Quantum Simulation of Cranked Zirconium Isotopes: A Fixed-N Approach with a Structured Number-Conserving Ansatz

We present a methodological study of quantum simulation of cranking in a Nilsson $+$ pairing Hamiltonian on a fixed deformation grid. The many-body Routhian is mapped to qubits via the Jordan--Wigner transformation and minimized using the Variational Quantum Eigensolver (VQE) in a truncated active space $(M)$. We employ a structured, number-conserving singles-and-doubles ansatz: double excitations implement pair transfer, while singles are restricted to the nonzero Coriolis-coupling graph of the active Nilsson basis. For $M=8$, this yields 42 parameters while preserving particle number exactly. Exact number conservation enforces $\langle P_k \rangle = 0$, so the conventional pairing gap $\Delta_\kappa \propto G\left|\sum_k \langle P_k \rangle\right|$ vanishes identically. We instead introduce a fixed-$N$ pairing-coherence diagnostic, $$\Delta_{\mathrm{coh}} = G \sqrt{\sum_{k \neq l} \left| \langle P_k^\dagger P_l \rangle \right|},$$ used as a scalar measure of off-diagonal pair coherence rather than a BCS gap. We study even-even $^{80,82,84}$Zr. $^{80}$Zr shows a stable oblate minimum at $\delta^\ast \approx -0.25$; $^{82}$Zr exhibits the strongest rotational evolution; $^{84}$Zr retains a robust prolate minimum with the largest neutron pairing coherence. These results reflect the present truncated model rather than converged spectroscopy. A cranked BCS calculation on the same grid serves as a qualitative baseline. Comparisons between $M=6$ and $M=8$ show stable trends but visible shifts, so no active-space convergence is claimed. The structured fixed-$N$ ansatz thus captures consistent isotope trends and provides a practical framework to analyze pairing via $\Delta_{\mathrm{coh}}$.