Local spreading of stabilizer Rényi entropy in a brickwork random Clifford circuit

TL;DR

The paper investigates how local non-stabilizerness (magic), quantified by stabilizer Rényi entropy (SRE), spreads in a many-body system evolving under brickwork random Clifford circuits. By injecting localized magic via a few T gates into an initially stabilizer product state and tracking single-qubit SRE in reduced states, the authors reveal that the total SRE decays exponentially while the normalized single-qubit SRE spreads diffusively inside the light cone, despite the lack of conserved charges. This diffusive spreading arises from the Pauli-operator dynamics under Clifford gates and is supported by analyses of two-qubit gate actions; when the gate set is restricted, the spreading becomes superdiffusive, suggesting a sensitivity to circuit structure. The work also corroborates similar non-ballistic interior spreading through appendices on robustness of magic (LROM), opening avenues for analytical understanding and exploration of universality in quantum magic transport.

Abstract

Nonstabilizerness, or magic, constitutes a fundamental resource for quantum computation and a crucial ingredient for quantum advantage. Recent progress has substantially advanced the characterization of magic in many-body quantum systems, with stabilizer Rényi entropy (SRE) emerging as a computable and experimentally accessible measure. In this work, we investigate the spreading of SRE in terms of single-qubit reduced density matrices, where an initial product state that contains magic in a local region evolves under brickwork random Clifford circuits. For the case with Haar-random local Clifford gates, we find that the spreading profile exhibits a diffusive structure within a ballistic light cone when viewed through a normalized version of single-qubit SRE, despite the absence of explicit conserved charges. We further examine the robustness of this non-ballistic behavior of the normalized single-qubit SRE spreading by extending the analysis to a restricted Clifford circuit, where we unveil a superdiffusive spreading.

Figures (9)

• Figure 1: An illustration of the results for local SRE spreading in a brickwork random Clifford circuit. A system of $L$ qubits at $t=0$ is prepared in an initial product state containing local magic (orange circle). The circuit comprises alternating layers of random two-qubit Clifford gates (blue square), uniformly chosen from the Clifford group $\mathcal{C}_2$, arranged in a staggered brickwork pattern. Under the time evolution, the spreading profile of the single-qubit averaged SRE exhibits a diffusive structure within the light cone. If we instead choose the two-qubit Clifford gates from restricted elements (see Fig. \ref{['fig:circuit_cnoths']}) in the Clifford group, we find a superdiffusive structure.
• Figure 2: Spreading of the averaged single-qubit SRE, $\overline{M_i(t)}$, as a function of site $i$ and depth (time) $t$ of the circuit. The initial state $\ket{\psi_0}$ contains local magic at site $i=15$ of a system of total size $L=30$. As the quantity $\overline{M_i(t)}$ is exponentially decaying with time, we plot its logarithmic value for better visualization. The average is taken over $10^6$ circuit realizations.
• Figure 3: (a) Exponential decay of the sum of averaged single-qubit SREs, $\mathcal{M}(t)$, as given in Eq. \ref{['eq:expo_decay']}. The decay rate $\Gamma=0.44$ is indicated by the black dashed reference line with slope $-0.44$. (b) Variation of the normalized single-qubit SREs, $a_i(t)$, as a function of site $i$ at different time steps $t=2, 4, 6, 8, 10$. The solid lines represent numerical data, while the dashed lines show the corresponding values obtained from the discrete diffusion equation (Eq. \ref{['eq:dis_diff']}) using the numerical data from the previous time step. The description by the diffusion equation almost agrees with the actual dynamics. The total system size is $L = 30$. In both panels, all results are averaged over $10^6$ realizations.
• Figure 4: Input (time $t$) and output (time $t+1$) single-qubit SREs across a Clifford gate $C_{i,i+1}^{(t)}$ acting on neighboring sites $i$ and $i+1$. The averaged output values are equal $\overline{M_i(t+1)}=\overline{M_{i+1}(t+1)}$. Similarly, at time step $t$, the outputs of the gates in the previous layers satisfy $\overline{M_{i-1}(t)}=\overline{M_{i}(t)}$ and $\overline{M_{i+1}(t)}=\overline{M_{i+2}(t)}$.
• Figure 5: Variation of $\alpha(i,t)$ as a function of the Clifford gate position $i$ within the lightcone at different (a) even time steps $t = 6,8,10,12$, and (b) odd time steps $t = 7, 9, 11, 13$. In both panels, the position of the light cone boundaries corresponding to each time step is shown by the vertical dashed lines of the same color. The results show that the quantity $\alpha(i,t)$ is approximately independent of both $i$ and $t$, and equal to $e^{-\Gamma}=0.644$ (indicated by the solid gray line), at least well inside the light cone.