Quantum magic dynamics in random circuits

TL;DR

This work introduces a statistical-mechanical framework to quantify quantum magic, via Wigner negativity and mana, in random quantum circuits. For Haar random circuits, it derives a precise relation between magic and entanglement, showing mana scales as ⟨M⟩ = 1/2[log d_A − S(A)], highlighting a competition where entanglement suppresses magic. In random Clifford circuits, the authors map magic dynamics to a richer spin model with boundary conditions, enabling exact descriptions of magic spreading, scrambling, and measurement-induced concentration/teleportation, all tied to coherent information. The results illuminate how non-stabilizer resources propagate in many-body systems and connect them to information-theoretic measures, with potential implications for fault-tolerant quantum computing and holography. The statistical-mechanics mapping provides a versatile tool for probing magic distributions in complex quantum dynamics and suggests avenues for extending the approach to noisy settings and field-theoretic contexts.

Abstract

Magic refers to the degree of "quantumness" in a system that cannot be fully described by stabilizer states and Clifford operations alone. In quantum computing, stabilizer states and Clifford operations can be efficiently simulated on a classical computer, even though they may appear complicated from the perspective of entanglement. In this sense, magic is a crucial resource for unlocking the unique computational power of quantum computers to address problems that are classically intractable. Magic can be quantified by measures such as Wigner negativity and mana that satisfy fundamental properties such as monotonicity under Clifford operations. In this paper, we generalize the statistical mechanical mapping methods of random circuits to the calculation of Renyi Wigner negativity and mana. Based on this, we find: (1) a precise formula describing the competition between magic and entanglement in many-body states prepared under Haar random circuits; (2) a formula describing the the spreading and scrambling of magic in states evolved under random Clifford circuits; (3) a quantitative description of magic "squeezing" and "teleportation" under measurements. Finally, we comment on the relation between coherent information and magic.

Figures (3)

• Figure 1: Haar random circuits. We act upon the initial state a circuit that consists of random two-qudit gates arranged in a brickwall pattern. After some depth $t$, we focus on a subregion $A$ and ask how much mana it has. $B$ is the complement region.
• Figure 2: Stat-mech mapping. (a) The random circuit with brickwall structure. (b) Integrating over the Haar random two-qudit gates, we get a honey-comb lattice on which the spins are placed. The spins are permutations in $S_{2n}$. They interact with each other due to the Weingarten functions (the dashed lines) and the contractions (the solid lines). (c) Integrating out the spin at the center of every downward pointing triangle, we get a triangular lattice with three-body weights on downward pointing triangles.
• Figure :

Theorems & Definitions (2)

• Definition 1: stochastic Lagrangian subspace
• Definition 2: Stochastic orthogonal group