Measuring nonstabilizerness via multifractal flatness

Abstract

Universal quantum computing requires nonstabilizer (magic) quantum states. Quantifying the nonstabilizerness and relating it to other quantum resources is vital for characterizing the complexity of quantum many-body systems. In this work, we prove that a quantum state is a stabilizer if and only if all states belonging to its Clifford orbit have a flat probability distribution on the computational basis. This implies, in particular, that multifractal states are nonstabilizers. We introduce multifractal flatness, a measure based on the participation entropy that quantifies the wave-function distribution flatness. We demonstrate that this quantity is analytically related to the stabilizer entropy of the state and present several examples elucidating the relationship between multifractality and nonstabilizerness. In particular, we show that the multifractal flatness provides an experimentally and computationally viable nonstabilizerness certification. Our work unravels the direct relation between the nonstabilizerness of a quantum state and its wave-function structure.

6 pages, Comments are welcome!

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• Efficient quantum algorithms for stabilizer entropies
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• Critical behaviors of non-stabilizerness in quantum spin chains
• Entanglement-magic separation in hybrid quantum circuits
• Pseudomagic Quantum States
• Magic in generalized Rokhsar-Kivelson wavefunctions
• Entanglement phases, localization and multifractality of monitored free fermions in two dimensions
• Nonstabilizerness of Permutationally Invariant Systems
• Magic Resources of the Heisenberg Picture
• Hilbert space delocalization under random unitary circuits
• Probing quantum complexity via universal saturation of stabilizer entropies
• Hybrid Stabilizer Matrix Product Operator
• Efficient mutual magic and magic capacity with matrix product states
• Non-stabilizerness in kinetically-constrained Rydberg atom arrays
• Independent stabilizer Rényi entropy and entanglement fluctuations in random unitary circuits
• Anticoncentration and state design of random tensor networks
• Quantum Algorithms for Inverse Participation Ratio Estimation in multi-qubit and multi-qudit systems
• Non-Stabilizerness of Sachdev-Ye-Kitaev Model
• Many-body entropies and entanglement from polynomially-many local measurements
• Nonstabilizerness dynamics in many-body localized systems
• Long-range nonstabilizerness and phases of matter
• Stabilizer Rényi Entropy and Conformal Field Theory
• Bridging Entanglement and Magic Resources within Operator Space
• Magic transition in measurement-only circuits
• Extracting randomness from magic quantum states
• Interplay of entanglement structures and stabilizer entropy in spin models
• A nonstabilizerness monotone from stabilizerness asymmetry
• The non-stabilizerness of fermionic Gaussian states
• Efficient witnessing and testing of magic in mixed quantum states
• Stabilizer Rényi Entropy Encodes Fusion Rules of Topological Defects and Boundaries
• Maximal Magic for Two-qubit States
• Computing quantum magic of state vectors
• Universal distributions of overlaps from generic dynamics in quantum many-body systems
• Non-stabilizerness in quantum-enhanced metrological protocols
• Anticoncentration and State Design of Doped Real Clifford Circuits and Tensor Networks
• Certifying nonstabilizerness in quantum processors
• The Magic Barrier before Thermalization
• Robustness of Magic in the quantum Ising chain via Quantum Monte Carlo tomography
• Non-stabilizerness and entanglement from cat-state injection
• Connecting Magic Dynamics in Thermofield Double States to Spectral Form Factors
• Anticoncentration in Clifford Circuits and Beyond: From Random Tensor Networks to Pseudo-Magic States
• Spectral signatures of nonstabilizerness and criticality in infinite matrix product states
• Estimation of multivariate traces of states given partial classical information
• Van Hove singularities in stabilizer entropy densities
• High-expressibility Quantum Neural Networks using only classical resources