Topological magic response in quantum spin chains

TL;DR

This work introduces topological magic response (TSRE), a framework to quantify how nonstabilizerness propagates under finite-depth non-Clifford perturbations in one-dimensional quantum spin chains. By defining the quadri-partition topological stabilizer Rényi entropy M^q_topo and deploying analytic fixed-point calculations together with Pauli-MPS simulations, it demonstrates a clear dichotomy: trivial and symmetry-broken phases lack non-local magic, while symmetry-protected topological (SPT) phases host robust, universal non-local magic under T-gate doping. The results span Ising-type chains, Cluster Ising, tri-critical Ising, and spin-1 AKLT models, highlighting universal TSRE values like 2 log2(4/3) in SPT sectors and showing resilience to disorder. Overall, the paper links magic to topology in a way that goes beyond entanglement, suggesting new avenues for invariants and resources in quantum computation and many-body physics.

Abstract

Topological matter provides natural platforms for robust, non-local information storage, central to quantum error correction. Yet, while the relation between entanglement and topology is well established, little is known about the role of nonstabilizerness (or magic), a pivotal concept in fault-tolerant quantum computation, in topological phases. We introduce the concept of topological magic response, the ability of a state to spread over stabilizer space when perturbed by finite-depth non-Clifford circuits. Unlike a topological invariant or order parameter, this response function probes how a phase reacts to non-Clifford perturbations, revealing the presence of non-local quantum correlations. In Ising-type spin chains, we show that symmetry-broken and paramagnetic phases lack such a response, whereas symmetry-protected topological (SPT) phases always display it. To capture this, we utilize a combination of stabilizer Rényi entropies that, in analogy with topological entanglement entropy, isolates non-locally stored information. Using exact analytic computations and matrix product states simulations based on an algorithmic technique we introduce, we show that SPT phases doped with $T$ gates support robust topological magic response, while trivial phases remain featureless.

Figures (12)

• Figure 1: (a) Schematic of the ground state $|\psi\rangle$ and the T-gate–doped state $|\psi_T\rangle$ with $N_T$ number of T-gates. The transverse field $h$ tunes between the symmetry-protected topological (SPT) and trivial phases, with the transition occurring at the critical point $h = h_c$. The topological magic ($M^q_{\rm topo}$) response, shown at the center, is nonzero in the SPT phase and vanishes in the trivial phase as the system size $L$ varies. (b) Spin-chain partitioning into four segments: $A = [1,L/4], B = (L/4, L/2], D = (L/2, 3L/4]$, and $C = (3L/4, L]$.
• Figure 2: The quadri-partition topological stabilizer Rényi entropy (TSRE), $M^q_\mathrm{topo}$, as a function of $g$ in the tri-critical Ising model (Eq. \ref{['TCIM_ham']}). The results are computed using the MPS representation of the fully T-gate doped state ($N_T=L$), $|\psi_T\rangle$ (Eq. \ref{['MPS_Tgate']}), for different system sizes $L$.
• Figure 3: Topological stabilizer Renyi entropy ($M^q_{topo}$) of the (a,b,c) Ising model and (d,e,f) Cluster Ising model with (a,d) $N_T=0$, (b,e) $N_T=1$ and (c,f) $N_T=L$ number of T-gates on the ground state of system size $L$ as a function of transverse field $h$ with $J=1$.
• Figure 4: The two edge modes correlation as a function of the magnetic field strength ($h$) in the Cluster Ising model for different system sizes $L$, in the absence of T-gates and $J=1$. The inset shows the effect of disorder strength ($\Delta$) on the correlation of a single edge mode for $L=32$ and $h=0.24$ (corresponding to the red dot in the main figure), averaged over $N_s=500$ disorder realizations.
• Figure 5: Topological SRE $M^q_{\rm topo}$ as a function of disorder strength $\Delta$ for the $N_T = L$ T-gates doped ground state of the Cluster Ising model, with $h=0.24$, $J=1$, and $L=32$. The inset shows the correlation of a single edge mode under the same conditions, averaged over $N_s=500$ disorder realizations, highlighting the effect of increasing disorder.