适用于有限维李对称架构，量化了保持采样难度所需的空间资源。
英文摘要
Boson sampling demonstrates quantum advantage through the interference of indistinguishable particles, with output probabilities governed by matrix permanents. Realizing it on deterministic, matter-based platforms requires encoding the bosonic modes in finite-dimensional local Hilbert spaces, which introduces a leakage channel absent in linear optics: multi-particle bunching beyond the local truncation $d$. We develop a unified framework for non-interacting sampling on the irreducible representations of compact Lie groups, in which the transition amplitude is the immanant of a submatrix of the single-particle transition matrix, recovering the permanent in the bosonic case. Within this framework we bound the bunching leakage through a Dyson-series analysis: decomposing the correlated many-body leakage operator into independent random matrices and applying non-commutative concentration inequalities, we prove, in a Gaussian model of the transition matrix, that its spectral norm concentrates at $\tilde{O}(\sqrt{n})$ rather than the $O(n)$ worst-case of prior spin-based emulations; the passage to the physical Haar ensemble is reduced to a single submatrix-comparison input, verified at leading order. Exact numerics across local dimensions $d=2$--$5$ indicate that the bound is tight, the Haar-ensemble norm matching the closed form $\sqrt{d(n-d+1)}$ to sub-percent accuracy. This tightens the required mode number from $m=Ω(n^4)$ to the near-optimal $m=\tildeΩ(n^{1+2/(d-1)})$; for a spin-1 representation ($d=3$) the overhead falls to $m=\tildeΩ(n^2)$, matching the collision-free threshold. The result is independent of particle statistics and applies across finite-dimensional Lie-symmetric architectures, quantifying the spatial resources needed to preserve sampling hardness.
