aiqbench analysis
This research establishes the precise sampling overhead for implementing quadratic collective phases using linear combinations of unitaries, revealing a dichotomy where costs are polynomial for rational angles but can become exponential otherwise. By analyzing continued fractions and symmetry constraints, the study identifies optimal decomposition strategies for quantum optimization layers and bosonic simulations without requiring ancilla qubits. These findings provide critical benchmarks for resource estimation in near-term quantum algorithms that rely on specific phase-shift operations.
The exact LCU sampling overhead of collective diagonal unitaries: resonances and a continued-fraction dichotomy
arXiv:2610.03595v1 Announce Type: new Abstract: We consider the cost of implementing the quadratic collective phase $e^{-i\gamma K^2}$, 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 $\Gamma$. 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 $\gamma/\pi$: rational angles $\pi p/q$ in lowest terms cost exactly $\Gamma=q$ for every $n\ge2q-2$; the cost is $\Theta(n)$ if and only if $\gamma/\pi$ 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 $\Gamma\le n+1$, 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 $\gamma/\pi$.
Abstract:We consider the cost of implementing the quadratic collective phase $e^{-i\gamma K^2}$, 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 $\Gamma$. 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 $\gamma/\pi$: rational angles $\pi p/q$ in lowest terms cost exactly $\Gamma=q$ for every $n\ge2q-2$; the cost is $\Theta(n)$ if and only if $\gamma/\pi$ 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 $\Gamma\le n+1$, 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 $\gamma/\pi$.
arXiv:2610.03595 [quant-ph]
(or arXiv:2610.03595v1 [quant-ph] for this version)
https://doi.org/10.48550/arXiv.2610.03595 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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Full text under the publisher's open licence · 374 words
