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Bridging Entanglement and Magic Resources within Operator Space

Neil Dowling, Kavan Modi, Gregory A. L. White

Abstract

Local-operator entanglement (LOE) dictates the complexity of simulating Heisenberg evolution using tensor network methods, {and bears witness to many-body chaos for local dynamics}. We show that LOE is also sensitive to how non-Clifford a unitary is: its magic resources. In particular, we prove that LOE is always upper-bound by three distinct magic monotones: $T$-count, unitary nullity, and operator stabilizer Rényi entropy. Moreover, in the average case for large, random circuits, LOE and magic monotones approximately coincide. Our results imply that an operator evolution that is expensive to simulate using tensor network methods must also be inefficient using both stabilizer and Pauli truncation methods. {In terms of a previous conjecture on the characteristic scaling of LOE, our results also mean that non-integrable spin chains cannot be simulated classically}. Entanglement in operator space therefore measures a unified picture of non-classical resources, in stark contrast to the Schrödinger picture.

Bridging Entanglement and Magic Resources within Operator Space

Abstract

Local-operator entanglement (LOE) dictates the complexity of simulating Heisenberg evolution using tensor network methods, {and bears witness to many-body chaos for local dynamics}. We show that LOE is also sensitive to how non-Clifford a unitary is: its magic resources. In particular, we prove that LOE is always upper-bound by three distinct magic monotones: -count, unitary nullity, and operator stabilizer Rényi entropy. Moreover, in the average case for large, random circuits, LOE and magic monotones approximately coincide. Our results imply that an operator evolution that is expensive to simulate using tensor network methods must also be inefficient using both stabilizer and Pauli truncation methods. {In terms of a previous conjecture on the characteristic scaling of LOE, our results also mean that non-integrable spin chains cannot be simulated classically}. Entanglement in operator space therefore measures a unified picture of non-classical resources, in stark contrast to the Schrödinger picture.

Paper Structure

This paper contains 6 sections, 11 theorems, 71 equations, 2 figures, 1 table.

Table of Contents

  1. Relation between local-operator entanglement and state entanglement
  2. Upper-bounding LOE by magic monotones
  3. Bounding Rényi LOE from average operator purity in replica space
  4. Clifford and Haar averages
  5. Average operator purity over the $\nu$-compressible ensemble
  6. Average operator purity over the $T$-doped Clifford ensemble

Key Result

Theorem 1

For Rényi indices of $\alpha \leq 2$, and $N_A=N/2$, the average LOE is bounded as Here, the lower-bounds are always positive, and $\mu_\tau$ and $\mu_\nu$ refer to the $T$-doped Clifford and $\nu$-compressible ensembles respectively.

Figures (2)

  • Figure 1: Depiction of key quantities investigated in this work. (a) The local-operator entanglement (LOE) is the entanglement $E_A^{(\alpha)}$ of the Choi state of an operator, $|O_U\rangle\!\rangle$, across some (doubled-space) spatial bipartition $A:\bar{A}$. (b) The extensive scaling of this quantity has been observed to be related to chaos in many-body systems, both in the sense of it necessarily indicating information scrambling dowling2023scrambling and witnessing the non-integrability of spin chains Prosen2007. (c) In addition to being sensitive to the entanglement of a unitary (cf. App. \ref{['ap:oe']}), we show that the LOE equally probes magic resources and operator stabilizer entropies, unifying the three corresponding simulation techniques under the one umbrella.
  • Figure 2: Any unitary can be decomposed as a global Clifford unitaries, $C_0,C_1 \in \mathcal{C}_N$, sandwiching a non-Clifford unitary on only $\nu(U)/2$ qubits, $V \in \mathcal{U}_\nu$ (top), where $\nu(U)$ is the unitary nullity Jiang2023. A unitary with $T$-count $\tau(U)$ can be decomposed as layers of $N$-qubit Clifford unitaries $C_i\in\mathcal{C}_N$ interspersed with $\tau(U)$ single-site $T$-gates (bottom). The $\nu$-compressible ($\mu_\nu$) and $T$-doped ($\mu_\tau$) ensembles are constructed from uniformly sampling these components over the Clifford and unitary groups as appropriate.

Theorems & Definitions (17)

  • Theorem 1
  • Theorem 2
  • Corollary 3
  • Proposition 4
  • proof
  • Proposition 5
  • proof
  • Proposition 6
  • proof
  • Proposition 7
  • ...and 7 more