通过分离作用位置与作用内容实现结构化哈密顿量的高效块编码
Efficient Block Encoding of Structured Hamiltonians by Separating Where and What
- Alice & Bob
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出通过分离算子作用位置与内容来优化结构化哈密顿量的容错块编码，利用置换网络和桥接CSWAP显著降低T门数量，在海森堡环和安德森模型中分别实现约3倍和1.7倍的T门减少。
AI中文摘要：
容错块编码结构化哈密顿量可以通过在SELECT操作中分离算子作用的位置与所施加的内容，从而降低其非克利福德成本。我们构造了置换-作用-逆置换电路，其CSWAP网络利用了支撑几何结构：选定的支撑被移动到固定的目标寄存器，一个共享的局域电路作用于该寄存器，然后撤销置换。对于固定大小的支撑，SELECT的T门数量随系统规模增长而非随项数增长，且无需平移对称性或因子化系数。对于双位点支撑，我们在位址控制的位点置换网络中确立了最小CSWAP数量并达到了该数量。编译为泡利项后，这些电路使用8SN+O(log N)个T门和O(log N)个置换工作量子比特，其中N是系统量子比特数，S由支撑几何决定，最近邻相互作用时S=1，全连接对时S=3/2。我们还引入了一种桥接CSWAP，它在目标作用期间保留并修复一个临时AND门。当修复为克利福德操作时，它以保留一个工作量子比特为代价，将匹配的正向-逆向CSWAP对的T门数量减半。对于海森堡环，完整的块编码查询在系统规模增大时使T门数量减少约3倍。对于五轨道安德森杂质模型（结合了最近邻和全连接支撑几何），在大浴池尺寸下减少约1.7倍。两种比较均使用所考虑的最低成本编译基线，块编码归一化不变，峰值辅助量子比特数相当。
英文摘要：
In fault-tolerant block encodings of structured Hamiltonians, the support geometry can set how the non-Clifford cost of SELECT scales with the system size, however many operators act on each support. We construct permute-act-unpermute circuits: a CSWAP network fixed by this geometry brings a selected support to a fixed target register, a shared local circuit acts there, and the permutation is undone. The operators add only an $XZ$ mask on this register, at a cost independent of the system size, and one query can mix several support geometries. For two-site supports, we establish and attain the minimum CSWAP count within address-controlled networks of site transpositions. Compiled for Pauli terms, these circuits use $8\mathrm{S}N+o(N)$ $T$ gates with $O(\log N)$ permutation work qubits, where $N$ is the number of system qubits and $\mathrm{S}$ is determined by the support geometry, with $\mathrm{S}=1$ for nearest-neighbour interactions and $\mathrm{S}=3/2$ for all-to-all pairs; in both cases the correction is $O(\log N)$. We also introduce a bridged CSWAP, which remembers the logical value of its temporary AND across the target action and repairs it from the prepared labels, so that the inverse CSWAP undoes the exchange without recomputing the value. When the repair is Clifford, it halves the $T$ count of a matched forward-inverse CSWAP pair at the cost of one retained work qubit. For a Heisenberg ring, the complete block-encoding query reduces the $T$ count by a factor approaching $3$ as the system size grows, relative to flat and product-wise Pauli walks. For a five-orbital Anderson impurity model, which combines nearest-neighbour and all-to-all support geometries, the reduction is about $1.7$ at large bath sizes, relative to the cheapest of three comparators, each built from the methods of the references it follows.
