Pauli Spectrum and Non-stabilizerness of Typical Quantum Many-Body States

TL;DR

This work introduces the Pauli spectrum as a detailed diagnostic of non-stabilizerness (magic) in quantum many-body states and develops a typicality-based framework predicting Haar-like behavior for typical states. It derives exact Pauli-spectrum distributions for Haar-random states, and defines the filtered stabilizer entropy to robustly separate typical from atypical states, including product and pseudomagic ensembles. Through analytical results and extensive numerics on random circuits and chaotic and disordered spin chains, the authors show that typical states exhibit maximal filtered magic density and that tails in the spectrum encode ETH-related fine structure and ergodicity-breaking phenomena. The study also demonstrates the practical utility of sampling Pauli strings and discusses extensions to qudits and future directions in constructing qubit-specific magic monotones and studying magic-driven phase transitions.

Abstract

An important question of quantum information is to characterize genuinely quantum (beyond-Clifford) resources necessary for universal quantum computing. Here, we use the Pauli spectrum to quantify how magic, beyond Clifford, typical many-qubit states are. We first present a phenomenological picture of the Pauli spectrum based on quantum typicality and then confirm it for Haar random states. We then introduce filtered stabilizer entropy, a magic measure that can resolve the difference between typical and atypical states. We proceed with the numerical study of the Pauli spectrum of states created by random circuits as well as for eigenstates of chaotic Hamiltonians. We find that in both cases the Pauli spectrum approaches the one of Haar random states, up to exponentially suppressed tails. Our results underscore differences between typical and atypical states from the point of view of quantum information.

Figures (7)

• Figure 1: (a) Pauli spectrum for the atypical product state $|\Theta\rangle$ (where we fix $\theta=2 \arctan(\sqrt{2-\sqrt{3}})$ and $\phi=2\arctan(1-\sqrt{2})$), a typical (Haar random) state $|\Psi_\mathrm{Haar}\rangle$, and for a stabilizer state $|\Psi_\mathrm{Stab}\rangle$ generated by a random Clifford unitary for $N=10$ qubits. These distributions manifest distinctive traits, characterizing structural properties of non-stabilizerness. (b) The (filtered) stabilizer entropy ($\tilde{M}_q$) $M_q$. Resolving between $M_q$ for $|\Theta\rangle$ and $|\Psi_\mathrm{Haar}\rangle$ requires precision growing exponentially with $q$, whereas the separation of the $\tilde{M}_q$ values is neat at any $q$. Both $\tilde{M}_q$ and $M_q$ are magic measures and $M_q=\tilde{M}_q=0$ for stabilizer states.
• Figure 2: Pauli spectra of typical states (red dashed lines, Eq. \ref{['eq:pheno']} and \ref{['eq:phenore']}) and the numerical results for: random unitary circuits (a), random orthogonal circuits (b), mid-spectrum eigenstates of chaotic Hamiltonian $H^\mathrm{sb}$ without TRI, (c), and with TRI $H$, (d), see Text. Only the regular parts of the distributions are plotted.
• Figure 3: The filtered stabilizer entropy $\tilde{M}_q$ at $q=1,\dots,6$ for the random Haar states (red dashed lines) and eigenstates of chaotic Hamiltonian for systems (markers) without (a) and with (b) time-reversal invariance. The data for distinct $q$ are shifted downwards by $(q-1)/2$ for clarity.
• Figure 4: Ergodicity breaking in disordered Ising model. Pauli spectrum $\Pi(x)$ in the ergodic (a), and MBL (b) regime for system size $N=10,12,14$ compared with the analytical prediction $\Pi_d^\mathrm{O}(x)$ for $d=2^{14}$. (b) The (filtered) stabilizer entropy ($\tilde{M}_q$) $M_q$ for the typical state $\ket{\Psi_{O}}$, and mid-spectrum eigenstates of $H_{\mathrm{dis}}$ in the ergodic regime ($W=0.25, 1.25$) and in MBL regime ($W=3$) shown as function of the Rényi index $q$. (c) The density of filtered stabilizer entropy $\tilde{M}_q/N$ as a function of disorder strength $W$, the red dashed lines denote the result for Haar-random real states.
• Figure 5: Non-stabilizerness in sparse phase states (SPS). (a) Pauli spectrum for SPS state $\ket{SPS}$ with $|S|=64$ for system size $N=14$ is compared with the typical state result $\Pi^{U}_d(x)$ for $d=2^{14}$. (b) The (filtered) stabilizer entropy ($\tilde{M}_q$) $M_q$ for the typical state $\ket{\Psi_{\mathrm{Haar}}}$ and SPS shown as a function of the Rényi index $q$.