Efficient mutual magic and magic capacity with matrix product states

Abstract

Stabilizer Rényi entropies (SREs) probe the non-stabilizerness (or magic) of many-body systems and quantum computers. Here, we introduce the mutual von-Neumann SRE and magic capacity, which can be efficiently computed in time $O(N{\chi }^3)$ for matrix product states (MPSs) of bond dimension $\chi$. We find that mutual SRE characterizes the critical point of ground states of the transverse-field Ising model, independently of the chosen local basis. Then, we relate the magic capacity to the anti-flatness of the Pauli spectrum, which quantifies the complexity of computing SREs. The magic capacity characterizes transitions in the ground state of the Heisenberg and Ising model, randomness of Clifford+T circuits, and distinguishes typical and atypical states. Finally, we make progress on numerical techniques: we design two improved Monte-Carlo algorithms to compute the mutual $2$-SRE, overcoming limitations of previous approaches based on local update. We also give improved statevector simulation methods for Bell sampling and SREs with $O(8^{N/2})$ time and $O(2^N)$ memory, which we demonstrate for $24$ qubits. Our work uncovers improved approaches to study the complexity of quantum many-body systems.

13+10 pages, 6+9 figures

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• Stabilizer Rényi Entropy Encodes Fusion Rules of Topological Defects and Boundaries
• Local spreading of stabilizer Rényi entropy in a brickwork random Clifford circuit
• Rise and fall of nonstabilizerness via random measurements
• Robustness of Magic in the quantum Ising chain via Quantum Monte Carlo tomography
• Analyzing the free states of one quantum resource theory as resource states of another
• Spectral signatures of nonstabilizerness and criticality in infinite matrix product states
• Optimal quantum reservoir learning in proximity to universality
• Connecting Magic Dynamics in Thermofield Double States to Spectral Form Factors
• Van Hove singularities in stabilizer entropy densities
• Bell sampling in Quantum Monte Carlo simulations
• Limits of Clifford Disentangling in Tensor Network States