发表机构
Alfréd Rényi Institute of Mathematics; SKLMS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（阿尔弗雷德·雷尼数学研究所; 中国科学院数学与系统科学研究院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
提出一种通过哈密顿模拟实现低辅助量子比特块编码的简单构造，仅需一个辅助量子比特，电路深度接近最优。
AI 中文摘要
块编码是量子算法中的核心原语，但标准构造通常需要对数级别的辅助量子比特和复杂的受控操作。最近的 lower bound 进一步表明，在广泛的电路模型中，精确构造的这种辅助量子比特开销是不可避免的。我们表明，在近似设置中可以绕过这一障碍。具体来说，我们提出了一种简单的单辅助量子比特构造，通过广义量子信号处理将哈密顿演化转换为底层哈密顿量的块编码。对于由厄米分解 $A=\sum_{j=1}^L \alpha_j H_j$ 给出的算子，我们以两种方式实例化这种块编码构造，它们的不同之处在于所需哈密顿演化的实现方式。使用高阶 Trotterization，我们得到 $A$ 的 $\varepsilon$-近似块编码，仅需一个辅助量子比特和电路深度 $\widetilde O\big(L(\alpha/\varepsilon)^{o(1)}\big)$，其中 $\alpha=\sum_j \alpha_j$。使用多乘积公式，我们得到电路深度 $\widetilde O(L)$，代价是 $O(\log\log(1/\varepsilon))$ 个辅助量子比特。我们的构造提供了标准 LCU 框架的替代方案，重点是在保持（接近）最优电路深度的同时减少辅助量子比特的数量。
英文摘要
Block encodings are a basic interface between quantum algorithms and linear algebra. Standard LCU constructions achieve optimal circuit depth but typically require logarithmically many ancilla qubits. We ask how much quantum workspace can be reduced without sacrificing circuit depth, and study this tradeoff from both algorithmic and lower-bound perspectives. For a Hermitian decomposition $A=\sum_{j=1}^L α_j H_j$, with $\|H_j\|=1$ and $α=\sum_j|α_j|$, we give two coherent $\varepsilon$-approximate block-encoding constructions. The first uses one ancilla qubit and has depth $\widetilde O(L(α/\varepsilon)^{o(1)})$, while the second uses $O(\log\log(α/\varepsilon))$ ancillas and achieves depth $\widetilde O(L)$. For a broad Suzuki-based coherent simulation architecture, we prove an ancilla-depth tradeoff. In the polynomial-resource regime and for a constant number of coherent rounds, $\log(1/\varepsilon)\le O((\log Q)^2+2^a\log Q)$, where $Q$ is depth normalized by the number of Hamiltonian terms and $a$ is the ancilla count. Thus polylogarithmic dependence on $1/\varepsilon$ requires more than constantly many ancillas within this architecture. In a separate repeated-query LCU model, for balanced coefficients $1/L$ and error $\varepsilon=η/L$ with fixed $0<η<1$, we prove $2^a=Ω_η(L^2/(T+L))$, where $T$ is the number of oracle queries. Hence $a=Ω(\log L)$ when $T=O(L^α)$ for some $α<2$. Moreover, in the exact case, $a\ge \log L$ regardless of $T$. We also extend this tradeoff to arbitrary nonnegative coefficients. Finally, we apply our low-ancilla constructions to normalized trace estimation in DQC1, obtaining an optimal algorithm linear in the approximate degree together with a matching query lower bound. Together, these results establish quantitative space-depth and space-query tradeoffs in two natural circuit models.
CommentsSubstantially revised and expanded version, with a new title, two new tradeoff results, and an application to normalized trace estimation in DQC1 model
