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| Fig. 1: Illustration of two-site unitary gates in the first model. (Source: J. H. Son) |
In this report, following von Keyserlingk et al., I present how three different diagnostics of scrambling and quantum chaos - operator spreading, out-of-time correlators, and entanglement growth - behave in three different models of one-dimensional quantum spin chains (See also Nahum et al. for the related content). [1,2] These models highlight how the quantities of interest can behave similarly/differently in different models, all of which share the concept of locality.
In the first model, time evolution is given by arrays of two-site unitary transformations, illustrated in Fig. 1, chosen from a uniform distribution given by the Haar measure. Interestingly, in this case, the problem of studying operator spreading is mapped to a random walk, and the quantities of interest can be computed analytically. One can establish from the result that the quantities of interest to us behave as:
In the Heisenberg picture, an operator that is local at the initial time broadens rapidly. Especially both ends of the broadened operators at the late time spread with some butterfly velocity and diffusion constant specific to the microsopic details of the model, showing hydrodynamical behavior.
The out-of-time correlators increase sharply near the time set by the butterfly velocity in the previous bullet point, and saturates to a constant.
The entanglement entropy of subsystems grows linearly and then saturates to a constant.
In the kicked Ising model, time evolution is given by a periodic time-dependent Hamiltonian: for the first half of the period, the Hamiltonian is identical to the time-independent (non-transverse) Ising model with external field terms. For the second half of the period, the Hamiltonian is equivalent to a sum of x-Pauli matrices acting on each site. While this model is not very amenable to any exact calculations, one can still numerically compute quantities that diagnose quantum chaos using tensor network methods.
Despite being a translationally invariant time-periodic system, how the operator spreads, how out-of-time correlators behave, and how the entanglement entropy grows over time are remarkably similar to those in Haar random unitary circuits. This leads to the conjecture that chaotic behaviors derived from Haar random unitary circuits are universal in a wide range of models.
In the last class of models, the time evolution is still given by translationally invariant sets of local unitary transformations, along with the condition that the unitary time evolution should be in the Clifford subgroup. Among the circuits that satisfy the above criteria, the specific class of circuits in which operators spread in a fractal manner will be studied, hence the adjective "fractal". This class of models is widely studied in the quantum information community, and many rigorous results are available. Furthermore, despite being quantum circuits, it is known that these models can be simulated efficiently by classical computers.
In fractal Clifford circuits, the entanglement entropy grows linearly, as in random unitary circuits and the kicked Ising model. However, the out-of-time correlators never saturate to a constant and continue oscilliating even at late times. Hence, while there is a notion of operator spreading in fractal Clifford circuits as well, observables in fractal Clifford circuits behave differently from ones in the kicked Ising model and Haar random unitary circuits.
© Jun Ho Son. The author warrants that the work is the author's own and that Stanford University provided no input other than typesetting and referencing guidelines. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.
[1] C. W. von Keyserlingk et al., "Operator Hydrodynamics, OTOCs, and Entanglement Growth in Systems Without Conservation Laws", Phys. Rev. X 8, 021013 (2018).
[2] A. Nahum, S. Vijay, and J. Haah, "Operator Spreading in Random Unitary Circuit", Phys. Rev. X 8, 021014 (2018).