Pauli Spectrum and Non-stabilizerness of Typical Quantum Many-Body States

Abstract

An important question of quantum information is to characterize genuinely quantum (beyond-Clifford) resources necessary for universal quantum computing. Here, we use the Pauli spectrum to quantify how magic, beyond Clifford, typical many-qubit states are. We first present a phenomenological picture of the Pauli spectrum based on quantum typicality and then confirm it for Haar random states. We then introduce filtered stabilizer entropy, a magic measure that can resolve the difference between typical and atypical states. We proceed with the numerical study of the Pauli spectrum of states created by random circuits as well as for eigenstates of chaotic Hamiltonians. We find that in both cases the Pauli spectrum approaches the one of Haar random states, up to exponentially suppressed tails. Our results underscore differences between typical and atypical states from the point of view of quantum information.

4.5 pages + 4 SM pages, Comments are welcome!

References in corpus (56)

• Thermalization and its mechanism for generic isolated quantum systems
• Quantum Many-Body Scars and Weak Breaking of Ergodicity
• Random Quantum Circuits
• Restrictions on Transversal Encoded Quantum Gate Sets
• Entanglement spectrum in one-dimensional systems
• Application of a resource theory for magic states to fault-tolerant quantum computing
• Stabilizer Rényi entropy
• Typicality for Generalized Microcanonical Ensembles
• Anomalous thermalization in ergodic systems
• Many-body localization dynamics from gauge invariance
• Polynomially filtered exact diagonalization approach to many-body localization
• Ergodic and non-ergodic dual-unitary quantum circuits with arbitrary local Hilbert space dimension
• Many-Body Localization in the Age of Classical Computing
• Universal behavior beyond multifractality of wave-functions at measurement--induced phase transitions
• Eigenstate Thermalization in a Locally Perturbed Integrable System
• Quantifying nonstabilizerness of matrix product states
• Stabilizer entropies and nonstabilizerness monotones
• Measuring magic on a quantum processor
• Measurement-induced phase transitions in $(d+1)$-dimensional stabilizer circuits
• Conformal field theories are magical
• Many-body magic via Pauli-Markov chains -- from criticality to gauge theories
• Eigenstate Thermalization Hypothesis and Free Probability
• Eigenstate thermalization hypothesis beyond standard indicators: Emergence of random-matrix behavior at small frequencies
• Magic-state resource theory for the ground state of the transverse-field Ising model
• Random quantum circuits are approximate unitary $t$-designs in depth $O\left(n{t}^{5+o(1)}\right)$
• Dynamical purification and the emergence of quantum state designs from the projected ensemble
• Low-frequency behavior of off-diagonal matrix elements in the integrable XXZ chain and in a locally perturbed quantum-chaotic XXZ chain
• Solvable model of deep thermalization with distinct design times
• Measuring nonstabilizerness via multifractal flatness
• Eigenstate thermalization hypothesis and its deviations from random-matrix theory beyond the thermalization time
• Phase transition in magic with random quantum circuits
• Non-negative Wigner functions in prime dimensions
• Efficient quantum algorithms for stabilizer entropies
• Dynamical Magic Transitions in Monitored Clifford+T Circuits
• Eigenstate thermalization in dual-unitary quantum circuits: Asymptotics of spectral functions
• Out-of-time-order correlations and the fine structure of eigenstate thermalisation
• Stability of many-body localization in Floquet systems
• Improved upper bounds on the stabilizer rank of magic states
• Learning t-doped stabilizer states
• Entanglement-magic separation in hybrid quantum circuits
• Critical behaviors of non-stabilizerness in quantum spin chains
• Transitions in entanglement complexity in random quantum circuits by measurements
• Pseudomagic Quantum States
• Lower bound for the T count via unitary stabilizer nullity
• Magic in generalized Rokhsar-Kivelson wavefunctions
• Single-particle eigenstate thermalization in quantum-chaotic quadratic Hamiltonians
• Phases and dynamics of ultracold bosons in a tilted optical lattice
• Statistical mechanics model for Clifford random tensor networks and monitored quantum circuits
• Error-resilience Phase Transitions in Encoding-Decoding Quantum Circuits
• Phase transition in Stabilizer Entropy and efficient purity estimation
• Magic of quantum hypergraph states
• Entanglement Growth and Minimal Membranes in $(d+1)$ Random Unitary Circuits
• Tight-binding billiards
• Microcanonical windows on quantum operators
• Anticoncentration and state design of random tensor networks
• Emergence of unitary symmetry of microcanonically truncated operators in chaotic quantum systems

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• Probing quantum complexity via universal saturation of stabilizer entropies
• Stabilizer Entropy and entanglement complexity in the Sachdev-Ye-Kitaev model
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• Efficient mutual magic and magic capacity with matrix product states
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• Independent stabilizer Rényi entropy and entanglement fluctuations in random unitary circuits
• Non-Stabilizerness of Sachdev-Ye-Kitaev Model
• Nonstabilizerness dynamics in many-body localized systems
• Nonstabilizerness in the unitary and monitored quantum dynamics of XXZ-staggered and SYK models
• Long-range nonstabilizerness and phases of matter
• Stabilizer Rényi Entropy and Conformal Field Theory
• Interplay of entanglement structures and stabilizer entropy in spin models
• Extracting randomness from magic quantum states
• The non-stabilizerness of fermionic Gaussian states
• Fermionic Magic Resources of Quantum Many-Body Systems
• Stabilizer Scars
• Efficient witnessing and testing of magic in mixed quantum states
• Operational interpretation of the Stabilizer Entropy
• Stabilizer Rényi Entropy Encodes Fusion Rules of Topological Defects and Boundaries
• Computing quantum magic of state vectors
• Anticoncentration and State Design of Doped Real Clifford Circuits and Tensor Networks
• Non-stabilizerness in quantum-enhanced metrological protocols
• Local spreading of stabilizer Rényi entropy in a brickwork random Clifford circuit
• A trace distance-based geometric analysis of the stabilizer polytope for few-qubit systems
• Momentum space magic for the transverse field quantum Ising model
• Nonstabilizerness generation in a multiparticle quantum walk
• Nonstabilizerness and Error Resilience in Noisy Quantum Circuits
• Rise and fall of nonstabilizerness via random measurements
• Robustness of Magic in the quantum Ising chain via Quantum Monte Carlo tomography
• The Magic Barrier before Thermalization
• Connecting Magic Dynamics in Thermofield Double States to Spectral Form Factors
• Quantum Magic in Discrete-Time Quantum Walk
• Anticoncentration in Clifford Circuits and Beyond: From Random Tensor Networks to Pseudo-Magic States
• Entanglement and magic on the light-front
• Complexity of Quadratic Quantum Chaos
• On the Hardness of Measuring Magic
• On the stabilizer complexity of Hawking radiation
• No-cost Bell nonlocality certification from quantum tomography and its applications in quantum-magic-resource witnessing
• High-expressibility Quantum Neural Networks using only classical resources
• Limits of Clifford Disentangling in Tensor Network States
• Van Hove singularities in stabilizer entropy densities
• Quantum Complexity in Rule-Based Constrained Many-Body Models: Scars, Fragmentation, and Chaos