Spectral signatures of nonstabilizerness and criticality in infinite matrix product states

Abstract

While nonstabilizerness (''magic'') is a key resource for universal quantum computation, its behavior in many-body quantum systems, especially near criticality, remains poorly understood. We develop a spectral transfer-matrix framework for the stabilizer Rényi entropy (SRE) in infinite matrix product states, showing that its spectrum contains universal subleading information. In particular, we identify an SRE correlation length -- distinct from the standard correlation length -- which diverges at continuous phase transitions and governs the spatial response of the SRE to local perturbations. We derive exact SRE expressions for the bond dimension $χ=2$ MPS ''skeleton'' of the cluster-Ising model, and we numerically probe its universal scaling along the $\mathbb{Z}_2$ critical lines in the phase diagram. These results demonstrate that nonstabilizerness captures signatures of criticality and local perturbations, providing a new lens on the interplay between computational resources and emergent phenomena in quantum many-body systems.

Figures (9)

• Figure 1: Our central result, Eq. (\ref{['Eq:Magic Universal']}), for the mixed state SRE, $\widetilde{M}^{(n)}$, of order $n$. The density matrix $\rho$ describes an $N$-qubit subsystem of an infinite MPS state. $\widetilde{M}^{(n)}$ splits into three boxed terms. The red box is an extensive term $\propto N m^{(n)}$ due to the dominant eigenvalue $\mu_1$ of the replica transfer matrix. This term is non-universal, as illustrated by its different behavior for the Ising-spin and Rydberg-atom realizations of the same $\mathbb{Z}_2$ critical point upon varying $\lambda$. The blue box represents correlations between the subsystem and its boundary, determined by the dominant eigenvector of the replica transfer matrix ($c_1$ term) and the Rényi entropy $S^{(n)}$. This term defines the mutual SRE, $L_\infty^{(n)}$, of two adjacent semi-infinite subsystems, which diverges logarithmically with the correlation length $\xi$. Finally, in the green box, $f(N)$ represents the subleading, exponentially-decaying contribution to the SRE. This defines the SRE correlation length, $\xi_\mathrm{SRE}^{(n)}$, which exhibits a power-law divergence near criticality.
• Figure 2: Nonstabilizer properties of the MPS skeleton in Eq. (\ref{['Eq:Skelton MPS']}). (a) The mixed-state SRE density $\tilde{m}^{(2)}$ over a subsystem of increasing size $N$, illustrating convergence to the thermodynamic limit value in Eq. (\ref{['eq:skeleton peak']}) (black line labeled $m^{(2)}$). (b) The mutual SRE, $L^{(2)}(A\,{:}\,B)$ in Eq. \ref{['eq:mutual magic']}, between two connected subsystems $A$ and $B$ of size $N$. Black line shows the thermodynamic limit value, $L^{(2)}_{\infty}$ [Eq. (\ref{['Eq:mutual SRE definition']}) with Eqs. (\ref{['eq:S2skeleton']})-(\ref{['eq:skeletonmutualanalytic']})]. Inset: Correction coefficients $c_i$ in the decomposition of the nonstabilizerness. (c): The maximum change in the SRE, $\delta M^{(2)}_U$, caused by an application of a single-qubit unitary. The optimal angles $\theta,\phi,\lambda$ were found by numerical optimization of Eq. \ref{['eq:oneunitary_SRE']}. The finite-size results in panels (a)-(b) were obtained by numerical evaluation of Eq. \ref{['eq:Mixed state magic']} and Eq. \ref{['eq:mutual magic']}.
• Figure 3: The SRE correlation length and response to perturbations of the MPS skeleton. (a): Correlation lengths of the standard transfer matrix $E$ and of the $n$th order SRE transfer matrix $\mathbb{E}$ for different numbers of replicas $n\leq5$. All correlation lengths diverge at the multicritical point, $g_c=0$. (b): Same data plotted on a log-log scale close to the critical point $g_c=0$. The ratio of the slopes of the blue and orange curves is $2$, consistent with Eq. (\ref{['eq:corrlengths_taylor']}). Different correlation lengths generally diverge at different rates. (c): SRE correlations between two applied $S$ gates for two values of $g$ close to $g_c=0$. The dashed lines are fits to the exponential decay with the SRE correlation length, which accurately describes the behavior of the correlators at large distances $r$ (see text).
• Figure 4: Phase diagram of nonstabilizerness in the cluster Ising model, Eq. (\ref{['eq:clustermodel']}), in the thermodynamic limit. (a) The SRE density, $m^{(2)}$. (b) The mutual SRE, $L^{(2)}_\infty$. (c) The SRE correlation length, $\xi^{(2)}_{\mathrm{SRE}}$. All results are for iMPS with $\chi=50$ and $\chi_t=60$. The magenta line indicates the trajectory of the MPS skeleton, Eq. (\ref{['eq:MPS skeleton Hamiltonian']}).
• Figure 5: (a): The logarithm of the SRE correlation length $\log\xi_\mathrm{SRE}^{(2)}$, plotted as a function of the standard MPS correlation length $\log\xi$, at $g_c=0, 0.25,0.5$ along the horizontal cluster-Ising critical line for a few $\chi_t$ values indicated in the legend. The gray shaded area is the range where these two quantities are proportional to one another for the given values of $\chi_t$. (b): The mutual SRE density $W^{(2)}_{\infty}$ at the points $g_c=0$ and $g_c=2$ on the horizontal cluster-Ising critical line, plotted as a function of $\log \xi_\mathrm{SRE}^{(2)}$. The linear fit to the data is shown in red dashed lines, with the expected value $1/8$ shown in black, demonstrating good agreement. (c): Analogous results for the points $g_c=0.1,0.25, 0.5$ along the horizontal critical line, where the numerical data shows a visible deviation from the $1/8$ scaling. In (b) and (c), the noticeable upturn of $W^{(2)}_{\infty}$ and departure from linear dependence is due to the saturation of the SRE correlation length $\xi_\mathrm{SRE}^{(2)}$ in panel (a) for finite values of $\chi_t$.