Spectral Properties Versus Magic Generation in $T$-doped Random Clifford Circuits

Abstract

We study the emergence of complexity in deep random $N$-qubit $T$-gate doped Clifford circuits, as reflected in their spectral properties and in magic generation, characterized by the stabilizer Rényi entropy distribution and the non-stabilizing power of the circuit. For pure (undoped) Clifford circuits, a unique periodic orbit structure in the space of Pauli strings implies peculiar spectral correlations and level statistics with large degeneracies. $T$-gate doping induces an exponentially fast transition to chaotic behavior, described by random matrix theory. We compare these complexity indicators with magic generation properties of the Clifford+$T$ ensemble, and determine the distribution of magic, as well as the average non-stabilizing power of the quantum circuit ensemble. In the dilute limit, $N_T\ll N$, magic generation is governed by single-qubit behavior. Magic is generated in approximate quanta, increases approximately linearly with the number of $T$-gates, $N_T$, and displays a discrete distribution for small $N_T$. At $N_T\approx N$, the distribution becomes quasi-continuous, and for $N_T\gg N$ it converges to that of Haar-random unitaries, and averages to a finite magic density, $m_2$, ${\lim }_{N\to \infty }⟨m_2{⟩}_{\text{Haar}}=1$. This is in contrast to the spectral transition, where $\mathcal{O}(1)$ $T$-gates suffice to remove spectral degeneracies and to induce a transition to chaotic behavior in the thermodynamic limit. Magic is therefore a more sensitive indicator of complexity.

13 pages, 9 figures

References in corpus (43)

• Efficient classical simulation of slightly entangled quantum computations
• Universal Quantum Computation with ideal Clifford gates and noisy ancillas
• Improved Simulation of Stabilizer Circuits
• Quantum Resource Theories
• A Theory of Fault-Tolerant Quantum Computation
• On the role of entanglement in quantum computational speed-up
• Quantum Entanglement Growth Under Random Unitary Dynamics
• Operator hydrodynamics, OTOCs, and entanglement growth in systems without conservation laws
• Ballistic spreading of entanglement in a diffusive nonintegrable system
• The Resource Theory of Stabilizer Computation
• Application of a resource theory for magic states to fault-tolerant quantum computing
• A Game of Surface Codes: Large-Scale Quantum Computing with Lattice Surgery
• Improved classical simulation of quantum circuits dominated by Clifford gates
• Stabilizer Rényi entropy
• Magic State Distillation: Not as Costly as You Think
• On entanglement spreading in chaotic systems
• Multiqubit Clifford groups are unitary 3-designs
• Onset of Random Matrix Behavior in Scrambling Systems
• Quantifying the magic of quantum channels
• Quantifying nonstabilizerness of matrix product states
• Stabilizer entropies and nonstabilizerness monotones
• Robustness of Magic and Symmetries of the Stabiliser Polytope
• Measuring magic on a quantum processor
• Hadamard-free circuits expose the structure of the Clifford group
• Qutrit Magic State Distillation
• Efficiently computable bounds for magic state distillation
• Nonstabilizerness via matrix product states in the Pauli basis
• Phase transition in magic with random quantum circuits
• Single T gate in a Clifford circuit drives transition to universal entanglement spectrum statistics
• Towards low overhead magic state distillation
• Magic spreading in random quantum circuits
• Efficient quantum algorithms for stabilizer entropies
• Dynamical Magic Transitions in Monitored Clifford+T Circuits
• Entanglement growth in diffusive systems
• A semi-classical limit for the many-body localization transition
• Classical simulatability, entanglement breaking, and quantum computation thresholds
• Efficient unitary designs with a system-size independent number of non-Clifford gates
• Stabilizer entropy dynamics after a quantum quench
• Transitions in Entanglement Complexity in Random Circuits
• A single $T$-gate makes distribution learning hard
• Stabilizer entropy of quantum tetrahedra
• Independent stabilizer Rényi entropy and entanglement fluctuations in random unitary circuits
• Extracting randomness from magic quantum states

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• Local spreading of stabilizer Rényi entropy in a brickwork random Clifford circuit
• Nonstabilizerness generation in a multiparticle quantum walk
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• Optimal quantum reservoir learning in proximity to universality
• Van Hove singularities in stabilizer entropy densities