The exact LCU sampling overhead of collective diagonal unitaries: resonances and a continued-fraction dichotomy
Abstract
The minimal sampling overhead of the quadratic collective phase e−iγK2 over one layer of single-qubit rotations is decided by the continued fraction of γ/π. Here K is a collective observable with spectrum {0,…,n}, such as the permutation-symmetric Hamming weight. This gate is the cardinality-penalty layer of constrained optimization, the one-axis-twisting gate of spin squeezing and the Kerr phase of a bosonic mode. Instead of compiling it to two-qubit gates, we sample it as an LCU over such layers at overhead Γ. Our results give an optimality theory for this overhead. Ancilla-free sampling reproduces the target's outcome probabilities up to the minimal Γ, and no smaller factor works for every input; for permutation-symmetric K, independent per-qubit angles or one layer of arbitrary single-qubit gates give no further reduction. Rational angles πp/q in lowest terms cost exactly q once n≥2q−2, attained uniquely by a uniform q-point sampler; the cost is Θ(n) if and only if γ/π is badly approximable, and bounded in n if and only if it is rational. A second result shows what the symmetry buys: without it, the squaring phase of Fourier arithmetic on m qubits costs at least 20.557m−O(1) over single-qubit gates, beyond the 2m/2 of every cut bound, by a new certificate-lifting argument. For constructions known only through upper bounds, it gives the optimum: the Fourier-based LCU of Carrera Vazquez, Egger and Woerner is, at fixed n, optimal up to a constant at typical angles and beaten by (n+1)/q at rational ones with an n-independent sampler; the Kerr decomposition of Upreti, Quesada and Chabaud is the unique optimum over phase shifters at rational parameters without photon-number cutoff; and it identifies the rational couplings where one-axis twisting admits optimal cat decompositions.
Related
- Exact damping-basis continued fractions for few-atom lasers: collective correlations beyond second-order cumulant closure
- Generating random unitaries by products of conjugated Hamiltonian evolutions
- Low-Depth Random Unitaries without Ancillae
- Optimal Quantum-Circuit Synthesis and Implementation of General Unitaries with Parallelized Bitstring-Conditioned Rotations
- A Catalytic Algorithm for Transposing Unknown Qutrit Unitaries with Tight Query Bounds
BibTeX
@misc{wang2026exact,
title = {{The exact LCU sampling overhead of collective diagonal unitaries: resonances and a continued-fraction dichotomy}},
author = {Zhihui Wang and Sujit Roy and Manil Maskey and Rahul Ramachandran},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.03595},
eprint = {2610.03595},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.03595},
}
RIS
TY - UNPB
TI - The exact LCU sampling overhead of collective diagonal unitaries: resonances and a continued-fraction dichotomy
AU - Zhihui Wang
AU - Sujit Roy
AU - Manil Maskey
AU - Rahul Ramachandran
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.03595
UR - https://arxiv.org/abs/2610.03595
AB - The minimal sampling overhead of the quadratic collective phase e^−iγK^2 over one layer of single-qubit rotations is decided by the continued fraction of γ/π. Here K is a collective observable with spectrum {0,…,n}, such as the permutation-symmetric Hamming weight. This gate is the cardinality-penalty layer of constrained optimization, the one-axis-twisting gate of spin squeezing and the Kerr phase of a bosonic mode. Instead of compiling it to two-qubit gates, we sample it as an LCU over such layers at overhead Γ. Our results give an optimality theory for this overhead. Ancilla-free sampling reproduces the target's outcome probabilities up to the minimal Γ, and no smaller factor works for every input; for permutation-symmetric K, independent per-qubit angles or one layer of arbitrary single-qubit gates give no further reduction. Rational angles πp/q in lowest terms cost exactly q once n≥2q−2, attained uniquely by a uniform q-point sampler; the cost is Θ(n) if and only if γ/π is badly approximable, and bounded in n if and only if it is rational. A second result shows what the symmetry buys: without it, the squaring phase of Fourier arithmetic on m qubits costs at least 2^0.557m−O(1) over single-qubit gates, beyond the 2^m/2 of every cut bound, by a new certificate-lifting argument. For constructions known only through upper bounds, it gives the optimum: the Fourier-based LCU of Carrera Vazquez, Egger and Woerner is, at fixed n, optimal up to a constant at typical angles and beaten by (n+1)/q at rational ones with an n-independent sampler; the Kerr decomposition of Upreti, Quesada and Chabaud is the unique optimum over phase shifters at rational parameters without photon-number cutoff; and it identifies the rational couplings where one-axis twisting admits optimal cat decompositions.
ER -