Magic-state resource theory for the ground state of the transverse-field Ising model

Ground states of quantum many-body systems are both entangled and possess a kind of quantum complexity, as their preparation requires universal resources that go beyond the Clifford group and stabilizer states. These resources—sometimes described as magic—are also the crucial ingredient for quantum advantage. We study the behavior of the stabilizer Rényi entropy in the integrable transverse field Ising spin chain. We show that the locality of interactions results in a localized stabilizer Rényi entropy in the gapped phase, thus making this quantity computable in terms of local quantities in the gapped phase, while measurements involving $L$ spins are necessary at the critical point to obtain an error scaling with $O\left(L^{-1}\right)$.

Figure 1

Numerical simulations of the stabilizer Rényi entropy of the ground state $|G\left(\lambda \right)\rangle$ of the Hamiltonian in Eq. (2) for $\lambda =0.1,0.3,0.6, \lambda =1$ and $\lambda =2,2.5,5$ as a function of the length of the chain $N\in [5,12]$. The curves are fitted to be straight lines for any $\lambda \mathrm{s}$, with slopes $\alpha \left(\lambda \right)$ and intercepts $\beta \left(\lambda \right)$ fitted in the top-left corner.

Figure 2

Plot of the single spin density of nonstabilizerness ${\alpha }_1\left(\lambda \right)$ for ${\lambda }^{-1}\le 0.6$, computed in Eq. (10), versus the density of nonstabilizerness $\alpha \left(\lambda \right)$ extracted through the fits in Fig. 1.

Figure 3

Comparison of the error ${\epsilon }_L\left(1\right)$ for $L\in \{5,11\}$ with the fit $\gamma L^{-1}$ with $\gamma =0.2034\pm 0.0002$.