AI 中文总结
该研究提出在量子相位估计（QPE）前用sigmoid谱滤波器预处理输入算子，通过放大谱隙压缩QPE电路深度，在LiH等案例中实现深度和门数显著减少，保真度提升。
AI 中文摘要
我们证明，在量子相位估计（QPE）电路中，可通过在估计前对输入算子应用sigmoid谱滤波器进行预处理，显著压缩电路深度。对于具有小谱隙Δλ的系统，标准QPE需要m=ceil(log₂(1/Δλ))个精度量子比特，电路深度为Θ(2ᵐ)。应用软阶跃变换f(λ;τ,w)可将有效谱隙放大至Δբ>Δλ（当w<1/4时），所需精度降至mբ=ceil(log₂(1/Δբ))，电路深度压缩倍数为2^(αΔₘ)，其中α=1对应LMR密度矩阵指数化框架，α∈[0.11,0.42]对应受控相位门电路。我们证明该压缩是精确的，上界为log₂(1/(4wΔλ))+1，且对于严格简并谱无法实现。进一步表明阈值参数τ仅需O(w)精度，因此协方差对角化或CASSCF等经典预处理不会形成循环。当4w²(2^Δₘ -1)>Δλlog(1/ε)时，可获得净资源优势。在LiH和BeH₂键长拉伸计算、经典协方差数据集及合成近简并案例上的验证显示，电路深度最多可压缩27倍，CX门数量最多可减少21倍。对于LiH，在1%硬件错误率下，QPE输出保真度从0.66提升至0.98。该方法保留主子空间至机器精度，无需修改QPE，可与读出阶段和态制备滤波结合。阴性对照测试确立了适用条件：m_raw≥2且Δλ>0。
英文摘要
We show that quantum phase estimation (QPE) circuits can be significantly compressed in depth by preprocessing the input operator with a sigmoid spectral filter before estimation. For systems with small spectral gaps Delta_lambda, standard QPE requires m = ceil(log2(1/Delta_lambda)) precision qubits and depth Theta(2^m). Applying a soft-step transformation f(lambda; tau,w) amplifies the effective gap to Delta_f > Delta_lambda (for w < 1/4), reducing the required precision to m_f = ceil(log2(1/Delta_f)) and compressing circuit depth by 2^(alpha Delta_m), where alpha = 1 for the LMR density-matrix exponentiation framework and alpha is in [0.11,0.42] for controlled-phase-gate circuits. We prove that this compression is exact, bounded above by log2(1/(4w Delta_lambda)) + 1, and impossible for exactly degenerate spectra. We further show that the threshold parameter tau requires only O(w) accuracy, so classical preprocessing such as covariance diagonalisation or CASSCF avoids circularity. A net resource advantage occurs when 4w^2(2^Delta_m - 1) > Delta_lambda log(1/epsilon). Validation on LiH and BeH2 bond-stretch calculations, classical covariance datasets, and synthetic near-degenerate cases demonstrates depth reductions of up to 27x and CX-gate reductions of up to 21x. For LiH, QPE output fidelity improves from 0.66 to 0.98 at a 1% hardware error rate. The method preserves the principal subspace to machine precision, requires no modification of QPE, and can be combined with readout-stage and state-preparation filtering. Negative-control tests establish the benefit condition: m_raw >= 2 and Delta_lambda > 0.
