Magic spreading under unitary Clifford dynamics

Abstract

Nonstabilizerness, or quantum magic, presents a valuable resource in quantum error correction and computation. We study the dynamics of locally injected magic in unitary Clifford circuits, where the total magic is conserved. However, the absence of physical observables quantifying magic precludes a direct microscopic or hydrodynamic description of its local distribution and dynamics. Using insights from stabilizer quantum error correcting codes, we rigorously show that the spatial distribution of magic can be inferred from a canonical representation of low-magic states, dubbed the bipartite magic gauge. Moreover, we propose two operationally relevant magic length scales. We numerically establish that, at early times, both length scales grow ballistically at distinct velocities set by the entanglement velocity, after which magic delocalizes. Our work sheds light on the spatiotemporal structure of quantum resources and complexity in many-body dynamics, opening up avenues for investigating their transport properties and further connections with quantum error correction.

Figures (13)

• Figure 1: Magic spreading in a $T$-doped random Clifford circuit with $L$ qubits, time $t$, and a brickwall of random 2-qubit Clifford gates (grey). Magic is locally injected by a $T$ gate acting on a short-range entangled stabilizer state with the two central qubits in a Bell pair. The typical size of the smallest contiguous region where magic can be extracted (red ellipse) spreads at the entanglement velocity $v_{\rm E}$ (red cone). The union of the smallest mutually independent such intervals (blue and red ellipses) indicates the region outside of which operations cannot reduce the magic (blue cone), which spreads at $2v_{\rm E}$.
• Figure 2: Magic length scales: typical magic length $\ell_{\rm typ}$ and full linear extent of magic $W$, defined in terms of mutually-irreducible and minimal-length contiguous subsystems $A_1, \dots, A_k$ where magic is extractable from $\ket{\psi}$.
• Figure 3: Growth of magic length scales over time in random Clifford circuits for $L=102$ and identity doping rate $p=0$. Blue (orange) circles show FLEOM (LML). At early times, both MLS grow ballistically at fitted rates $2v_W$ and $2v_{\rm \ell}$, respectively (dashed lines). Inset: $p$-dependence of $v_W$ and $v_{\rm \ell}$ compared to the entanglement velocity $v_{\rm E}(p)$. Data averaged over 2$\times 10^3$ circuit realizations, negligible error bars not shown.
• Figure 4: Channel capacity proxy $\tilde{\cal C}$ of the channel that traces out $fL=|B|$ randomly chosen qubits after evolution up to time $t$ under a random brickwork Clifford circuit with identity doping $p=0.1$ (circles) and a global random Clifford (crosses). For $t\gtrsim t_{\rm sat}$, $\tilde{\cal C} = {\cal O}(1)$ for $f \leq 0.5$ indicating the preservation of magic; for $f \gtrsim 0.5$, $\tilde{\cal C}$ quickly drops to zero and magic is lost (see inset). Data averaged over $10^4$ realizations of $B$ for a single random circuit instance, negligible error bars not shown.
• Figure 5: Representation of a $[[n,k]]$ stabilizer code with $n$ qubits and $k$ logical qubits as a pure stabilizer state $\ket{\Phi_{ABR}}$ on an enlarged system $ABR$, with $n=|AB|$ and $k=|R|$. $\ket{\Phi_{ABR}}$ is the Choi state of the isometry $V_{\rm encoding}$. The spatial structure of the logical operators, for any subsystem $A$ and its complement $B$, can be inferred from the spatial structure of the stabilizers of $\ket{\Phi_{ABR}}$, as discussed in the proof of \ref{['lemma:bip']}.

Theorems & Definitions (2)

• Theorem 1
• Lemma 1