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: $T$-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.

5 pages + appendices, 2 figures, comments welcome! v2/3: added references, restructured results

References in corpus (66)

• Entanglement Entropy and Quantum Field Theory
• Area laws for the entanglement entropy - a review
• Entanglement in quantum critical phenomena
• Improved Simulation of Stabilizer Circuits
• Evolution of Entanglement Entropy in One-Dimensional Systems
• An Area Law for One Dimensional Quantum Systems
• Matrix product states represent ground states faithfully
• Quantum Entanglement Growth Under Random Unitary Dynamics
• Integration with respect to the Haar measure on unitary, orthogonal and symplectic group
• Quantum Computation as Geometry
• Entropy scaling and simulability by Matrix Product States
• Application of a resource theory for magic states to fault-tolerant quantum computing
• Entanglement Entropy dynamics in Heisenberg chains
• Stabilizer Rényi entropy
• Simulation of quantum circuits by low-rank stabilizer decompositions
• Operator space entanglement entropy in transverse Ising chain
• Matchgates and classical simulation of quantum circuits
• On entanglement spreading in chaotic systems
• Operator Entanglement in Interacting Integrable Quantum Systems: the Case of the Rule 54 Chain
• Linear growth of quantum circuit complexity
• Entanglement scaling of operators: a conformal field theory approach, with a glimpse of simulability of long-time dynamics in 1+1d
• Is efficiency of classical simulations of quantum dynamics related to integrability?
• Introduction to Haar Measure Tools in Quantum Information: A Beginner's Tutorial
• Operator Entanglement in Local Quantum Circuits I: Chaotic Dual-Unitary Circuits
• Operator entanglement entropy of the time evolution operator in chaotic systems
• Stabilizer entropies and nonstabilizerness monotones
• Schur-Weyl Duality for the Clifford Group with Applications: Property Testing, a Robust Hudson Theorem, and de Finetti Representations
• Operator Space Entanglement Entropy in XY Spin Chains
• Conformal field theories are magical
• Quantum Chaos is Quantum
• Emergent irreversibility and entanglement spectrum statistics
• Density Matrix Renormalization Group in the Heisenberg Picture
• Stabilizer entropies are monotones for magic-state resource theory
• Fast and converged classical simulations of evidence for the utility of quantum computing before fault tolerance
• Resource theory of non-Gaussian operations
• Operator Entanglement in Local Quantum Circuits II: Solitons in Chains of Qubits
• Scrambling is Necessary but Not Sufficient for Chaos
• Measuring nonstabilizerness via multifractal flatness
• Phase transition in magic with random quantum circuits
• Nonstabilizerness determining the hardness of direct fidelity estimation
• Magic spreading in random quantum circuits
• Dynamical Magic Transitions in Monitored Clifford+T Circuits
• Recovering quantum gates from few average gate fidelities
• Critical properties of homogeneous binary trees
• Resource theory of quantum scrambling
• Efficient unitary designs with a system-size independent number of non-Clifford gates
• Non-stabilizerness versus entanglement in matrix product states
• Augmenting Density Matrix Renormalization Group with Clifford Circuits
• Learning t-doped stabilizer states
• Entanglement-magic separation in hybrid quantum circuits
• Diffusion and operator entanglement spreading
• Lower bound for the T count via unitary stabilizer nullity
• Learning efficient decoders for quasi-chaotic quantum scramblers
• Stabilizer Tensor Networks: universal quantum simulator on a basis of stabilizer states
• Fast estimation of outcome probabilities for quantum circuits
• Entanglement of Local Operators and the Butterfly Effect
• Unscrambling Quantum Information with Clifford decoders
• Magic-induced computational separation in entanglement theory
• Magic Resources of the Heisenberg Picture
• Quantifying non-stabilizerness via information scrambling
• Complexity of quantum circuits via sensitivity, magic, and coherence
• Real-time operator evolution in two and three dimensions via sparse Pauli dynamics
• Stabilizer Tensor Networks with Magic State Injection
• Improved simulation of quantum circuits dominated by free fermionic operations
• Absence of localization in two-dimensional Clifford circuits
• Polynomially restricted operator growth in dynamically integrable models

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