## Dynamics of quantum information

Entanglement is the non-classical correlation that powers quantum computing, and a key ingredient for quantum technologies in general. It is also a cornerstone in our understanding of quantum matter: the pattern of entanglement of a many-body state can serve as the definition of a phase (for example in topological order); out of equilibrium, the growth and propagation of entanglement is the key to fundamental questions about thermalization in isolated systems and the onset of quantum chaos.

Recently I have been interested in the dynamics of quantum entanglement in many-body systems modeled by *quantum circuits *. These models are interesting for practical reasons, as the natural kinds of dynamics implemented on digital quantum simulators; but they are also interesting for fundamental reasons, as powerful “toy models” of quantum dynamics that capture the universal properties of *locality* and *unitarity* while dropping all the additional structure one may have in quantum dynamics (eigenspectra, energy conservation, symmetry, *etc*). These models are often quite tractable analytically, at least in a statistical sense, and have been instrumental in the derivation of many results on quantum chaos, growth of entanglement and operator spreading in many-body systems, among others.

Recently, this direction has extended to include “monitored” systems, whose unitary evolution is interspersed with measurements by an outside observer. These systems give rise to a new kind of out-of-equilibrium phases of matter—*entanglement phases*, defined by the amount of entanglement in the states produced by the dynamics at late times. Intuitively these phases, and the transitions that separate them, arise from a competition between the chaotic dynamics of the unobserved system (which tends to generate entanglement) and the measurements (which may destroy it). The existence of an “entangling” phase, where entanglement is robust against measurements, is particularly surprising, and can be understood as the spontaneous, dynamical formation of a quantum error correcting code.

#### Recent work

PRX **12**, 011045 (2022) and PRL **126**, 060501 (2021) (arXiv) | *Space-time duality *is a transformation that exchanges the roles of space and time in (1+1)-dimensional quantum dynamics. Typically, it breaks unitarity—that is, an evolution that is unitary in time won’t be unitary in space. This creates a surprising connection with *monitored dynamics*: the non-unitary “space evolution” can be viewed as a trajectory of a monitored system. In these works we used this insight to better understand known entanglement phases, discover new ones, and propose practical realizations of these phases on digital quantum simulators. This includes a protocol to measure entanglement in these states that side-steps the need to solve a “decoding” problem or apply post-selection that generally arises in monitored dynamics.

PRX **11**, 011030 (2021) | We introduced a new type of entanglement phase transitions in *measurement-only dynamics*. Even without chaotic dynamics, measurements alone can dynamically generate a quantum error correcting code and protect information *from themselves*. This is enabled by Heisenberg’s uncertainty principle: the outcomes of *incompatible* (non-commuting) measurements can’t be known at the same time. As a result, sufficiently “frustrated” ensembles of measurements may end up supporting an entangling phase. This new paradigm for monitored phases has interesting connections to some recent developments in the theory of quantum error correction.

## Nonequilibrium phases, localization and prethermalization

Measurement-induced entanglement phases are only one example of phase structure out of equilibrium. It is possible to give a sharp definition of phases in interacting, periodically driven (also known as “Floquet”) many-body systems. A paradigmatic example of this is the *discrete time crystal*, a phase of matter that spontaneously breaks discrete time-translation symmetry.

That these phases exist at all is surprising. Driven, interacting systems tend to absorb energy from the drive and heat up to “infinite temperature”, reaching a completely featureless state where any possibility of quantum order is lost. The only way to truly avoid thermalization, as far as we know, is *many-body localization *(MBL): a robustly non-thermalizing phase supported by strong disorder (or detuning). A Floquet MBL system doesn’t absorb energy from the drive, retaining local memory of its initial conditions for infinitely long time. Aside from MBL, there are a variety of strategies to postpone heating until very late times, so that while the system eventually thermalizes, it can still realize interesting nonequilibrium states over a long, tunable *“prethermal” *time scale.

My interests in quantum dynamics include both fundamental questions (expanding the scope of quantum phases to new settings including driving, dissipation, decoherence, *etc.*) and applied ones (quantum simulation in noisy, intermediate-scale quantum [NISQ] devices, experimental implementation of dynamical protocols in noisy conditions, *etc.*).

#### Recent work

Nature **601**, 531 (2022) (arXiv) and PRX Quantum **2**, 030346 (2021) (see also Physics article) | In this collaboration with the Google Quantum AI team, we made a proposal for the realization and detection of a many-body-localized discrete time crystal (DTC) on the Sycamore superconducting quantum processor, and then collaborated on its experimental implementation. Convincingly observing a nonequilibrium phase on a NISQ device poses a number of challenges, since these devices are by definition exposed to decoherence and limited in size, while a phase of matter demands a thermodynamically-large system—and, for a dynamical phase, an infinite duration in time. We addressed these limitations by using the extreme programmability of the device to “turn back time”, vary the size of the system, and efficiently probe the many-body Hilbert space.

PRX **10**, 021044 (2020) | In this collaboration with Immanuel Bloch’s lab (LMU/MPIQ Munich), we gave the first demonstration of Floquet prethermalization in a system of optically trapped atoms subject to periodic driving by varying the amplitude of the optical lattice. Our experimental collaborators found evidence of an impressive slow-down of thermalization as they increased the frequency of the drive. Intuitively, as the driving frequency increases, the system must absorb energy in bigger and bigger quantized packets; this requires more and more atoms to change their state in a coordinated way, which suppressed the transition amplitude exponentially (see this Physics Viewpoint article, too). On top of this general suppression of heating, we also saw interesting spectral features related to the system’s superfluid and Mott-insulator phases.

## Topological and strongly-correlated matter

Much of my PhD work revolved around interacting or disordered low-dimensional electron systems, particularly in the quantum Hall regime, and I remain actively interested in these topics. When a 2D electron gas is placed in a strong perpendicular magnetic field, its kinetic energy becomes frozen into flat bands (the Landau levels); the system is then dominated by interactions (or disorder). This is a recipe for plenty of interesting phenomena, and has been a source of ideas in condensed matter physics for four decades.

Recently, flat, topological bands have appeared in a different setting: that of *moire materials*, layers of 2D materials stacked on top of each other with a small relative twist. Here, too, these ingredients give rise to lots of interesting phases and phenomena.

#### Recent work

PRB **106**, 035421 (2022) (arXiv) | With large-scale numerical simulations we explored the viability of a recently proposed mechanism for superconductivity in moire materials and beyond, based on the pairing of Skyrmions (magnetic textures). To do so we used a model based on a multilayered quantum Hall system (the different layers model the spin and sub-band degrees of freedom in a basic model of twisted bilayer graphene), wrapped around an infinite cylinder, and used the density matrix renormalization group (DMRG) method to study the strongly-correlated ground state at various levels of doping. The results are quite remarkable—the pairing appears to be stable in a wide (and realistic) range of parameters, even though all the underlying electron-electron interactions are repulsive.

PRL **124**, 086602 (2020) (arXiv) | We studied what happens to the integer quantum Hall plateau transition in the quasi-1D limit, when the system approaches a cylinder rather than a plane. Contrary to the 2D case, where topological states concentrate around a vanishing window near the band center, here we observed a surprising proliferation of topological states across the Landau level. This is explained by thinking of the system as a “topological pump” (a well-known idea, dating back to Thouless, to understand the quantization of the Hall effect). In the presence of strong disorder, we find that during each pump cycle the electrons hop through a network of nonlocal resonances, describing random walks through the length of the system. This novel interplay of topology and disorder is unlikely to be visible in mesoscopic systems, where the relevant adiabatic time scales would be astronomical—but it may be realized in microscopic quantum simulators with tunable disorder.