# Quantum Chaos is Quantum

Lorenzo Leone1, Salvatore F. E. Oliviero1, You Zhou2,3, and Alioscia Hamma1

1Physics Department, University of Massachusetts Boston, 02125, USA
2School of Physical and Mathematical Sciences, Nanyang Technological University, 637371, Singapore
3Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA

### Abstract

It is well known that a quantum circuit on $N$ qubits composed of Clifford gates with the addition of $k$ non Clifford gates can be simulated on a classical computer by an algorithm scaling as $\text{poly}(N)\exp(k)$[1]. We show that, for a quantum circuit to simulate quantum chaotic behavior, it is both necessary and sufficient that $k=\Theta(N)$. This result implies the impossibility of simulating quantum chaos on a classical computer.

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### Cited by

[1] Lorenzo Leone, Salvatore F. E. Oliviero, and Alioscia Hamma, "Isospectral Twirling and Quantum Chaos", Entropy 23 8, 1073 (2021).

[2] You Zhou and Alioscia Hamma, "Entanglement of random hypergraph states", Physical Review A 106 1, 012410 (2022).

[3] Lorenzo Leone, Salvatore F. E. Oliviero, Stefano Piemontese, Sarah True, and Alioscia Hamma, "Retrieving information from a black hole using quantum machine learning", Physical Review A 106 6, 062434 (2022).

[4] Sarah True and Alioscia Hamma, "Transitions in Entanglement Complexity in Random Circuits", Quantum 6, 818 (2022).

[5] Troy J. Sewell and Christopher David White, "Mana and thermalization: Probing the feasibility of near-Clifford Hamiltonian simulation", Physical Review B 106 12, 125130 (2022).

[6] Tobias Haug and Lorenzo Piroli, "Quantifying nonstabilizerness of matrix product states", Physical Review B 107 3, 035148 (2023).

[7] Lorenzo Leone, Salvatore F. E. Oliviero, and Alioscia Hamma, "Stabilizer Rényi Entropy", Physical Review Letters 128 5, 050402 (2022).

[8] Salvatore F. E. Oliviero, Lorenzo Leone, and Alioscia Hamma, "Magic-state resource theory for the ground state of the transverse-field Ising model", Physical Review A 106 4, 042426 (2022).

[9] Salvatore F. E. Oliviero, Lorenzo Leone, Alioscia Hamma, and Seth Lloyd, "Measuring magic on a quantum processor", npj Quantum Information 8 1, 148 (2022).

[10] Kanato Goto, Tomoki Nosaka, and Masahiro Nozaki, "Probing chaos by magic monotones", Physical Review D 106 12, 126009 (2022).

[11] Tobias Haug and M.S. Kim, "Scalable Measures of Magic Resource for Quantum Computers", PRX Quantum 4 1, 010301 (2023).

[12] Jonas Haferkamp, "Random quantum circuits are approximate unitary t-designs in depth O(nt5+o(1))", Quantum 6, 795 (2022).

[13] J. Haferkamp, F. Montealegre-Mora, M. Heinrich, J. Eisert, D. Gross, and I. Roth, "Efficient Unitary Designs with a System-Size Independent Number of Non-Clifford Gates", Communications in Mathematical Physics (2022).

[14] Roy J. Garcia, Kaifeng Bu, and Arthur Jaffe, "Quantifying scrambling in quantum neural networks", Journal of High Energy Physics 2022 3, 27 (2022).

[15] Salvatore F. E. Oliviero, Lorenzo Leone, and Alioscia Hamma, "Transitions in entanglement complexity in random quantum circuits by measurements", Physics Letters A 418, 127721 (2021).

[16] Piotr Dulian and Adam Sawicki, "A random matrix model for random approximate $t$-designs", arXiv:2210.07872, (2022).

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