The exact LCU sampling overhead of collective diagonal unitaries: resonances and a continued-fraction dichotomy

Abstract

We consider the cost of implementing the quadratic collective phase $e^{-iγK^2}$, with $K$ a collective observable of evenly spaced spectrum, such as the permutation-symmetric Hamming weight. This gate is the cardinality-penalty layer of constrained quantum optimization, the one-axis-twisting gate of spin squeezing and the Kerr phase of a bosonic mode. Instead of two-qubit gates, it can be sampled as a linear combination of single-qubit rotation layers (an LCU) at an overhead $Γ$. We determine the minimal overhead. Ancilla-free sampling of the layers reproduces the target's outcome probabilities up to this factor, and no smaller factor works for every input; when $K$ is permutation symmetric, arbitrary single-qubit gates do no better. The minimal overhead's growth with the register size $n$ is set by the continued fraction of $γ/π$: rational angles $πp/q$ in lowest terms cost exactly $Γ=q$ for every $n\ge2q-2$; the cost is $Θ(n)$ if and only if $γ/π$ is badly approximable; and a master formula, proved two-sided near low-denominator rationals, reads it off the nearest one. Without the symmetry the cost over single-qubit gates can be exponential: for the binary-weighted $K=\sum_j2^jx_j$ the squaring phase of Fourier arithmetic on $m$ qubits costs at least $2^{0.557m-O(1)}$, beyond the $2^{m/2}$ ceiling of every cut bound, by a new certificate-lifting argument. The Fourier-based LCU of Carrera Vazquez, Egger and Woerner (2026), which guarantees $Γ\le n+1$, is optimal up to a constant factor at typical angles at fixed $n$; at rational angles the unique optimizer is the fractional-revival decomposition of Kerr optics, a recipe independent of $n$. In bosonic simulation, the Kerr-gate phase-shifter decomposition of Upreti, Quesada and Chabaud (2026) is the unique optimum over phase shifters, and a finite-cost one exists only at rational $γ/π$.