One of the main topological invariants that characterizes several topologically-ordered phases is the many-body Chern number (MBCN). Paradigmatic examples include several fractional quantum Hall phases, which are expected to be realized in different atomic and photonic quantum platforms in the near future. Experimental measurement and numerical computation of this invariant is conventionally based on the linear-response techniques which require having access to a family of states, as a function of an external parameter, which is not suitable for many quantum simulators. Here, we propose an ancilla-free experimental scheme for the measurement of this invariant, without requiring any knowledge of the Hamiltonian. Specifically, we use the statistical correlations of randomized measurements to infer the MBCN of a wavefunction. Remarkably, our results apply to disk-like geometries that are more amenable to current quantum simulator architectures.

We develop a general kinetic theory framework to describe the hydrodynamics of strongly interacting, nonequilibrium quantum systems in which integrability is weakly broken, leaving a few residual conserved quantities. This framework is based on a generalized relaxation-time approximation; it gives a simple, but surprisingly accurate, prescription for computing nonequilibrium transport even in strongly interacting systems. This approximation reproduces the crossover from generalized to conventional hydrodynamics in interacting one-dimensional Bose gases with integrability-breaking perturbations, both with and without momentum conservation. It also predicts the hydrodynamics of chaotic quantum spin chains, in good agreement with matrix product operator calculations.

With the advent of hybrid quantum classical algorithms using parameterized quantum circuits the question of how to optimize these algorithms and circuits emerges. In this paper we show that the number of single-qubit rotations in parameterized quantum circuits can be decreased without compromising the expressibility or entangling capability of the circuit. We also compare expressibility and entangling capability across different number of qubits in parameterized quantum circuits. We also consider a parameterized photonics quantum circuit, without any single-qubit rotations, which yields an expressibility and entangling capability comparable to the best regular parameterized quantum circuits.

In this paper, our prime objective is to apply the techniques of {\it parameter estimation theory} and the concept of {\it Quantum Metrology} in the form of {\it Fisher Information} to investigate the role of certain physical quantities in the open quantum dynamics of a two entangled qubit system under the Markovian approximation. There exist various physical parameters which characterize such system, but can not be treated as any quantum mechanical observable. It becomes imperative to do a detailed parameter estimation analysis to determine the physically consistent parameter space of such quantities. We apply both Classical Fisher Information (CFI) and Quantum Fisher Information (QFI) to correctly estimate these parameters, which play significant role to describe the out-of-equilibrium and the long range quantum entanglement phenomena of open quantum system. {\it Quantum Metrology}, compared to {\it classical parameter estimation theory}, plays a two-fold superior role, improving the precision and accuracy of parameter estimation. Additionally, in this paper, we present a new avenue in terms of {\it Quantum Metrology}, which beats the classical parameter estimation. We also present an interesting result of \textit{revival of out-of-equilibrium feature at the late time scales, arising due to the long-range quantum entanglement at early time scale and provide a physical interpretation for the same in terms of Bell's Inequality Violation in early time scale giving rise to non-locality.

Long-lived dark states, in which an experimentally accessible qubit is not in thermal equilibrium with a surrounding spin bath, are pervasive in solid-state systems. We explain the ubiquity of dark states in a large class of inhomogenous central spin models using the proximity to integrable lines with exact dark eigenstates. At numerically accessible sizes, dark states persist as eigenstates at large deviations from integrability, and the qubit retains memory of its initial polarization at long times. Although the eigenstates of the system are chaotic, exhibiting exponential sensitivity to small perturbations, they do not satisfy the eigenstate thermalization hypothesis. Rather, we predict long relaxation times that increase exponentially with system size. We propose that this intermediate chaotic but non-ergodic regime characterizes mesoscopic quantum dot and diamond defect systems, as we see no numerical tendency towards conventional thermalization with a finite relaxation time.

We study the Anderson transition in lattices with the connectivity of a random-regular graph. Our results indicate that fractal dimensions are continuous across the transition, but a discontinuity occurs in their derivatives, implying the non-ergodicity of the metal near the Anderson transition. A critical exponent $\nu = 1.00 \pm0.02$ and critical disorder $W= 18.2\pm 0.1$ are found via a scaling approach. Our data support that the predictions of the relevant Gaussian Ensemble are only recovered at zero disorder.

We consider discrete-time quantum walks of many interacting particles. For a specific kind of pairwise interaction, we find peculiar multipartite bound states which fall apart if any subsystem is removed. This provides a conceptually simple physical model of structures known as Borromean rings. Interestingly, our approach highlights the role of entanglement in such systems. In order to form a Borromean bound state, the particles need to exhibit Greenberger-Horne-Zeillinger (GHZ) entanglement. Moreover, we discuss our findings in the context of formation of composite particles.

The resilience of quantum entanglement to a classicality-inducing environment is tied to fundamental aspects of quantum many-body systems. The dynamics of entanglement has recently been studied in the context of measurement-induced entanglement transitions, where the steady-state entanglement collapses from a volume-law to an area-law at a critical measurement probability $p_{c}$. Interestingly, there is a distinction in the value of $p_{c}$ depending on how well the underlying unitary dynamics scramble quantum information. For strongly chaotic systems, $p_{c} > 0$, whereas for weakly chaotic systems, such as integrable models, $p_{c} = 0$. In this work, we investigate these measurement-induced entanglement transitions in a system where the underlying unitary dynamics are many-body localized (MBL). We demonstrate that the emergent integrability in an MBL system implies a qualitative difference in the nature of the measurement-induced transition depending on the measurement basis, with $p_{c} > 0$ when the measurement basis is scrambled and $p_{c} = 0$ when it is not. This feature is not found in Haar-random circuit models, where all local operators are scrambled in time. When the transition occurs at $p_{c} > 0$, we use finite-size scaling to obtain the critical exponent $\nu = 1.3(2)$, close to the value for 2+0D percolation. We also find a dynamical critical exponent of $z = 0.98(4)$ and logarithmic scaling of the R\'{e}nyi entropies at criticality, suggesting an underlying conformal symmetry at the critical point. This work further demonstrates how the nature of the measurement-induced entanglement transition depends on the scrambling nature of the underlying unitary dynamics. This leads to further questions on the control and simulation of entangled quantum states by measurements in open quantum systems.

We propose a wave operator method to calculate eigenvalues and eigenvectors of large parameter-dependent matrices, using an adaptative active subspace. We consider a hamiltonian which depends on external adjustable or adiabatic parameters, using adaptative projectors which follow the successive eigenspaces when the adjustable parameters are modified. The method can also handle non-hermitian hamiltonians. An iterative algorithm is derived and tested through comparisons with a standard wave operator algorithm using a fixed active space and with a standard block-Davidson method. The proposed approach is competitive, it converges within a few dozen iterations at constant memory cost. We first illustrate the abilities of the method on a 4-D coupled oscillator model hamiltonian. A more realistic application to molecular photodissociation under intense laser fields with varying intensity or frequency is also presented. Maps of photodissociation resonances of H${}_2^+$ in the vicinity of exceptional points are calculated as an illustrative example.

This article considers quantum systems described by a finite-dimensional complex Hilbert space $H$. We first define the concept of a finite observable on $H$. We then discuss ways of combining observables in terms of convex combinations, post-processing and sequential products. We also define complementary and coexistent observables. We then introduce finite instruments and their related compatible observables. The previous combinations and relations for observables are extended to instruments and their properties are compared. We present four types of instruments; namely, identity, trivial, L\"uders and Kraus instruments. These types are used to illustrate different ways that instruments can act. We next consider joint probabilities for observables and instruments. The article concludes with a discussion of measurement models and the instruments they measure.

The single-particle spectral function of a strongly correlated system is an essential ingredient to describe its dynamics and transport properties. We develop a general method to calculate the exact spectral function of a strongly interacting one-dimensional Bose gas in the Tonks-Girardeau regime, valid for any type of confining potential, and apply it to bosons on a lattice to obtain the full spectral function, at all energy and momentum scales. We find that it displays three main singularity lines. The first two can be identified as the analogs of Lieb-I and Lieb-II modes of a uniform fluid; the third one, instead, is specifically due to the presence of the lattice. We show that the spectral function displays a power-law behaviour close to the Lieb-I and Lieb-II singularities, as predicted by the non-linear Luttinger liquid description, and obtain the exact exponents. In particular, the Lieb-II mode shows a divergence in the spectral function, differently from what happens in the dynamical structure factor, thus providing a route to probe it in experiments with ultracold atoms.

Consider a sequence in a finite-dimensional complex (resp. real) vector space arising as the iterates of an arbitrary point under the composition of a unitary (resp. orthogonal) map with the orthogonal projection on the hyperplane orthogonal to the starting point. We show that, generically, the series of the squared norms of those points sums to the dimension of the underlying space. The exact formula for this series in non-generic cases is provided as well, along with the quantum-mechanical interpretation of this result.

We compute the Rydberg spectrum of a single Ca$^+$ ion in a Paul trap by incorporating various internal and external coupling terms of the ion to the trap in the Hamiltonian. The coupling terms include spin-orbit coupling in Ca$^+$, charge (electron and ionic core) coupling to the radio frequency and static fields, ion-electron coupling in the Paul trap, and ion center-of-mass coupling. The electronic Rydberg states are precisely described by a one-electron model potential for e$^-$+Ca$^{2+}$, and accurate eigenenergies, quantum defect parameters, and static and tensor polarizabilities for a number of excited Rydberg states are obtained. The time-periodic rf Hamiltonian is expanded in the Floquet basis, and the trapping-field-broadened Rydberg lines are compared with recent observations of Ca$^+(23P)$ and Ca$^+(52F)$ Rydberg lines.

The quantized Hall conductivity of integer and fractional quantum Hall (IQH and FQH) states is directly related to a topological invariant, the many-body Chern number. The conventional calculation of this invariant in interacting systems requires a family of many-body wave functions parameterized by twist angles in order to calculate the Berry curvature. In this paper, we demonstrate how to extract the Chern number given a single many-body wave function, without knowledge of the Hamiltonian.

For FQH states, our method requires one additional integer invariant as input: the number of $2\pi$ flux quanta, $s$, that must be inserted to obtain a topologically trivial excitation. As we discuss, $s$ can be obtained in principle from the degenerate set of ground state wave functions on the torus, without knowledge of the Hamiltonian.

We perform extensive numerical simulations involving IQH and FQH states to validate these methods.

In two dimensions, the topological order described by $\mathbb{Z}_2$ gauge theory coupled to free or weakly interacting fermions with a nonzero spectral Chern number $\nu$ is classified by $\nu \; \mathrm{mod}\; 16$ as predicted by Kitaev [Ann. Phys. 321, 2 (2006)]. Here we provide a systematic and complete construction of microscopic models realizing this so-called sixteenfold way of anyon theories. These models are defined by $\Gamma$ matrices satisfying the Clifford algebra, enjoy a global $\mathrm{SO}(\nu)$ symmetry, and live on either square or honeycomb lattices depending on the parity of $\nu$. We show that all these models are exactly solvable by using a Majorana representation and characterize the topological order by calculating the topological spin of an anyonic quasiparticle and the ground-state degeneracy. The possible relevance of the $\nu=2$ and $\nu=3$ models to materials with Kugel-Khomskii-type spin-orbital interactions is discussed.

Entanglement of formation is a fundamental measure that quantifies the entanglement of bipartite quantum states. This measure has recently been extended into multipartite states taking the name $\alpha$-entanglement of formation. In this work, we follow an analogous multipartite extension for the Gaussian version of entanglement of formation, and focusing on the the finest partition of a multipartite Gaussian state we show this measure is fully additive and computable for 3-mode Gaussian states.

Recent quantum technologies utilize complex multidimensional processes that govern the dynamics of quantum systems. We develop an adaptive diagonal-element-probing compression technique that feasibly characterizes any unknown quantum processes using much fewer measurements compared to conventional methods. This technique utilizes compressive projective measurements that are generalizable to arbitrary number of subsystems. Both numerical analysis and experimental results with unitary gates demonstrate low measurement costs, of order $O(d^2)$ for $d$-dimensional systems, and robustness against statistical noise. Our work potentially paves the way for a reliable and highly compressive characterization of general quantum devices.

We introduce a quantity, called pseudo entropy, as a generalization of entanglement entropy via post-selection. In the AdS/CFT correspondence, this quantity is dual to areas of minimal area surfaces in time-dependent Euclidean spaces which are asymptotically AdS. We study its basic properties and classifications in qubit systems. In specific examples, we provide a quantum information theoretic meaning of this new quantity as an averaged number of Bell pairs when the post-section is performed. We also present properties of the pseudo entropy for random states. We then calculate the pseudo entropy in the presence of local operator excitations for both the two dimensional free massless scalar CFT and two dimensional holographic CFTs. We find a general property in CFTs that the pseudo entropy is highly reduced when the local operators get closer to the boundary of the subsystem. We also compute the holographic pseudo entropy for a Janus solution, dual to an exactly marginal perturbation of a two dimensional CFT and find its agreement with a perturbative calculation in the dual CFT. We show the linearity property holds for holographic states, where the holographic pseudo entropy coincides with a weak value of the area operator. Finally, we propose a mixed state generalization of pseudo entropy and give its gravity dual.

A two-dimensional array of Kerr-nonlinear parametric oscillators (KPOs) with local four-body interactions is a promising candidate for realizing an Ising machine with all-to-all spin couplings, based on adiabatic quantum computation in the Lechner-Hauke-Zoller (LHZ) scheme. However its performance has been evaluated for only a few KPOs. By numerically simulating more KPOs, here we show that the performance can be dramatically improved by reducing inhomogeneity in photon numbers induced by the four-body interactions. The discrepancies of the photon numbers can be corrected by tuning the detunings of KPOs depending on their positions, without monitoring their states during adiabatic time evolution. This correction can be used independent of the number of KPOs in the LHZ scheme and thus can be applied to large-scale implementation.

Exceptional points (EPs), the degeneracy point of non-Hermitian systems, have recently attracted great attention after its ability to greatly enhance the sensitivity of micro-cavities is demonstrated experimentally. Unlike the usual degeneracies in Hermitian systems, at EPs, both the eigenenergies and eigenvectors coalesce. Although some of the exotic properties and potential applications of EPs are explored, the range of EPs studied is largely limited by the experimental capability. Some of the systems, e.g. with higher-order EPs, are hard to achieve with conventional simulations. Here we propose an extendable method to simulate non-Hermitian systems on the quantum circuits, where a wide range of EPs can be studied. The system is inherently parity-time (PT) broken due to the non-symmetric controlling effects and post-selection. Inspired by the quantum Zeno effect, the circuit structure grantees the success rate of the post-selection. A sample circuit is implemented in a quantum programming framework, and the phase transition at EP is demonstrated. Considering the scalable and flexible nature of quantum circuits, our model is capable of simulating large scale systems with higher-order EPs. We believe this work may lead to broader applications of quantum computers and provide a tool to the studies for non-Hermitian systems and the associated EPs.