Nonstabilizerness dynamics in many-body localized systems

TL;DR

The paper investigates how nonstabilizerness, or magic, spreads in disordered many-body localized systems using stabilizer Rényi entropy ($\mathcal{M}_2$). It develops an $\ell$-bit phenomenology and validates it with numerical simulations of the disordered TFIM, revealing a power-law growth of $\mathcal{M}_2$ that slowly saturates due to dephasing and interactions, in contrast to rapid saturation in ergodic systems. A key result is a universal relationship between $\mathcal{M}_2$ and the entanglement entropy $\mathcal{S}$ in the MBL regime, with $\mathcal{M}_2$ collapsing onto a master curve (or requiring a disorder-dependent rescaling for some initial states), indicating a deep connection between quantum complexity and entanglement in localized dynamics. These findings elucidate how disorder and interactions constrain the generation of magic resources, with implications for the classical simulability of MBL dynamics and the resource theory of quantum computation in disordered many-body systems.

Abstract

Nonstabilizerness, also known as ``magic'', quantifies the deviation of quantum states from stabilizer states, capturing the complexity necessary for quantum computational advantage. In this study, we investigate the dynamics of nonstabilizerness in disordered many-body localized (MBL) systems using the stabilizer Rényi entropy (SRE). Leveraging a phenomenological description based on the $\ell$-bit model, we analytically and numerically demonstrate that interactions profoundly influence nonstabilizerness spreading, inducing a power-law growth of SRE that markedly contrasts with the rapid saturation observed in ergodic systems. We validate our theoretical predictions through numerical simulations of the disordered transverse-field Ising model, showing excellent agreement across various disorder strengths, system sizes, and initial states. Additionally, we uncover a universal relationship between SRE and entanglement entropy, revealing their common scaling in the MBL regime independent of disorder strength and system size. Our results offer critical insights into the interplay of disorder, interactions, and complexity in quantum many-body systems.

Figures (11)

• Figure 1: Nonstabilizerness spreading in disordered quantum systems. In the non-interacting case (III, red dashed line), $\mathcal{M}_2$ is analytically tractable (see supmat). It saturates rapidly to a finite value due to the absence of spin dephasing. In the MBL regime, dephasing induces a power-law growth of $\mathcal{M}_2$ toward a saturation value described by Eq. \ref{['Eq:SRE_MBL']}. For the initial X-polarized state (I, orange dashed line), nonstabilizerness grows rapidly and saturates to the Haar value. For a generic product state (II, orange dashed line), the growth is slower and saturates at a lower value, revealing the dependence of nonstabilizerness spreading on the choice of initial state.
• Figure 2: Dynamics of nonstabilizerness in disordered TFIM. Evolution of SRE for initial states (a) $|\Psi_Z\rangle$, (b) $|\Psi_X\rangle$, (c) $|\Psi_Y\rangle$, and (d) $|\Psi_R\rangle$ (see text). The results are for $L=16$ and averaged over $1000$ realizations, considering the product state close to the middle of the spectrum. To demonstrate the validity of \ref{['Eq:SRE_MBL']}, we performed a numerical fit at $W=5$ for all states. The saturation value $\mathcal{M}_2^{\mathrm{sat}}$ depends on the initial state: $\mathcal{M}_2^{\mathrm{sat}}\approx7.3$ for $|\Psi_Z\rangle$, $\mathcal{M}_2^{\mathrm{sat}}\approx11.53$ for $|\Psi_R\rangle$, while the SRE for $|\Psi_{X}\rangle$ and $|\Psi_{Y}\rangle$ states saturates to the Haar value $\mathcal{M}_2^{\mathrm{Haar}}$. The power-law growth exponents are $\beta\approx0.16$ for $|\Psi_Z\rangle$, $\beta\approx0.29$ for $|\Psi_{X}\rangle$, $\beta\approx0.39$ for $|\Psi_{Y}\rangle$, and $\beta\approx0.19$ for $|\Psi_{R}\rangle$. Similar behavior is obtained for other disorder strengths within the MBL regime.
• Figure 3: Initial state dependence of SRE and weight of Z gates in MBL regime. (a) SRE dynamics in the $\ell$-bit model for different choices of $\alpha$. (b) Power-law exponent $\beta$ characterizing the growth of $\mathcal{M}_2$ as a function of $\alpha$; time evolution of $\mathcal{W}_Z$ for different initial states for the (c) $\ell$-bit model and (d) TFIM. The saturation value of $\mathcal{M}_2$ depends on the degree of localization of the initial state in the $\ell$-bit basis and it is intrinsically connected to $\mathcal{W}_Z$.
• Figure 4: Nonstabilizerness versus entanglement in the MBL regime. The SRE $\mathcal{M}_2$ is plotted as a function of the half-chain entanglement entropy $\mathcal{S}$ for different disorder strengths $W$ and system size $L$. For the $Y$-polarized state $|\Psi_Y\rangle$, (a), $\mathcal{M}_2(\mathcal{S})$ collapse, without any fitting parameters, on a single master curve both for $\ell$-bit model and TFIM. For the random product state $|\Psi_R\rangle$, (b), the collapse occurs when $\mathcal{M}_2(\mathcal{S})$ are rescaled by an $L$-independent factor $f(W)$.
• Figure 5: (a) Nonstabilizerness spread in the $\ell$-bit model for different system sizes, with $|\Psi_{X}^{+}\rangle$ as the initial state. The dashed lines show the analytical solution (Eq. \ref{['Eq:SRE_MBL']} of the main text), which accurately describes the SRE growth after $t\sim 1$. Inset: Dependence of the power-law exponent $\beta^{\prime}$ with the system size for $\xi=0.5$; (b) Time evolution of $\Delta \mathcal{M}_2$ for different system sizes $L$. It decays polynomially in $t$ until it eventually saturates, except for $L=18$, for which the Heisenberg time is beyond the considered here. The saturation value exhibits an exponential decay with exponent $\lambda \approx -\ln{2}$, as shown in the inset.