Universal entanglement spectra in random quantum circuits

Chao‐Ming Jian

Andrew M. Projansky, Joshuah T. Heath, James Daniel Whitfield

The entanglement level statistics of a quantum state have recently been proposed to be a signature of universality in the underlying quantum circuit. This is a consequence of level repulsion in the entanglement spectra being tied to the integrability of entanglement generated. However, such studies of the level-spacing statistics in the entanglement spectrum have thus far been limited to the output states of Clifford and Haar random circuits on product state inputs. In this work, we provide the first example of a circuit which is composed of a simulable gate set, yet has a Wigner-Dyson distributed entanglement level spectrum without any perturbing universal element. We first show that, for matchgate circuits acting on random product states, Wigner-Dyson statistics emerge by virtue of a single SWAP gate, in direct analog to previous studies on Clifford circuits. We then examine the entanglement spectrum of matchgate circuits with varied input states, and find a sharp jump in the complexity of entanglement as we go from two- to three-qubit entangled inputs. Studying Clifford and matchgate hybrid circuits, we find examples of classically simulable circuits whose output states exhibit Wigner-Dyson entanglement level statistics in the absence of universal quantum gate elements. Our study thus provides strong evidence that entanglement spectrum is not strongly connected to notions of simulability in any given quantum circuit.

Raúl O. Vallejos, Fernando de Melo, Gabriel G. Carlo

We test the principle of majorization J. I. Latorre and M. A. Mart\'{\i}n-Delgado, Phys. Rev. A 66, 022305 (2002). in random circuits. Three classes of circuits were considered: (i) universal, (ii) classically simulatable, and (iii) neither universal nor classically simulatable. The studied families are: {cnot, H, T}, {cnot, H, not}, {cnot, H, S} (Clifford), matchgates, and IQP (instantaneous quantum polynomial-time). We verified that all the families of circuits satisfy on average the principle of decreasing majorization. In most cases the asymptotic state (number of gates $\ensuremath{\rightarrow}\ensuremath{\infty}$) behaves like a random vector. However, clear differences appear in the fluctuations of the Lorenz curves associated with asymptotic states. The fluctuations of the Lorenz curves discriminate between universal and nonuniversal classes of random quantum circuits, and they also detect the complexity of some nonuniversal but not classically efficiently simulatable quantum random circuits. We conclude that majorization can be used as an indicator of complexity of quantum dynamics, as an alternative to, e.g., entanglement spectrum and out-of-time-order correlators.

Salvatore F. E. Oliviero, Lorenzo Leone, Alioscia Hamma

Andrew C. Potter, Romain Vasseur

Ben T. McDonough, Claudio Chamon, Justin H. Wilson +1 more

Universal features of chaotic quantum dynamics underlie our understanding of thermalization in closed quantum systems and the complexity of quantum computations. Reversible automaton circuits, comprised of classical logic gates, have emerged as a tractable means to study such dynamics. Despite generating no entanglement in the computational basis, these circuits nevertheless capture many features expected from fully quantum evolutions. In this work, we demonstrate that the differences between automaton dynamics and fully quantum dynamics are revealed by the operator entanglement spectrum, much like the entanglement spectrum of a quantum state distinguishes between the dynamics of states under Clifford and Haar random circuits. While the operator entanglement spectrum under random unitary dynamics is governed by the eigenvalue statistics of random Gaussian matrices, we show evidence that under random automaton dynamics it is described by the statistics of Bernoulli random matrices, whose entries are random variables taking values 0 or 1 . We study the crossover between automaton and generic unitary operator dynamics as the automaton circuit is doped with gates that introduce superpositions, namely Hadamard or R x = e − i π 4 X gates. We find that a constant number of superposition-generating gates is sufficient to drive the operator dynamics to the random-circuit universality class, similar to earlier results on Clifford circuits doped with T gates. This establishes the operator entanglement spectrum as a useful tool for probing the chaoticity and universality class of quantum dynamics.