Quantum Chaos is Quantum
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
Published: | 2021-05-04, volume 5, page 453 |
Eprint: | arXiv:2102.08406v3 |
Doi: | https://doi.org/10.22331/q-2021-05-04-453 |
Citation: | Quantum 5, 453 (2021). |
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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.

Featured image: $\mathit{Left}$ : Scheme of the $4$-Doped Clifford circuit. $\mathit{Right}$ : Detail of $K_{C_{4}}$, a unitary single-qubit non Clifford gate $K$ evolved adjointly by a Clifford circuit $C_{4}$. Note that the set formed by these circuits is equivalent to the set of doped Clifford circuits, i.e circuits composed by Clifford unitaries $C_{i}$ interspersed with single-qubit non Clifford gates $K_{i}$.
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