Periodically-driven or Floquet systems can realize anomalous topological phenomena that do not exist in any equilibrium states of matter, whose classification and characterization require new theoretical ideas that are beyond the well-established paradigm of static topological phases. In this work, we provide a general framework to understand anomalous Floquet higher-order topological insulators (AFHOTIs), the classification of which has remained a challenging open question. In two dimensions (2D), such AFHOTIs are defined by their robust, symmetry-protected corner modes pinned at special quasienergies, even though all their Floquet bands feature trivial band topology. The corner-mode physics of an AFHOTI is found to be generically indicated by 3D Dirac/Weyl-like topological singularities living in the phase spectrum of the bulk time-evolution operator. Physically, such a phase-band singularity is essentially a "footprint" of the topological quantum criticality, which separates an AFHOTI from a trivial phase adiabatically connected to a static limit. Strikingly, these singularities feature unconventional dispersion relations that cannot be achieved on any static lattice in 3D, which, nevertheless, resemble the surface physics of 4D topological crystalline insulators. We establish the above higher-order bulk-boundary correspondence through a dimensional reduction technique, which also allows for a systematic classification of 2D AFHOTIs protected by point group symmetries. We demonstrate applications of our theory to two concrete, experimentally feasible models of AFHOTIs protected by $C_2$ and $D_4$ symmetries, respectively. Our work paves the way for a unified theory for classifying and characterizing Floquet topological matters.

We explain how gapped quantum spin liquids, both conventional and 'fractonic', may be unambiguously diagnosed experimentally using the technique of multidimensional coherent spectroscopy. 'Conventional' gapped quantum spin liquids (e.g. $Z_2$ spin liquid) do not have clear signatures in linear response, but do have clear fingerprints in non-linear response, accessible through the already existing experimental technique of two dimensional coherent spectroscopy. Type I fracton phases (e.g. X-cube) are (surprisingly) even easier to distinguish, with strongly suggestive features even in linear response, and unambiguous signatures in non-linear response. Type II fracton systems, like Haah's code, are most subtle, and may require consideration of high order non-linear response for unambiguous diagnosis.

We consider the binary fragmentation problem in which, at any breakup event, one of the daughter segments either survives with probability $p$ or disappears with probability $1\!-\!p$. It describes a stochastic dyadic Cantor set that evolves in time, and eventually becomes a fractal. We investigate this phenomenon, through analytical methods and Monte Carlo simulation, for a generic class of models, where segment breakup points follow a symmetric beta distribution with shape parameter $\alpha$, which also determines the fragmentation rate. For a fractal dimension $d_f$, we find that the $d_f$-th moment $M_{d_f}$ is a conserved quantity, independent of $p$ and $\alpha$. We use the idea of data collapse -- a consequence of dynamical scaling symmetry -- to demonstrate that the system exhibits self-similarity. In an attempt to connect the symmetry with the conserved quantity, we reinterpret the fragmentation equation as the continuity equation of a Euclidean quantum-mechanical system. Surprisingly, the Noether charge corresponding to dynamical scaling is trivial, while $M_{d_f}$ relates to a purely mathematical symmetry: quantum-mechanical phase rotation in Euclidean time.

Strongly correlated quantum systems give rise to many exotic physical phenomena, including high-temperature superconductivity. Simulating these systems on quantum computers may avoid the prohibitively high computational cost incurred in classical approaches. However, systematic errors and decoherence effects presented in current quantum devices make it difficult to achieve this. Here, we simulate the dynamics of the one-dimensional Fermi-Hubbard model using 16 qubits on a digital superconducting quantum processor. We observe separations in the spreading velocities of charge and spin densities in the highly excited regime, a regime that is beyond the conventional quasiparticle picture. To minimize systematic errors, we introduce an accurate gate calibration procedure that is fast enough to capture temporal drifts of the gate parameters. We also employ a sequence of error-mitigation techniques to reduce decoherence effects and residual systematic errors. These procedures allow us to simulate the time evolution of the model faithfully despite having over 600 two-qubit gates in our circuits. Our experiment charts a path to practical quantum simulation of strongly correlated phenomena using available quantum devices.

In a recent paper (Commun. Phys. 3, 100) Znidaric studies the growth of higher Renyi entropies in diffusive systems and claims that they generically grow ballistically in time, except for spin-1/2 models in d=1 dimension. Here, we point out that the necessary conditions for sub-ballistic growth of Renyi entropies are in fact much more general, and apply to a large class of systems, including experimentally relevant ones in arbitrary dimension and with larger local Hilbert spaces.

The term randomized benchmarking refers to a collection of protocols that in the past decade have become the gold standard for characterizing quantum gates. These protocols aim at efficiently estimating the quality of a set of quantum gates in a way that is resistant to state preparation and measurement errors, and over the years many versions have been devised. In this work, we develop a comprehensive framework of randomized benchmarking general enough to encompass virtually all known protocols. Overcoming previous limitations on e.g. error models and gate sets, this framework allows us to formulate realistic conditions under which we can rigorously guarantee that the output of a randomized benchmarking experiment is well-described by a linear combination of matrix exponential decays. We complement this with a detailed discussion of the fitting problem associated to randomized benchmarking data. We discuss modern signal processing techniques and their guarantees in the context of randomized benchmarking, give analytical sample complexity bounds and numerically evaluate their performance and limitations. In order to reduce the resource demands of this fitting problem, we moreover provide scalable post-processing techniques to isolate exponential decays, significantly improving the practical feasibility of a large set of randomized benchmarking protocols. These post-processing techniques generalize several previously proposed methods such as character benchmarking and linear-cross entropy benchmarking. Finally we discuss, in full generality, how and when randomized benchmarking decay rates can be used to infer quality measures like the average fidelity. On the technical side, our work significantly extends the recently developed Fourier-theoretic perspective on randomized benchmarking and combines it with the perturbation theory of invariant subspaces and ideas from signal processing.

Quantum control over a physical system requires thermal fluctuations and thermal decoherence to be negligible, which becomes more challenging with decreasing natural frequencies of the target system. For microwave circuits, the quantum regime can be reached simply by cooling them to mK temperatures. Radio-frequency (RF) systems in the MHz regime, however, require further cooling or have to be coupled to an auxiliary quantum system with a coupling rate exceeding their thermal decoherence rate. A powerful tool to cool below the thermodynamic bath temperature is sideband-cooling, a technique that originated from the field of trapped ions and cold atoms and that has been applied in cavity optomechanics for groundstate cooling of mechanical motion. Here, we engineer a system of two superconducting LC circuits coupled by a current-mediated photon-pressure interaction and demonstrate sideband-cooling of a hot RF circuit using a microwave cavity and the regime of quantum-coherent coupling between the circuits. Due to a dramatically increased coupling strength, we obtain a large single-photon quantum cooperativity $\mathcal{C}_{\mathrm{q}0} \sim 1$ and reduce the residual thermal RF occupancy by 75% through sideband-cooling with less than a single pump photon. For larger pump powers, the photon-pressure coupling rate exceeds the RF thermal decoherence rate by a factor of three and the RF circuit is cooled into the quantum groundstate. Our results demonstrate photon-pressure coupling with a hot radio-frequency circuit in the quantum regime and lay the foundation for radio-frequency quantum photonics.

This article points out that observables and instruments can be combined in many ways that have natural and physical interpretations. We shall mainly concentrate on the mathematical properties of these combinations. Section~1 reviews the basic definitions and observables are considered in Section~2. We study parts of observables, post-processing, generalized convex combinations, sequential products and tensor products. These combinations are extended to instruments in Section~3. We consider properties of observables measured by combinations of instruments. We introduce four special types of instruments, namely Kraus, L\"uders, trivial and semitrivial instruments. We study when these types are closed under various combinations. In this work, we only consider finite-dimensional quantum systems. A few of the results presented here have appeared in the author's previous articles.

We describe an experimental effort designing and deploying error-robust single-qubit operations using a cloud-based quantum computer and analog-layer programming access. We design numerically-optimized pulses that implement target operations and exhibit robustness to various error processes including dephasing noise, instabilities in control amplitudes, and crosstalk. Pulse optimization is performed using a flexible optimization package incorporating a device model and physically-relevant constraints (e.g. bandwidth limits on the transmission lines of the dilution refrigerator housing IBM Quantum hardware). We present techniques for conversion and calibration of physical Hamiltonian definitions to pulse waveforms programmed via Qiskit Pulse and compare performance against hardware default DRAG pulses on a five-qubit device. Experimental measurements reveal default DRAG pulses exhibit coherent errors an order of magnitude larger than tabulated randomized-benchmarking measurements; solutions designed to be robust against these errors outperform hardware-default pulses for all qubits across multiple metrics. Experimental measurements demonstrate performance enhancements up to: $\sim10\times$ single-qubit gate coherent-error reduction; $\sim5\times$ average coherent-error reduction across a five qubit system; $\sim10\times$ increase in calibration window to one week of valid pulse calibration; $\sim12\times$ reduction gate-error variability across qubits and over time; and up to $\sim9\times$ reduction in single-qubit gate error (including crosstalk) in the presence of fully parallelized operations. Randomized benchmarking reveals error rates for Clifford gates constructed from optimized pulses consistent with tabulated $T_{1}$ limits, and demonstrates a narrowing of the distribution of outcomes over randomizations associated with suppression of coherent-errors.

Overcoming the detrimental effect of disorder at the nanoscale is very hard since disorder induces localization and an exponential suppression of transport efficiency. Here we unveil novel and robust quantum transport regimes achievable in nanosystems by exploiting long-range hopping. We demonstrate that in a 1D disordered nanostructure in presence of long-range hopping, transport efficiency, after decreasing exponentially with disorder at first, is then enhanced by disorder (Disorder-Enhanced Transport, DET regime) until, counter-intuitively, it reaches a Disorder-Independent Transport (DIT) regime, persisting over several orders of disorder magnitude in realistic systems. To enlighten the relevance of our results, we demonstrate that an ensemble of emitters in a cavity can be described by an effective long-range Hamiltonian. The specific case of a disordered molecular wire placed in an optical cavity is discussed, showing that the DIT and DET regimes can be reached with state-of-the-art experimental set-ups.

In this work we answer the question: how sudden or adiabatic is a change in the frequency of a quantum harmonic oscillator (HO)? To do this, we study a frequency transition with a continuous parameter enabling to tune the speed of the transition from the sudden to the adiabatic limit. We assume the HO is in its fundamental state in the remote past and compute numerically the time evolution operator by employing an iterative method. The resulting state of the system is a vacuum squeezed state, presented in two different bases related by Bogoliubov transformations, of which we fully characterize and discuss squeezing and adiabaticity along the frequency transition. Finally, we obtain analytical approximate expressions relating squeezing with the transition speed as well as the initial and final frequencies. We think our results may shed some light on subtleties and common inaccuracies in the literature related to the adiabatic theorem interpretation for this system.

Out-of-time-order correlators (OTOCs) have proven to be a useful tool for studying thermalisation in quantum systems. In particular, the exponential growth of OTOCS, or scrambling, is sometimes taken as an indicator of chaos in quantum systems, despite the fact that saddle points in integrable systems can also drive rapid growth in OTOCs. Here we use the simple model of the two-site Bose-Hubbard model to demonstrate how the OTOC growth driven by chaos can nonetheless be distinguished from that driven by saddle points. Besides quantitative differences in the long term average as predicted by the eigenstate thermalisation hypothesis, the saddle point gives rise to oscillatory behaviour not observed in the chaotic case. The differences are also highlighted by entanglement entropy, which in the chaotic Floquet case matches the Page curve prediction. These results illustrate additional markers that can be used to distinguish chaotic behaviour in quantum systems, beyond the initial exponential growth in OTOCs.

An accelerated detector will see thermal radiation around with temperature T which is termed as Unruh effect. The two accelerated observers can detect quantum correlations between them through the vacuum field. A natural question arises: is this equilibrium quantum correlation equivalent to the one between the two stationary detectors under a thermal bath? A further question is what if the two accelerations or the couplings between the detectors and vacuum field are different and how this nonequilibrium effect can influence the quantum correlations. In this study, we examine the above scenarios. We found that as the acceleration of the detectors increase, the quantum correlations (entanglement, mutual information, discord) will increase first and eventually decay depending on the redshift effect. The temperature and acceleration characterizing the equivalent space-time curvature show different impacts on the global quantum correlations in this equilibrium (equal temperatures or accelerations of the two detectors) scenario. When the equal acceleration is in the opposite direction, as the distance increases, the correlation will decay, and the entanglement appears to be more robust than the quantum mutual information and discord. Importantly, we have uncovered that the nonequilibriumness characterized by the degree of the detailed balance breaking due to the different couplings of the detectors to the vacuum field can increase the quantum correlations. We also uncovered both the nonequilibrium (different accelerations of the two detectors) and distance effects on quantum correlations between the detectors.

Choosing an appropriate representation of the molecular Hamiltonian is one of the challenges faced by simulations of the nonadiabatic quantum dynamics around a conical intersection. The adiabatic, exact quasidiabatic, and strictly diabatic representations are exact and unitary transforms of each other, whereas the approximate quasidiabatic Hamiltonian ignores the residual nonadiabatic couplings in the exact quasidiabatic Hamiltonian. A rigorous numerical comparison of the four different representations is difficult because of the exceptional nature of systems where the four representations can be defined exactly and the necessity of an exceedingly accurate numerical algorithm that avoids mixing numerical errors with errors due to the different forms of the Hamiltonian. Using the quadratic Jahn-Teller model and high-order geometric integrators, we are able to perform this comparison and find that only the rarely employed exact quasidiabatic Hamiltonian yields nearly identical results to the benchmark results of the strictly diabatic Hamiltonian, which is not available in general. In this Jahn-Teller model and with the same Fourier grid, the commonly employed approximate quasidiabatic Hamiltonian led to inaccurate wavepacket dynamics, while the Hamiltonian in the adiabatic basis was the least accurate, due to the singular nonadiabatic couplings at the conical intersection.

We consider a railway dispatching problem: delay and conflict management on a single-track railway line. We examine the issue of train dispatching consequences caused by the arrival of an already delayed train to the segment being considered. This is a computationally hard problem and its solution is needed in a very short time in practice. We introduce a quadratic unconstrained binary optimization (QUBO) model of the problem that is suitable for solving on quantum annealer devices: an emerging alternative computational hardware technology. These offer a scalability of computationally hard problems better than that of classical computers. We provide a proof-of-principle demonstration of applicability through solving selected real-life problems from the Polish railway network. We carry out the actual calculation on the D-Wave 2000Q machine. We also include solutions of our model using classical algorithms for solving QUBO, including those based on tensor networks. These are of potential practical relevance in case of smaller instances and serve as a comparison for understanding the behavior of the actual quantum device.

The approximate numerical method for a calculation of a quantum wave impedance in a case of a potential energy with a complicated spatial structure is considered. It was proved that the approximation of a real potential by a piesewise constant function is also reasonable in a case of using a quantum impedance approach.The dependence of an accuracy of numerical calculations on a number of cascads by which a real potential is represented was found. The method of including into a consideration of zero-range singular potentials was developed.

We study space-inhomogeneous quantum walks (QWs) on the integer lattice, which we assign three different coin matrices to the positive part, negative part, and to the origin, respectively. We call the model the two-phase QW with one defect. It covers the one-defect and the two-phase QW, which have been intensively researched. Localization is one of the most characteristic properties of QWs, and various types of two-phase QW with one-defect occurs localization. Moreover, the existence of eigenvalues is deeply related to localization. In this paper, we obtain the necessary and sufficient condition for the existence of eigenvalues. Our analytical methods are mainly based on the transfer matrix, a useful tool to generate the generalized eigenfunctions. Furthermore, we explicitly derive eigenvalues for some classes of two-phase QW with one defect.

Photon thermalisation and condensation in dye-filled microcavities is a growing area of scientific interest, at the intersection of photonics, quantum optics and statistical physics. We give here a short introduction to the topic, together with an explanation of some of our more important recent results. A key result across several projects is that we have a model based on a detailed physical description which has been used to accurately describe experimental observations. We present a new open-source package in Python called PyPBEC which implements this model. The aim is to enable the reader to readily simulate and explore the physics of photon condensates themselves, so this article also includes a working example code which can be downloaded from the GitHub repository.

Disregarding degeneracies and considering only different atomic levels, is it possible that two different transitions in hydrogen atom give the same frequency of radiation? That is, can different energy level transitions in a hydrogen atom have the same photon radiation frequency? The answer is definitely yes, and infinitely many transitions have been found, but to our knowledge a generalization is still lacking. In this note, we will show a general solution, i.e., how all equifrequency transition pairs can be obtained. This puzzle is a simple yet concrete example of how number theory can help understanding quantum systems, a curious theme that emerges in theoretical physics3 but usually inaccessible to high school and college students.

Analyzing general uncertainty relations one can find that there can exist such pairs of non-commuting observables $A$ and $B$ and such vectors that the lower bound for the product of standard deviations $\Delta A$ and $\Delta B$ calculated for these vectors is zero: $\Delta A\,\cdot\,\Delta B \geq 0$. Here we discuss examples of such cases and some other inconsistencies which can be found performing a rigorous analysis of the uncertainty relations in some special cases. As an illustration of such cases matrices $(2\times 2)$ and $(3 \times 3)$ and the position--momentum uncertainty relation for a quantum particle in the box are considered. The status of the uncertainty relation in $\cal PT$--symmetric quantum theory and the problems associated with it are also studied.