Nonstabilizerness generation in a multiparticle quantum walk

TL;DR

This work investigates the generation and propagation of nonstabilizerness (magic) in single- and multiparticle quantum walks within the XXZ Heisenberg model by tracking the stabilizer Rényi entropy $M_2$ and the Pauli spectrum. Analytically and numerically, magic is shown to spread within the light-cone dictated by the dynamics, with easy-plane ($\Delta<1$) behavior governed by single-particle motion and easy-axis ($\Delta>1$) behavior dominated by slow doublon propagation, yielding a logarithmic time growth of $M_2$ in both cases and a substantial slowdown in the doublon regime. The Pauli spectrum exhibits Poissonian level statistics in the stationary magic regime, independently of interaction strength, particle number, or integrability-breaking perturbations. Together, these results clarify how interactions modulate nonstabilizerness in many-body quantum systems and suggest universality in the asymptotic statistical properties of Pauli coefficients for driven quantum walks.

Abstract

We investigate the generation of non-stabilizerness, or magic, in a multi-particle quantum walk by analyzing the time evolution of the stabilizer Rényi entropy $M_2$. Our study considers both single- and two-particle quantum walks in the framework of the XXZ Heisenberg model with varying interaction strengths. We demonstrate that the spread of magic follows the light-cone structure dictated by the system's dynamics, with distinct behaviors emerging in the easy-plane ($Δ< 1$) and easy-axis ($Δ> 1$) regimes. For $Δ< 1$, magic generation is primarily governed by single-particle dynamics, while for $Δ> 1$, doublon propagation dominates, resulting in a significantly slower growth of $M_2$. Furthermore, the magic exhibits logarithmic growth in time for both one and two-particle dynamics. Additionally, by examining the Pauli spectrum, we show that the statistical distribution of level spacings exhibits Poissonian behavior, independent of interaction strength or particle number. Our results shed light on the role of interactions on magic generation in a many-body system.

Figures (5)

• Figure 1: Time evolution of the stabilizer Rényi entropy $M_2$ in the single-particle quantum walk, shown as a function of time for different system sizes. The MPS data for $L\in\{50,100,200,400\}$Haug2023Tarabunga2024 is compared against (coeff) results from wave-function coefficients \ref{['eq:magic_SP']} for $L=600$, and (asymp) the asymptotic expression \ref{['eq:magic_SP_asympt']}.
• Figure 2: (a) Time evolution of magic $M_2$ in the two-particle quantum walk for different interaction strengths $\Delta$, using the MPS approach. (b)--(d) Density plots showing the local magnetization dynamics $\langle Z\rangle$ for $\Delta = \{0.5, 1, 2\}$, obtained using exact diagonalization. In each plot, the green lines indicate the boundaries of the single-particle light cone, while the orange lines mark the doublon light cone with $v^{(D)}_F=0.5J/\Delta$. The density intensity is truncated near $\langle Z\rangle=1$ for better visibility of the single-particle light cones at $\Delta\geq 1$. In all the panels the system size is fixed to $L=128$.
• Figure 3: The time evolution of $M_2$ for large $\Delta$ values. Solid lines show the total $M_2$ computed using the MPS algorithm, while dashed lines represent the doublon contribution. The dash-dotted lines represent the effective doublon contribution shifted by a constant, which reproduces at large $t$ the MPS results. The system size is $L=128$ for MPS simulations.
• Figure 4: The cumulative average of magic as a function of $\Delta$ for the two-particle quantum walk, measured before the single-particle light cone reaches the system boundary at $tJ \approx L/2$. The inset shows the logarithmic growth of $\langle M_2(t)\rangle_c$ at large $t$ for several $\Delta$ values. The system size is $L=128$.
• Figure 5: The distribution of the ratios $P(\tilde{r})$ for the Pauli string values in the long-time limit for the single and two-particle quantum walk. The dynamics is governed by (a) the XXZ Hamiltonian \ref{['eq:XXZ']} or (b) the nonintegrable next-nearest-neighbor hopping XXZ model \ref{['eq:H_nnn']} with $J'=0.5J$. The red dashed line corresponds to the Poisson distribution, $P(\tilde{r}) =2/(1+\tilde{r})^2\theta(\tilde{r}) \theta(1-\tilde{r})$Atas2013.