Exact Operator Complexity Measures from Random Matchgate Unitaries

arXiv:2609.38300 · quant-ph, cond-mat.stat-mech, math-ph, math.MP · Submitted 2026-09-29 · Read on arXiv

quant-ph, cond-mat.stat-mech, math-ph, math.MP

Submitted: 2026-09-29

Updated: 2026-09-29

Comments: 19 pages main + 16 pages appendix; many figures. Comments welcome!

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Terminology

• Entanglement — 69× in this paper · explained in 4 episodes
• Operator Entanglement — 34× in this paper · explained in 2 episodes
• Hilbert Space — 9× in this paper · explained in 2 episodes
• Non-Gaussianity — 9× in this paper · explained in 2 episodes
• Trace — 8× in this paper · explained in 2 episodes
• Quantum Circuits — 4× in this paper · explained in 2 episodes
• Entanglement Entropy — 3× in this paper · explained in 3 episodes
• Conservation Laws — 2× in this paper · explained in 3 episodes
• SYK Model — 2× in this paper · explained in 2 episodes
• Admissibility — 1× in this paper · explained in 2 episodes
• Chaotic Dynamics — 1× in this paper · explained in 2 episodes
• Shadow Tomography — 1× in this paper · explained in 4 episodes

Sources

• Classical Simulability from Operator Entanglement Scaling
• Learning, Optimizing, and Simulating Fermions with Quantum Computers
• Optimal Haar random fermionic linear optics circuits
• Theory of the Matchgate Commutant
• The commutant of fermionic Gaussian unitaries
• Fermionic Averaged Circuit Eigenvalue Sampling
• Page Curve for Local-Operator Entanglement from Free Probability
• Noise-induced Simulability Transition from Operator Scrambling
• Free Independence and Unitary Design from Random Matrix Product Unitaries
• Exact Enumeration and Sampling of Matrices with Specified Margins
• Coarse-grained dynamics of operator and state entanglement
• Natural super-orbitals representation of many-body operators
• Entangling power and quantum circuit complexity
• Simulation of Fermionic circuits using Majorana Propagation
• Resolving space-time structures of quantum impurities with a numerically exact few-body algorithm
• Classical shadows over symmetric spaces
• Continuum field theory of matchgate tensor network ensembles

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