发表机构
The University of Tokyo; OptQC Corp.; University of Copenhagen; University of Amsterdam（东京大学; OptQC公司; 哥本哈根大学; 阿姆斯特丹大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出基于随机纯化和稀释的协变量子态与信道集体层析成像协议，实现最优拷贝复杂度，并解决Hayashi测量高效电路构造问题，获得门高效学习器。
AI 中文摘要
我们给出了具有已知对称性的量子态和信道的集体层析成像协议，利用随机纯化和稀释将学习简化为纯态估计。对于与紧致群表示对易且具有重数 $m_\lambda$ 的量子态，在迹距离误差 $\varepsilon$ 足够小且失败概率为 $\eta$ 的非平凡情况 $\sum_\lambda m_\lambda^2>1$ 下，最优拷贝复杂度为 $\Theta((\sum_\lambda m_\lambda^2+\log\eta^{-1})/\varepsilon^2)$。对于具有紧致群 $G$ 的有限维酉表示的 $G$-协变信道，并行 $\widetilde{O}(D_G/\varepsilon^2)$ 次查询在固定成功概率 $2/3$ 下达到金刚石距离误差 $\varepsilon$，其中 $D_G$ 统计在施加迹保持约束前协变 Choi 算子的实参数个数。对于固定局域维度 $d$ 的 $k$ 个 qudit 上的置换协变信道，在足够小的固定误差和固定成功概率 $2/3$ 下，我们获得了金刚石距离的最优查询标度 $\Theta_d(k^{d^4-1})$ 和 Choi 迹距离的最优查询标度 $\Theta_d(k^{d^4-d^2})$。此外，我们解决了量子层析成像中的一个开放问题，即构造逼近最优纯态估计的 Hayashi 测量的高效量子电路。我们将对称子空间中的态编码为玻色子占据模式，通过外差探测和归一化实现测量，并利用量子 Hermite 变换在量子比特上近似此过程。结合我们与对称性兼容的纯化和稀释电路，这产生了查询最优且门高效的置换协变态和信道学习器，其门复杂度为 $O_d(\mathrm{poly}(k,\varepsilon^{-1},\log\eta^{-1}))$。
英文摘要
We give collective tomography protocols for quantum states and channels with known symmetries, using random purification and dilation to reduce learning to pure-state estimation. For states commuting with a compact-group representation with multiplicities $m_λ$, the optimal copy complexity is $Θ((\sum_λm_λ^2+\logη^{-1})/\varepsilon^2)$ for sufficiently small trace-distance error $\varepsilon$ and failure probability $η$ for the nontrivial case $\sum_λm_λ^2>1$. For $G$-covariant channels with finite-dimensional unitary representations of a compact group $G$, parallel $\widetilde{O}(D_G/\varepsilon^2)$ queries achieve a diamond-distance error $\varepsilon $ at fixed success probability $2/3$, where $D_G$ counts the real parameters of covariant Choi operators before imposing trace preservation. For permutation-covariant channels on $k$ qudits of fixed local dimension $d$, at sufficiently small fixed error and fixed success probability $2/3$, we obtain the optimal query scalings $Θ_d(k^{d^4-1})$ for diamond distance and $Θ_d(k^{d^4-d^2})$ for Choi trace distance. Furthermore, we resolve an open problem in quantum tomography of constructing efficient quantum circuits that approximate the Hayashi measurement for optimal pure-state estimation. We encode states in the symmetric subspace into bosonic occupation modes, realize the measurement by heterodyne detection and normalization, and approximate this procedure on qubits using the quantum Hermite transform. Combined with our symmetry-compatible purification and dilation circuits, this yields query-optimal and gate-efficient learners of permutation-covariant states and channels with gate complexity $O_d(\mathrm{poly}(k,\varepsilon^{-1},\logη^{-1}))$.
Comments95 pages, 6 figures
