费米子和自旋哈密顿量块编码的最优T计数
Optimal T-Count for Block Encodings of Fermionic and Spin Hamiltonians
- North Carolina State University（北卡罗来纳州立大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文确定了费米子和自旋哈密顿量块编码的最优T门计数，提出辅助比特压缩定理，并给出Kitaev蜂窝模型紧的Θ(n+log(1/ε))最坏情况缩放。
AI中文摘要：
我们确定了在酉Clifford+T模型中，当允许任意数量的干净辅助比特和不受限制的块编码子归一化时，构造结构化费米子和自旋哈密顿量的块编码所需的非Clifford T门成本，但不允许电路中途测量或经典前馈。我们的主要技术工具是一个辅助比特压缩定理：任何使用a个干净辅助比特和至多s个T门的n量子比特算子的块编码，都可以压缩为使用至多min{a,n+2s}个辅助比特，而不增加绝对误差或T计数。对于具有有界单体和双体系数的通用二次量子化哈密顿量，在算子范数块编码误差ε下，体积覆盖论证结合电路计数给出了最坏情况下的下界Ω(n²√log(n⁴/ε))，在固定精度下与现有上界匹配。对于n个自旋上的键依赖Kitaev蜂窝族，我们通过稳定子零度和单量子比特态制备分别独立得到下界Ω(n)和Ω(log(1/ε))，这些下界使用不同的哈密顿量实例建立。结合一个显式的LCU构造，它们给出了紧的最坏情况缩放Θ(n+log(1/ε))。作为应用，我们评估了基于量子奇异值变换的哈密顿量模拟电路的T计数，其中每个块编码查询分别编译。当相位合成和控制查询增加至多常数因子开销时，模拟T计数按查询次数乘以每次查询的最优T计数缩放。
英文摘要：
We determine the non-Clifford $T$-gate cost of constructing block encodings of structured fermionic and spin Hamiltonians in a unitary Clifford$+T$ model, when arbitrarily many clean ancillas and unrestricted block-encoding subnormalization are allowed, but without mid-circuit measurements or classical feed-forward. Our main technical tool is an ancilla-compression theorem: any block encoding of an $n$-qubit operator with $a$ clean ancillas and at most $s$ $T$ gates can be compressed to use at most $\min\{a,n+2s\}$ ancillas, without increasing the absolute error or $T$-count. For general second-quantized Hamiltonians with bounded one- and two-body coefficients, at operator-norm block-encoding error $ε$, a volume-covering argument combined with circuit counting gives the worst-case lower bound $Ω(n^2\sqrt{\log(n^4/ε)})$, matching the existing upper bound at fixed precision. For the bond-dependent Kitaev honeycomb family on $n$ spins, we obtain independent lower bounds $Ω(n)$ from stabilizer nullity and $Ω(\log(1/ε))$ from one-qubit state preparation, established using different Hamiltonian instances. Together with an explicit LCU construction, they give the tight worst-case scaling $Θ(n+\log(1/ε))$. As an application, we evaluate the $T$-count of a Hamiltonian simulation circuit based on quantum singular value transformation, with each block-encoding query compiled separately. When phase synthesis and controlled queries add at most constant-factor overhead, the simulation $T$-count scales as the query count times the optimal $T$-count per query.
