AI 中文总结
研究针对哈密顿量模拟中乘积公式电路规模与精度成多项式关系的问题，提出高阶嵌套对易子补偿（HNCC）算法，通过特定展开和随机采样补偿误差，实现多对数精度依赖，降低门数，优于其他基于乘积公式的方法。
AI 中文摘要
乘积公式是哈密顿量模拟中最实用的方法之一，无需辅助量子比特，误差界由嵌套对易子控制。但其电路规模与精度成多项式关系。我们开发了高阶嵌套对易子补偿（HNCC）算法，保留乘积公式优点，实现电路规模与多对数精度相关，标准采样成本为\(\mathcal{O}(\varepsilon^{-2})\)。HNCC用截断的Baker–Campbell–Hausdorff展开表示高阶Trotter误差，通过随机采样的泡利旋转通道在超算子层面补偿误差，避免哈达玛测试和辅助量子比特。对于固定的\(K\)阶乘积公式应用于\(N\)个量子比特上的\(k\)局部哈密顿量，HNCC使用\(\mathcal{O}(\varepsilon^{-2})\)次重复和每个电路最大门数\(\mathcal{O}(N^{\frac{2}{2K + 1}} (kg_0t\log(1/\varepsilon))^{1+\frac{1}{2K + 1}}k(\Gamma+\log(1/\varepsilon)))\)将\(\operatorname{tr}[Oe^{-\mathrm{i}tH}\rho e^{\mathrm{i}tH}]\)估计到加法精度\(\varepsilon\|O\|\)。结果时间依赖性与\(2K + 1\)阶乘积公式匹配。有限尺寸资源估计表明，HNCC在基于乘积公式的方法中实现了每个电路最低的CNOT和\(T\)门数。
英文摘要
Product formulas are among the most practical approaches to Hamiltonian simulation, requiring no ancillary qubits and exhibiting error bounds governed by nested commutators rather than only by Hamiltonian norms. Their circuit size, however, scales polynomially with the inverse precision. We develop a high-order nested-commutator compensation (HNCC) algorithm that preserves the main advantages of product formulas while achieving polylogarithmic precision dependence in the circuit size and the standard $\mathcal{O}(\varepsilon^{-2})$ sampling cost. HNCC uses a truncated Baker--Campbell--Hausdorff expansion to represent high-order Trotter errors by products of nested commutators and compensates these errors at the channel level through randomly sampled Pauli-rotation channels, avoiding Hadamard tests and ancillary qubits. For a fixed $K$-th order product formula applied to a $k$-local Hamiltonian on $N$ qubits with $Γ$ Pauli terms and local interaction strength $g_0$, HNCC estimates $\operatorname{tr}[Oe^{-i tH}ρe^{i tH}]$ to additive precision $\varepsilon\|O\|$ using $\mathcal{O}(\varepsilon^{-2})$ repetitions. Its maximum gate count per circuit is $\mathcal{O}\bigl( kN^{\frac{1}{2K+1}} Γ^{1-\frac{1}{2K+1}} \max\{Γ,N\log(1/\varepsilon)\}^{\frac{1}{2K+1}} (kg_0t\log(1/\varepsilon))^{1+\frac{1}{2K+1}} \bigr)$. Finite-size resource estimates for the periodic Heisenberg chain indicate that HNCC has the lowest estimated $T$-gate count per circuit among the product-formula-based methods considered.
Comments23 pages, 4 figures, 2 tables
