Magic Resources of the Heisenberg Picture

Abstract

We study a non-stabilizerness resource theory for operators, which is dual to that describing states. We identify that the stabilizer Rényi entropy analog in operator space is a good magic monotone satisfying the usual conditions while inheriting efficient computability properties and providing a tight lower bound to the minimum number of non-Clifford gates in a circuit. Operationally, this measure quantifies how well an operator can be approximated by one with only a few Pauli strings -- analogous to how entanglement entropy relates to tensor-network truncation. A notable advantage of operator stabilizer entropies is their inherent locality, as captured by a Lieb-Robinson bound. This feature makes them particularly suited for studying local dynamical magic resource generation in many-body systems. We compute this quantity analytically in two distinct regimes. First, we show that under random evolution, operator magic typically reaches near-maximal value for all Rényi indices, and we evaluate the Page correction. Second, harnessing both dual unitarity and ZX graphical calculus, we solve the operator stabilizer entropy for interacting integrable XXZ circuit, finding that it quickly saturates to a constant value. Overall, this measure sheds light on the structural properties of many-body non-stabilizerness generation and can inspire Clifford-assisted tensor network methods.

Figures (4)

• Figure 1: Schematic of the comparison of magic resources for states versus operators; cf. Eq. \ref{['eq:fundamental']}. (a) In the Schrödinger picture, magic resources $M$ tend to grow exponentially fast for a local circuit or dynamics. (b) In the Heisenberg picture, the growth of magic resource $\mathcal{M}$ is bounded by a Lieb-Robinson light cone for an initially local operator $O$. (c) Operationally, any operator which can be well-approximated by an operator with only a polynomial number $\chi$ of Pauli coefficients must have slowly growing operator magic resource: $\mathcal{M}\sim \mathcal{O}(\log(t))$.
• Figure 2: Tensor network diagram for the generalized Pauli purity, leading to the OSE \ref{['eq:zio']}, for an initially local operator $O$ under brickwork circuit dynamics. Time goes from top to bottom, and each brick represents a doubled-picture two-site unitary, $U \otimes U^*$. At the bottom we have $\Lambda$ tensors from Eq. \ref{['eq:Lambda']}, and at the light cone edges white bullets representing vectorised identity $\ket{\circ} = \sum_i \ket{ii}$, connecting the copies $U$ to $U^*$. Black dots represent the initial local operator $O$, and the normalization is not shown.
• Figure 3: A summary of graphical rewrite rules in ZX calculus. These are: $(f)$usion of spiders, the $(c)$opy rule, ($\pi$)-commutation, $(id)$entity, $(b)$ialbegra rule, and the $(H)$opf rule. The Hopf rule can be derived from the others but turns out to be particularly useful.
• Figure 4: Plot of the growth of $\mathcal{M}^{(2)}(U^\dagger_t O_j U_t)$ in the dual unitary XXZ model for $(a_x,a_y,a_z)= (\sqrt{0.7},\sqrt{0.2},\sqrt{0.1})$ and $J=\pi/8$. The plot asymptotes to the value given in Eq. \ref{['eq:asymptotic']}; $\lim_{t \to \infty} \mathcal{M}^{(2)}(U^\dagger_t O_j U_t) \approx 3.22$ for these parameters.

Theorems & Definitions (2)

• Theorem
• proof