Mixing and localization in random time-periodic quantum circuits of Clifford unitaries

Abstract

How much do local and time-periodic dynamics resemble a random unitary? In the present work, we address this question by using the Clifford formalism from quantum computation. We analyze a Floquet model with disorder, characterized by a family of local, time-periodic, and random quantum circuits in one spatial dimension. We observe that the evolution operator enjoys an extra symmetry at times that are a half-integer multiple of the period. With this, we prove that after the scrambling time, namely, when any initial perturbation has propagated throughout the system, the evolution operator cannot be distinguished from a (Haar) random unitary when all qubits are measured with Pauli operators. This indistinguishability decreases as time goes on, which is in high contrast to the more studied case of (time-dependent) random circuits. We also prove that the evolution of Pauli operators displays a form of mixing. These results require the dimension of the local subsystem to be large. In the opposite regime, our system displays a novel form of localization, produced by the appearance of effective one-sided walls, which prevent perturbations from crossing the wall in one direction but not the other.

Random Quantum Circuits

“…A large class of chaotic many-body Floquet systems without conservation laws thermalize to infinite temperature, like the fully random circuits discussed in Section 2.3. Although we focus here on chaotic Floquet dynamics, it should be noted that random Floquet circuits can also exhibit MBL (102,(208)(209)(210)(211)(212). In particular, Reference 208 emphasized the quantum circuit language, constructing a random Floquet circuit of Clifford gates (Section 3.3.4) that showed a solvable localization transition, related to classical percolation of spatial puddles.…”

Scrambling in quantum cellular automata

“…In this letter, we propose that quantum cellular automata (QCAs) [27][28][29] are a useful tool for studying scrambling (cf. [30][31][32][33][34][35][36]). These are lattice systems equipped with local Hilbert spaces on each site and a discrete timeevolution operation.…”

“…One can change the rule (even making them random [54]), the local Hilbert space, or the structure of the lattice (cf. [35,36]), thereby implementing alternate types of many-body systems. We can include multiple species, asymmetries, nonlocal interactions, or even various types of boundary conditions (reflecting, periodic, etcetera).…”