Single T gate in a Clifford circuit drives transition to universal entanglement spectrum statistics

Abstract

Clifford circuits are insufficient for universal quantum computation or creating tt-designs with t\ge 4t≥4. While the entanglement entropy is not a telltale of this insufficiency, the entanglement spectrum of a time evolved random product state is: the entanglement levels are Poisson-distributed for circuits restricted to the Clifford gate-set, while the levels follow Wigner-Dyson statistics when universal gates are used. In this paper we show, using finite-size scaling analysis of different measures of level spacing statistics, that in the thermodynamic limit, inserting a single T (\pi/8)(π/8) gate in the middle of a random Clifford circuit is sufficient to alter the entanglement spectrum from a Poisson to a Wigner-Dyson distribution.

Quantum Chaos is Quantum

“…A related question is the onset of irreversibility in a closed quantum system in the sense of entanglement complexity [16] is driven by the doping of a Clifford circuit. Similarly, it would be interesting to show how the entanglement spectrum statistics converges with the doping [11].…”

“…We consider a subgroup of the unitary group -the Clifford group -which only reproduces up to the four-point OTOC [3] and thus it is not sufficient to simulate quantum chaotic evolutions. In [11], it was shown numerically that a Clifford circuit on a d = 2 N -dimensional system of N qubits doped by a single T gate can bring a typical product state in an entangled state with the same entanglement spectrum statistics resulting from the random matrix theory for U(d). This result opens the question of whether it would be possible to simulate quantum chaos with classical resources.…”

Random matrix theory of the isospectral twirling

“…Finally, a very interesting subject is the transition from an integrable dynamics to a chaotic one. In the context of quantum circuits, it has been understood that one can dope a random Clifford circuit and obtain higher order unitary t-designs or transitions to universal entanglement [140,207]. In the context of quantum many-body systems, one could study integrable spin chains perturbed by an integrability-breaking term [153] and study the typical behavior in the sense of the isospectral twirling of the probes to quantum chaos.…”

Quantum Chaos and Circuit Parameter Optimization

“…Its relations with quantum chaos was studied in [46]. It would be interesting to better investigate the role of magic in the VQA problems, following [47]. In all cases, the SFFs are very different from the RMT predictions at 10 layers, while starting from 30 layers the agreement is excellent.…”