Momentum space magic for the transverse field quantum Ising model

TL;DR

This work probes the momentum-space nonstabilizerness of the one-dimensional transverse-field Ising chain by mapping the model to momentum-space qubits and analyzing the distribution of Pauli strings and stabilizer entropies. The ground state factorizes across momentum sectors, yielding a broad Pauli-spectrum in the ferromagnetic phase that evolves to a two-peaked structure near the paramagnetic phase, with a vanishing magic gap in all regimes. Stabilizer entropies are non-analytic at the critical point $|g|=1$ and saturate to zero as the field grows, while the ferromagnetic phase exhibits a gapped-free momentum-space spectrum and a constant per-site magic in the thermodynamic limit. Compared to real-space, momentum-space magic at criticality is significantly smaller, suggesting momentum-space simulations can be more classically tractable and offering complementary insights into nonstabilizerness and quantum phase transitions.

Abstract

Stabilizer entropies and quantum magic have been extensively explored in real-space formulations of quantum systems within the framework of resource theory. However, interesting and transparent physics often emerges in momentum space, such as Cooper pairing. Motivated by this, we investigate the momentum-space structure of Pauli strings and stabilizer entropies in the one-dimensional transverse-field quantum Ising model. By mapping the Ising chain onto momentum-space qubits, where the stabilizer state corresponds to the paramagnetic state, we analyze the evolution of the Pauli string distribution. In the ferromagnetic phase, the distribution is broad, whereas in the paramagnetic phase, it develops a two-peaked structure. We demonstrate that all ferromagnetic states possess the same degree of magic in the thermodynamic limit, while stabilizer entropies are non-analytic at the critical point and vanish with increasing transverse field. The momentum-space approach to quantum magic not only complements its real-space counterpart but also provides advantages in terms of analyzing nonstabilizerness and classical simulability.

Figures (3)

• Figure 1: The regular part of the distribution of non-zero Pauli strings, $|\langle\Phi|\Sigma|\Phi\rangle|\neq 0$ is plotted (normalized to unity here) across the quantum phase transition for $N=40$. The $g=0.4$ and $g=1$ data fall on top of each other. Since there are 3 different matrix elements for a $(k,-k)$ pair from Eq. \ref{['matrixelements']}, effectively $3^{N/2}$ contributions were taken into account. The Dirac delta peak at the origin from the vanishing strings, whose weight would be $1-(3/8)^{N/2}$, is not shown for clarity. The overall weight of the non-zero strings would be $(3/8)^{N/2}$, and out of these, the weight of Pauli string expectation values of 1 is $8^{-N/2}$ from Eq. \ref{['matrixelements']}.
• Figure 2: The stabilizer Rényi entropies are plotted for $N=2000$ and several Rényi index $n$, displaying completely flat behaviour within the ferromagnetic phase, a broad peak upon entering into the paramagnetic phase and vanishing in the $g\rightarrow\infty$ limit.
• Figure 3: The finite size scaling of the magic is shown for various system sizes. The inset zooms in to the derivative of the magic with respect to the transverse field at the quantum critical point.