Magic spreading in random quantum circuits

TL;DR

This work investigates how magic, i.e., nonstabilizerness, spreads under local chaotic quantum dynamics by introducing generalized stabilizer entropies (GSE) that quantify non-Clifford resources. Using brick-wall Haar-random circuits and a replica-Weingarten framework, the authors map the annealed averages of GSEs to tensor-network contractions, enabling simulations up to N≈1024 qudits. They find that magic saturates to its Haar value on a timescale t^{mag}_{sat} that scales logarithmically with system size, t^{mag}_{sat} ∝ \ln N, in stark contrast to linear entanglement growth, suggesting a universal log-depth relaxation of nonstabilizerness in chaotic many-body dynamics. The results, corroborated by tensor-network numerics and supplementary analyses (e.g., mana, monotone properties), imply broad relevance for chaotic quantum systems and offer a scalable route to quantify magic generation in large-scale quantum circuits.

Abstract

Magic is the resource that quantifies the amount of beyond-Clifford operations necessary for universal quantum computing. It bounds the cost of classically simulating quantum systems via stabilizer circuits central to quantum error correction and computation. How rapidly do generic many-body dynamics generate magic resources under the constraints of locality and unitarity? We address this central question by exploring magic spreading in brick-wall random unitary circuits. We explore scalable magic measures intimately connected to the algebraic structure of the Clifford group. These metrics enable the investigation of the spreading of magic for system sizes of up to $N=1024$ qudits, surpassing the previous state-of-the-art, which was restricted to about a dozen qudits. We demonstrate that magic resources equilibrate on timescales logarithmic in the system size, akin to anti-concentration and Hilbert space delocalization phenomena, but qualitatively different from the spreading of entanglement entropy. As random circuits are minimal models for chaotic dynamics, we conjecture that our findings describe the phenomenology of magic resources growth in a broad class of chaotic many-body systems.

Figures (9)

• Figure 1: A system of $N$ qudits is prepared at time $t=0$ in a product state $\ket{\Psi_0}$ with low-magic resources. Evolution under a quantum circuit comprising local Haar-random gates increases the nonstabilizerness of the state (denoted by the red gradient) and scrambles quantum information (symbolized by the light-cone). The nonstabilizerness approaches its long-time saturation value up to a given tolerance $\epsilon \ll 1$ at time $t^{\mathrm{mag}}_{\mathrm{sat}} \propto \ln N$, scaling logarithmically with $N$, while distant qudits become entangled only after a longer time $t^{\mathrm{ent}}_{\mathrm{sat}}\propto N$.
• Figure 2: Time evolution of the SRE $M_2$ for $N$ qubits ($d=2$) under the brick-wall Haar random circuits. (a) $M_2$ abruptly saturates to $M^{\mathrm{Haar}}_2$. (b) The difference $\Delta M_2 = M^{\mathrm{Haar}}_2 - M_2$ approaches exponential decay $\Delta M_2 \propto N e^{-\alpha_{2,2} t}$, where $\alpha_{2,2} = 0.43(3)$, see the inset. The annealed average $\tilde{M}_2$ obtained via \ref{['eq:Ydtensor']} (denoted "TN") and the quenched average $\overline{M}_2$ (denoted "ED") coincide within the error bars already for $N=8$.
• Figure 3: Dynamics of the SRE $M_2$ for $N$ qutrits ($d=3$) under random circuits. (a) Saturation of $M_2$ to $M^{\mathrm{Haar}}_2$ occurs similarly to the qubit case. (b) The difference $\Delta M_2 = M^{\mathrm{Haar}}_2 - M_2$ follows $\Delta M_2 \propto N e^{-\alpha_{3,2} t}$ with $\alpha_{3,2} = 1.03(3)$ at $t \gtrsim 5$; see the inset. The quenched $\tilde{M}_2$ and annealed $\overline{M}_2$ averages are indistinguishable on the scale of the figure for any $N$.
• Figure 4: Dynamics of the GSE $M_Y$ for $N$ qutrits ($d=3$) under random circuits. (a) Saturation of $M_Y$ to $M^{\mathrm{Haar}}_Y$ occurs similarly to the qubit case. (b) The difference $\Delta M_Y = M^{\mathrm{Haar}}_Y - M_Y$ follows $\Delta M_Y \propto N e^{-\alpha_{3,Y} t}$ with $\alpha_{3,Y} = 0.98(2)$ at $t \gtrsim 5$; see the inset. The quenched $\tilde{M}_Y$ and annealed $\overline{M}_Y$ averages approach each other with the increase of $N$.
• Figure 5: Self-averaging of SRE and GSE. For qubits, (a), and qutrits, (b), the difference $\delta M_2$ between the annealed and quenched averages of SRE scales linearly with the system size $N$, c.f., \ref{['eq:deltaM']} (for presentation purposes $\delta M_2$ is rescaled by a factor $e^{t/k_d}$ with $k_2=3$ and $k_3=5/7$). The coefficient $a_t$, shown in (c) and (d), describing the leading term in this dependence, decays exponentially in time $t$, with characteristic rates $\beta_{2,2}=0.83(3)$ and $\beta_{3,2} = 1.97(5)$. The relative error decays exponentially in time, \ref{['eq:delta2']}, see (e) and (f).

Theorems & Definitions (2)

• Theorem 1
• proof