Advantage of utilizing nonlocal magic resource in Haar-random circuits

TL;DR

The paper identifies an intrinsic logarithmic scaling law between the nonlocal magic resource and bond dimension in Haar-random circuits, showing that $M_n$ converges at $ agchi sim O( obreak )$, which substantially improves classical simulability of nonstabilizerness. By separating magic injection from entanglement through limited bond dimensions and using MPS with Pauli-based sampling, it demonstrates that entanglement acts as a container for nonlocal magic but does not drive its growth; $t_{ ext{SRE}} obreak o obreak O( obreak )$ while $t_{ ext{ENT}} obreak o obreak O( obreak )$, leading to no dynamical relationship between the two resources. The results suggest universal phenomena for chaotic many-body dynamics and provide practical routes for efficient classical simulations of quantum magic in highly entangled regimes. The work also introduces the concept of specific capacity and supports a broader conceptual link between entanglement structure and nonlocal nonstabilizerness across generic quantum circuits.

Abstract

Magic resources and entanglement are fundamental components for achieving the universal quantum computation, so is the interplay between them. Herein, we uncover an intrinsic scaling law of the magic resource and bond dimension of matrix product states in Haar-random quantum circuits, that is, the magic resource is converged on a bond dimension in logarithmic scale with the system size. From a practical perspective, this finding substantially enhances the classical simulability of nonstabilizerness. It also allows us to utilize the bond dimension as a bridge to link the entanglement and the nonlocal magic resource, which extends the capacity perspective that the entanglement plays the role of container for the nonlocal magic resource. Furthermore, the intrinsic scaling enables an information separation between the nonlocal magic resource and the extra entanglement. This, in turn, leads to the conclusion that, any dynamical relation between magic and entanglement resources is ruled out. In other words, it is inappropriate to regard the entanglement as the driving force of the growth and spreading of nonlocal magic resource.

Figures (6)

• Figure 1: Structure of Haar-random brick-wall circuits, with only two-qubit gates. The initial state is product state. Each layer contributes a depth (time step) of unity.
• Figure 2: The averaged SRE $\bar{M}_{n}(N)$ of Haar-random states as a function of the system size $N$ for the variable bond dimension $\chi$. (a)(b) The different convergence curves of SRE $\bar{M}_{n}$ and entanglement $\bar{S}$. (c)(d) The SRE $\bar{M}_{n}$ with more system sizes, all of which universally exhibit sharp increase to ${M}_{n}^{\mathrm{Sat}}$. The dashed lines represent the SRE of $N$-qubit Haar-random states. For $n=2$, ${M}_{2}^{\mathrm{Sat}}={M}_{2}^{\mathrm{Haar}}$. For $n=1$, the maximum value of $\bar{M}_{1}$ is selected as the approaching convergence value ${M}_{1}^{\mathrm{Sat}}$.
• Figure 3: Scaling of the SRE deviation $\Delta {M}_{n}$ with the bond dimension $\chi$. The different panels correspond to various Rényi rank $n$ and different system sizes $N$: (a) $n=2,N=11,15,30$, (b) $n=2,N=40,50,60,80$, (c) $n=1,N=11,13,15,19$, (d) $n=1,N=30,40,60,80$. The dashed lines represent fittings.
• Figure 4: Fitting parameters in the exponential decline of SRE deviation with various system sizes $N$. (a) $\beta$ increases linearly with the system size $N$. For $n=2$, the slope is ${\lambda}_{2}=0.29$, and for $n=1$, ${\lambda}_{1}=0.17$. (b) As $N$ increasing, $\alpha$ remains at around ${\alpha}_{2}\approx0.29$ for $n=2$ and ${\alpha}_{1}\approx0.33$ for $n=1$.
• Figure 5: Evolution of Haar-random circuits on an 11-qubit system with various bond dimensions. Wherein, "finite" stands for finite bond dimension, and "infinite" stands for infinite bond dimension. (a) Bond dimension for magic resource convergence $\chi_{\text{SRE}}=15$ can faithfully establish the dynamics of $\bar{M}_{2}$, but fail to completely establish entanglement. (b) The dynamics of $\bar{M}_{1}$ and the change of required bond dimensions under random circuits are displayed.