Circuit Depth Compression via Spectral Gap Amplification in Quantum Phase Estimation

Abstract

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.