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

Abstract

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 $Δ_κ\propto G\left|\sum_k \langle P_k \rangle\right|$ vanishes identically. We instead introduce a fixed-$N$ pairing-coherence diagnostic, \[ Δ_{\mathrm{coh}} = G \sqrt{\sum_{k \neq l} \left| \langle P_k^\dagger P_l \rangle \right|}, \] 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 $δ^\ast \approx -0.25$; $^{82}$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 $Δ_{\mathrm{coh}}$.

Figures (8)

• Figure 1: Illustrative 6-qubit schematic of the structured number-conserving ansatz. The green $X$ gates prepare the reference occupation, the orange four-wire blocks $G^2$ denote pair-transfer double excitations, and the blue two-wire blocks $G$ denote single excitations restricted to the nonzero $j_x$ coupling graph. The figure is schematic: only the logical block structure is shown, not the internal decomposition of the excitation operators.
• Figure 2: (a) Total quantum Routhian surfaces of $^{80}$Zr for the sampled cranking frequencies, all referenced to the $\omega=0$ minimum energy. (b) Total quantum Routhian surfaces at $\omega=0$ for $^{80,82,84}$Zr, shifted to their respective minima.
• Figure 3: Quantum rotational observables evaluated at the deformation minimum for $^{80,82,84}$Zr. Panel (a) shows the aligned angular momentum $J_x(\omega)$, where $^{80}$Zr retains the symmetric proton--neutron pattern of the $N=Z$ case, $^{82}$Zr shows the strongest total alignment, and $^{84}$Zr remains neutron dominated but more moderate in total rotational response. Panel (b) shows the dynamical moment of inertia $J^{(2)}(\omega)$ extracted on the same frequency mesh and plotted only at interior frequencies, since both the $\omega=0$ and $\omega=1.0$ endpoints are one-sided finite-difference estimates.
• Figure 4: Species-resolved quantum pair-transfer coherence at the deformation minimum, compared with the classical cranked-BCS gap, for $^{80,82,84}$Zr. The quantum pairing signal remains finite even though the anomalous gap $\Delta_\kappa$ vanishes by symmetry in the number-conserving ansatz.
• Figure 5: Quantum versus classical cranked-BCS comparison at the deformation minimum in $^{80,82,84}$Zr. Panel (a) shows the deformation path $\delta^\ast(\omega)$, while panel (b) shows the aligned angular momentum. The classical baseline follows a different low-frequency deformation path and predicts substantially larger high-frequency alignment, but should be read as a qualitative reference rather than a precision benchmark.