We show that a variant of the surface code---the XZZX code---offers remarkable performance for fault-tolerant quantum computation. The error threshold of this code matches what can be achieved with random codes (hashing) for every single-qubit Pauli noise channel; it is the first explicit code shown to have this universal property. We present numerical evidence that the threshold even exceeds this hashing bound for an experimentally relevant range of noise parameters. Focusing on the common situation where qubit dephasing is the dominant noise, we show that this code has a practical, high-performance decoder and surpasses all previously known thresholds in the realistic setting where syndrome measurements are unreliable. We go on to demonstrate the favorable sub-threshold resource scaling that can be obtained by specializing a code to exploit structure in the noise. We show that it is possible to maintain all of these advantages when we perform fault-tolerant quantum computation. We finally suggest some small-scale experiments that could exploit noise bias to reduce qubit overhead in two-dimensional architectures.

Cavity resonators are promising resources for quantum technology, while native nonlinear interactions for cavities are typically too weak to provide the level of quantum control required to deliver complex targeted operations. Here we investigate a scheme to engineer a target Hamiltonian for photonic cavities using ancilla qubits. By off-resonantly driving dispersively coupled ancilla qubits, we develop an optimized approach to engineering an arbitrary photon-number dependent (PND) Hamiltonian for the cavities while minimizing the operation errors. The engineered Hamiltonian admits various applications including canceling unwanted cavity self-Kerr interactions, creating higher-order nonlinearities for quantum simulations, and designing quantum gates resilient to noise. Our scheme can be implemented with coupled microwave cavities and transmon qubits in superconducting circuit systems.

Both photonic quantum computation and the establishment of a quantum internet require fiber-based measurement and feed-forward in order to be compatible with existing infrastructure. Here we present an all-fiber scheme for measurement and feed-forward, whose performance is benchmarked by carrying out remote preparation of single-photon polarization states at telecom-wavelengths. The result of a projective measurement on one photon deterministically controls the path a second photon takes with ultrafast optical switches. By placing well-calibrated passive polarization optics in the paths, we achieve a measurement and feed-forward fidelity of (99.0$\pm$ 0.5)%. Our methods are useful for photonic quantum experiments including computing, communication, and teleportation.

With advent of quantum internet, it becomes crucial to find novel ways to connect distributed quantum testbeds and develop novel technologies and research that extend innovations in managing the qubit performance. Numerous emerging technologies are focused on quantum repeaters and specialized hardware to extend the quantum distance over special-purpose channels. However, there is little work that utilizes current network technology, invested in optic technologies, to merge with quantum technologies. In this paper we argue for an AI-enabled control that allows optimized and efficient conversion between qubit and photon energies, to enable optic and quantum devices to work together. Our approach integrates AI techniques, such as deep reinforcement learning algorithms, with physical quantum transducer to inform real-time conversion between the two wavelengths. Learning from simulated environment, the trained AI-enabled transducer will lead to optimal quantum transduction to maximize the qubit lifetime.

We consider a finite two-dimensional Heisenberg triangular spin lattice at different degrees of anisotropy coupled to a dissipative Lindblad environment obeying the Born-Markovian constrain at finite temperature. We show how applying an inhomogeneous magnetic field to the system may significantly affect the entanglement distribution and properties among the spins in the asymptotic steady state of the system. Particularly, applying an inhomogeneous field with an inward (growing) gradient toward the central spin is found to considerably enhance the nearest neighbor entanglement and its robustness to the thermal dissipative decay effect in the completely anisotropic (Ising) system, whereas all the beyond nearest neighbor entanglements vanish entirely. Applying the same field to a partially anisotropic (XYZ) system, does not only enhance the nearest neighbor entanglements and their robustness but also all the beyond nearest neighbor ones. Nevertheless, the inhomogeneity of the field shows no effect on the asymptotic behavior of the entanglement in the isotropic (XXX) system, which vanishes under any system configuration. Moreover, the same inhomogeneous field exhibits the most influential impact, compared with the other ones, on the the spin dynamics as well. Although in the isotropic system the spins relax to a separable (disentangled) steady state with all the spins reaching a common spin state regardless of the field inhomogeneity, the spins in the steady state of the completely anisotropic system reach different distinguished spin states depending on their positions in the lattice. However, in the XYZ system, though the anisotropy is lower, the spin states become even more distinguished, accompanying the long range quantum correlation across the system, which is a sign of a critical behavior taking place at this combination of system anisotropy and field inhomogeneity.

We present a quantum algorithm for simulation of quantum field theory in the light-front formulation and demonstrate how existing quantum devices can be used to study the structure of bound states in relativistic nuclear physics. Specifically, we apply the Variational Quantum Eigensolver algorithm to find the ground state of the light-front Hamiltonian obtained within the Basis Light-Front Quantization framework. As a demonstration, we calculate the mass, mass radius, decay constant, electromagnetic form factor, and charge radius of the pion on the IBMQ Vigo chip. We consider two implementations based on different encodings of physical states, and propose a development that may lead to quantum advantage. This is the first time that the light-front approach to quantum field theory has been used to enable simulation of a real physical system on a quantum computer.

We introduce a novel Bayesian phase estimation technique based on adaptive grid refinement method. This method automatically chooses the number particles needed for accurate phase estimation using grid refinement and cell merging strategies such that the total number of particles needed at each step is minimal. The proposed method provides a powerful alternative to traditional sampling based sequential Monte Carlo method which tend to fail in certain instances such as when the posterior distribution is bimodal. We also combine grid based and sampling based methods as hybrid particle filter where grid based method can be used to estimate a small but dominant set of parameters and Liu-West (LW) based SMC for the remaining set of parameters. Principal kurtosis analysis can be used to decide the choice of parameters for grid refinement method and for sampling based methods. We provide numerical results comparing the performance of the proposed grid refinement method with Liu-West resampling based SMC. Numerical results suggest that the proposed method is quite promising for quantum phase estimation. It can be easily adapted to Hamiltonian learning which is a very useful technique for estimating unknown parameters of a Hamiltonian and for characterizing unknown quantum devices.

We introduce a new approach to Gottesman-Kitaev-Preskill (GKP) states that treats their finite-energy version in an exact manner. Based on this analysis, we develop new qubit-oscillator circuits that autonomously stabilize a GKP manifold, correcting errors without relying on qubit measurements. Finally, we show numerically that logical information encoded in GKP states is very robust against typical oscillator noise sources when stabilized by these new circuits.

We evaluate the exact dipole coupling strength between a single emitter and the radiation field within an optical cavity, taking into account the effects of multilayer dielectric mirrors. Our model allows one to freely vary the resonance frequency of the cavity, the frequency of light or atomic transition addressing it and the design wavelength of the dielectric mirror. The coupling strength is derived for an open system with unbound frequency modes. For very short cavities, the effective length used to determine their mode volume and the lengths defining their resonances are different, and also found to diverge appreciably from their geometric length, with the radiation field being strongest within the dielectric mirror itself. Only for cavities much longer than their resonant wavelength does the mode volume asymptotically approach that normally assumed from their geometric length.

Weak invariants are time-dependent observables with conserved expectation values. Their fluctuations, however, do not remain constant in time. On the assumption that time evolution of the state of an open quantum system is given in terms of a completely positive map, the fluctuations monotonically grow even if the map is not unital, in contrast to the fact that monotonic increases of both the von Neumann entropy and R\'enyi entropy require the map to be unital. In this way, the weak invariants describe temporal asymmetry in a manner different from the entropies. A formula is presented for time evolution of the covariance matrix associated with the weak invariants in the case when the system density matrix obeys the Gorini-Kossakowski-Lindblad-Sudarshan equation.

In this tutorial, we introduce basic conceptual elements to understand and build a gate-based superconducting quantum computing system.

A nonlinear phase shift is introduced to a Mach-Zehnder interferometer (MZI), and we present a scheme for enhancing the phase sensitivity. In our scheme, one input port of a standard MZI is injected with a coherent state and the other input port is injected with one mode of a two-mode squeezed-vacuum state. The final interference output of the MZI is detected with the method of active correlation output readout. Based on the optimal splitting ratio of beam splitters, the phase sensitivity can beat the standard quantum limit and approach the quantum Cram\'{e}r-Rao bound. The effects of photon loss on phase sensitivity are discussed. Our scheme can also provide some estimates for units of $\chi^{(3)}$, due to the relation between the nonlinear phase shift and the susceptibility $\chi^{(3)}$ of the Kerr medium.

We study the time-fluctuating magnetic gradient noise mechanisms in pairs of Si/SiGe quantum dots using exchange echo noise spectroscopy. We find through a combination of spectral inversion and correspondence to theoretical modeling that quadrupolar precession of the $^{73}$Ge nuclei play a key role in the spin-echo decay time $T_2$, with a characteristic dependence on magnetic field and the width of the Si quantum well. The $^{73}$Ge noise peaks appear at the fundamental and first harmonic of the $^{73}$Ge Larmor resonance, superimposed over $1/f$ noise due to $^{29}$Si dipole-dipole dynamics, and are dependent on material epitaxy and applied magnetic field. These results may inform the needs of dynamical decoupling when using Si/SiGe quantum dots as qubits in quantum information processing devices.

The output field from a continuously driven linear parametric oscillator may exhibit considerably more squeezing than the intracavity field. Inspired by this fact, we explore the nonclassical features of the steady-state output field of a driven nonlinear Kerr parametric oscillator using a temporal wave packet mode description. Utilizing a new numerical method, we have access to the density matrix of arbitrary wave packet modes. Remarkably, we find that even though the steady-state cavity field is always characterized by a positive Wigner function, the output may exhibit Wigner negativity, depending on the properties of the selected mode.

Being a very promising technology, with impressive advances in the recent years, it is still unclear how quantum computing will scale to satisfy the requirements of its most powerful applications. Although continued progress in the fabrication and control of qubits is required, quantum computing scalability will depend as well on a comprehensive architectural design considering a multi-core approach as an alternative to the traditional monolithic version, hence including a communications perspective. However, this goes beyond introducing mere interconnects. Rather, it implies consolidating the full communications stack in the quantum computer architecture. In this paper, we propose a double full-stack architecture encompassing quantum computation and quantum communications, which we use to address the monolithic versus multi-core question with a structured design methodology. For that, we revisit the different quantum computing layers to capture and model their essence by highlighting the open design variables and performance metrics. Using behavioral models and actual measurements from existing quantum computers, the results of simulations suggest that multi-core architectures may effectively unleash the full quantum computer potential.

We propose an efficient protocol to fully reconstruct a set of high-fidelity quantum gates. Usually, the efficiency of reconstructing high-fidelity quantum gates is limited by the sampling noise. Our protocol is based on a perturbative approach and has two stages. In the first stage, the initial part of noisy quantum gates is reconstructed by measuring traces of maps, and the trace can be measured by amplifying the noise in a way similar to randomised benchmarking and quantum spectral tomography. In the second stage, by amplifying the non-unital part using the unital part, we can efficiently reconstruct the non-unital part. We show that the number of measurements needed in our protocol scales logarithmically with the error rate of gates.

Understanding the intricate properties of one-dimensional quantum systems coupled to multiple reservoirs poses a challenge to both analytical approaches and simulation techniques. Fortunately, density matrix renormalization group-based tools, which have been widely used in the study of closed systems, have also been recently extended to the treatment of open systems. We present an implementation of such method based on state-of-the-art matrix product state (MPS) and tensor network methods, that produces accurate results for a variety of combinations of parameters. Unlike most approaches, which use the time-evolution to reach the steady-state, we focus on an algorithm that is time-independent and focuses on recasting the problem in exactly the same language as the standard Density Matrix Renormalization Group (DMRG) algorithm, initially put forward by M. C. Ba\~nuls et al. in Phys. Rev. Lett. 114, 220601 (2015). Hence, it can be readily exported to any of the available DMRG platforms. We show that this implementation is suited for studying thermal transport in one-dimensional systems. As a case study, we focus on the XXZ quantum spin chain and benchmark our results by comparing the spin current and magnetization profiles with analytical results. We then explore beyond what can be computed analytically. Our code is freely available on github at https://www.github.com/heitorc7/oDMRG.

The nitrogen-vacancy (NV) center in diamond has been established as a prime building block for quantum networks. However, scaling beyond a few network nodes is currently limited by low spin-photon entanglement rates, resulting from the NV center's low probability of coherent photon emission and collection. Integration into a cavity can boost both values via the Purcell effect, but poor optical coherence of near-surface NV centers has so far prevented their resonant optical control, as would be required for entanglement generation. Here, we overcome this challenge, and demonstrate resonant addressing of individual, fiber-cavity-coupled NV centers, and collection of their Purcell-enhanced coherent photon emission. Utilizing off-resonant and resonant addressing protocols, we extract Purcell factors of up to 4, consistent with a detailed theoretical model. This model predicts that the probability of coherent photon detection per optical excitation can be increased to 10% for realistic parameters - an improvement over state-of-the art solid immersion lens collection systems by two orders of magnitude. The resonant operation of an improved optical interface for single coherent quantum emitters in a closed-cycle cryogenic system at T $\sim$ 4 K is an important result towards extensive quantum networks with long coherence.

Quantum state tomography is a powerful, but resource-intensive, general solution for numerous quantum information processing tasks. This motivates the design of robust tomography procedures that use relevant resources as sparingly as possible. Important cost factors include the number of state copies and measurement settings, as well as classical postprocessing time and memory. In this work, we present and analyze an online tomography algorithm that is designed to optimize all the aforementioned resources at the cost of a worse dependence on accuracy. The protocol is the first to give optimal performance in terms of rank and dimension for state copies, measurement settings and memory. Classical runtime is also reduced substantially. Further improvements are possible by executing the algorithm on a quantum computer, giving a quantum speedup for quantum state tomography.

By developing a `two-crystal' method for color erasure, we can broaden the scope of chromatic interferometry to include optical photons whose frequency difference falls outside of the 400 nm to 4500 nm wavelength range, which is the passband of a PPLN crystal. We demonstrate this possibility experimentally, by observing interference patterns between sources at 1064.4 nm and 1063.6 nm, corresponding to a frequency difference of about 200 GHz.

The one-dimensional infinite square well is the simplest solution of quantum mechanics, and consequently one of the most important. In this article, we provide this solution using the real Hilbert space approach to quaternic quantum mechanics ($\mathbbm{H}$QM). We further provide the one-dimensional finite as well and a method to generate quaternic solutions from non-degenerate complex solutions.

The role of the spatiotemporal degrees of freedom in the preparation and observation of squeezed photonic states, produced by parametric down-conversion, is investigated. The analysis is done with the aid of a functional approach under the semi-classical approximation and the thin-crystal approximation. It is found that the squeezed state loses its minimum uncertainty property as the efficiency of down-conversion is increased, in a way that depends on the conditions of the homodyne measurements with which the amount of squeezing is determined.

The conceptual divide between classical physics and quantum mechanics has not been satisfactorily bridged as yet. The purpose of this paper is to show that such a bridge exists naturally in the Green-Wolf complex scalar representation of electromagnetic fields and its extension to massive fields.

We present an experimental realization of a 16 element, temporal-array, photon-number-resolving (PNR) detector, which is a multiplexed single-photon detector that splits an input signal over multiple time-bins, and the time-bins are detected using two superconducting nanowire single-photon detectors (SNSPD). A theoretical investigation of the PNR capabilities of the detector is performed and it is concluded that compared to a single-photon detector, our array detector can resolve one order of magnitude higher mean photon numbers, given the same number of input pulses to measure. This claim is experimentally verified and we show that the detector can accurately predict photon numbers between $10^{-3}$ to $10^{2}$. Our present detector is incapable of single-shot photon-number measurements with high precision since its effective quantum efficiency is $49\,\%$. Using SNSPDs with a higher quantum efficiency the PNR performance will improve, but the photon-number resolution will still be limited by the array size.

General purpose quantum computers can, in principle, entangle a number of noisy physical qubits to realise composite qubits protected against errors. Architectures for measurement-based quantum computing intrinsically support error-protected qubits and are the most viable approach for constructing an all-photonic quantum computer. Here we propose and demonstrate an integrated silicon photonic architecture that both entangles multiple photons, and encodes multiple physical qubits on individual photons, to produce error-protected qubits. We realise reconfigurable graph states to compare several schemes with and without error-correction encodings and implement a range of quantum information processing tasks. We observe a success rate increase from 62.5% to 95.8% when running a phase estimation algorithm without and with error protection, respectively. Finally, we realise hypergraph states, which are a generalised class of resource states that offer protection against correlated errors. Our results show how quantum error-correction encodings can be implemented with resource-efficient photonic architectures to improve the performance of quantum algorithms.

We investigate many-body localization of interacting spinless fermions in a one-dimensional disordered and tilted lattice. The fermions undergo energy-dependent transitions from ergodic to Stark many-body localization driven by the tilted potential, which are manifested by the appearance of mobility edges between delocalized states and Stark many-body localized states even when the disorder is weak. We can concretely diagnose these transitions rather than crossovers by finite-size scaling of energy-level statistics. Moreover, in the Stark many-body localization, the entanglement entropy obeys the area law scaling, in analogy to that in the conventional many-body localization.

Boosting is a general method to convert a weak learner (which generates hypotheses that are just slightly better than random) into a strong learner (which generates hypotheses that are much better than random). Recently, Arunachalam and Maity gave the first quantum improvement for boosting, by combining Freund and Schapire's AdaBoost algorithm with a quantum algorithm for approximate counting. Their booster is faster than classical boosting as a function of the VC-dimension of the weak learner's hypothesis class, but worse as a function of the quality of the weak learner. In this paper we give a substantially faster and simpler quantum boosting algorithm, based on Servedio's SmoothBoost algorithm.

The variance of (relative) surprisal (also known as varentropy) so far mostly plays a role in information theory as quantifying the leading order corrections to asymptotic i. i. d. limits. Here, we comprehensively study the use of it to derive single-shot results in (quantum) information theory. We show that it gives genuine sufficient and necessary conditions for approximate single-shot state-transitions in generic resource theories without the need for further optimization. We also clarify its relation to smoothed min- and max-entropies, and construct a monotone for resource theories using only the standard (relative) entropy and variance of (relative) surprisal. This immediately gives rise to enhanced lower bounds for entropy production in random processes. We establish certain properties of the variance of relative surprisal which will be useful for further investigations, such as uniform continuity and upper bounds on the violation of sub-additivity. Motivated by our results, we further derive a simple and physically appealing axiomatic single-shot characterization of (relative) entropy which we believe to be of independent interest. We illustrate our results with several applications, ranging from interconvertibility of ergodic states, over Landauer erasure to a bound on the necessary dimension of the catalyst for catalytic state transitions and Boltzmann's H-theorem.

It is known that distillation in continuous variable resource theories is impossible when restricted to Gaussian states and operations. To overcome this limitation, we enlarge the theories to include convex mixtures of Gaussian states and operations. This extension is operationally well-motivated since classical randomness is easily accessible. We find that resource distillation becomes possible for convex Gaussian resource theories-albeit in a limited fashion. We derive this limitation by studying the convex roof extension of a Gaussian resource measure and then go on to show that our bound is tight by means of example protocols for the distillation of squeezing and entanglement.

The Einstein-Podolsky-Rosen (EPR) paradox plays a fundamental role in our understanding of quantum mechanics, and is associated with the possibility of predicting the results of non-commuting measurements with a precision that seems to violate the uncertainty principle. This apparent contradiction to complementarity is made possible by nonclassical correlations stronger than entanglement, called steering. Quantum information recognises steering as an essential resource for a number of tasks but, contrary to entanglement, its role for metrology has so far remained unclear. Here, we formulate the EPR paradox in the framework of quantum metrology, showing that it enables the precise estimation of a local phase shift and of its generating observable. Employing a stricter formulation of quantum complementarity, we derive a criterion based on the quantum Fisher information that detects steering in a larger class of states than well-known uncertainty-based criteria. Our result identifies useful steering for quantum-enhanced precision measurements and allows one to uncover steering of non-Gaussian states in state-of-the-art experiments.

Using methods borrowed from machine learning we detect in a fully algorithmic way long range effects on local physical properties in a simple covalent system of carbon atoms. The fact that these long range effects exist for many configurations implies that atomistic simulation methods, such as force fields or modern machine learning schemes, that are based on locality assumptions, are limited in accuracy. We show that the basic driving mechanism for the long range effects is charge transfer. If the charge transfer is known, locality can be recovered for certain quantities such as the band structure energy.

A major test of the capabilities of modern quantum simulators and NISQ devices is the reliable realization of gauge theories, which constitute a gold standard of implementational efficacy. In addition to unavoidable unitary errors, realistic experiments suffer from decoherence, which compromises gauge invariance and, therefore, the gauge theory itself. Here, we study the effect of decoherence on the quench dynamics of a lattice gauge theory. Rigorously identifying the gauge violation as a divergence measure in the gauge sectors, we find at short times that it first grows diffusively $\sim\gamma t$ due to decoherence at environment-coupling strength $\gamma$, before unitary errors at strength $\lambda$ dominate and the violation grows ballistically $\sim\lambda^2t^2$. We further introduce multiple quantum coherences in the context of gauge theories to quantify decoherence effects. Both experimentally accessible measures will be of independent interest beyond the immediate context of this work.

For ultracold neutrons with a kinetic energy below 10 neV, strong scattering, characterized by $2\pi l_{c} / \lambda\leq 1$, can be obtained in metamaterials of C and $^7$Li. Here $l_{c}$ and $\lambda$ are the coherent scattering mean free path and the neutron wavelength, respectively. UCN interferometry and high-resolution spectroscopy (nano-electronvolt to pico-electronvolt resolution) in parallel waveguide arrays of neutronic metamaterials are given as examples of new experimental possibilities.

It is widely recognized that a magnetic system can only respond to a periodic driving significantly when the driving frequency matches the normal mode frequency of the magnet, which leads to magnetic resonance. Off-resonant phenomena are rarely considered for the diffculty to realize strong coupling between magnons and off-resonant waves. Here we examine the response of a magnetic system to squeezed light and surprisingly find that the magnons are maximally excited when the effective driving frequency is several orders of magnitude larger than the resonant frequency. The generated magnons are squeezed which brings the advantage of tunable squeezing through an external magnetic field. Further, we demonstrate that such off-resonant quasi-particle excitation is purely a quantum effect, which is rooted in the quantum fluctuations of particles in the squeezed vacuum. Our findings may provide an unconventional route to study off-resonant phenomena in a magnetic system and may further benefit the use of hybrid magnet-light systems in continuous variable quantum information.

We develop a unified framework for understanding the sign of fermion-mediated interactions by exploiting the symmetry classification of Green's functions. In particular, we establish a theorem regarding the sign of fermion-mediated interactions in systems with chiral symmetry. The strength of the theorem is demonstrated within multiple examples with an emphasis on electron-mediated interactions in materials.

We discuss how the Bohr-Sommerfeld quantization condition permeates the relationships between the quantization of Hall effect, the Berezenskii-Kosterlitz-Thouless vortex quantization, the Dirac magnetic monopole, the Haldane phase, the contact resistance in closed mesoscopic circuits of quantum physics. This paper is motivated by the recent derivation by one of the authors of the topological Chern number of the integer quantum Hall effect in electrical conductivity using a novel phase-space nonequilibirum quantum transport approach. The topological invariant in (~p, ~q; E, t)-phase space occurs to first-order in the gradient expansion of the nonequilibrium quantum transport equation. The Berry curvature related to orbital magnetic moment is also calculated leading to the quantization of orbital motion and edge states for 2-D systems. All of the above physical phenomena maybe unified simply from the geometric point of view of the old Bohr-Sommerfeld quantization, as a theory of Berry connection or a U (1) gauge theory

We show that conditional photon detection induces instantaneous phase synchronization between two decoupled quantum limit-cycle oscillators. We consider two quantum van der Pol oscillators without mutual coupling, each with an additional linearly coupled bath, and perform continuous measurement of photon counting on the output fields of the two baths interacting through a beam splitter. It is observed that in-phase or anti-phase coherence of the two decoupled oscillators instantaneously increases after the photon detection and then decreases gradually in the weak quantum regime or quickly in the strong quantum regime until the next photon detection occurs. In the strong quantum regime, quantum entanglement also increases after the photon detection and quickly disappears. We derive the analytical upper bounds for the increases in the quantum entanglement and phase coherence by the conditional photon detection in the quantum limit.

In this paper, we study quantum phase transitions and magnetic properties of a one-dimensional spin-1/2 Gamma model, which describes the off-diagonal exchange interactions between edge-shared octahedra with strong spin-orbit couplings along the sawtooth chain. The competing exchange interactions between the nearest neighbors and the second neighbors stabilize semimetallic ground state in terms of spinless fermions, and give rise to a rich phase diagram, which consists of three gapless phases. We find distinct phases are characterized by the number of Weyl nodes in the momentum space, and such changes in the topology of the Fermi surface without symmetry breaking produce a variety of Lifshitz transitions, in which the Weyl nodes situating at $k=\pi$ interchange from type I to type II. A coexistence of type-I and type-II Weyl nodes is found in phase II. The information measures including concurrence, entanglement entropy and relative entropy can effectively signal the second-order transitions. The results indicate that the Gamma model can act as an exactly solvable model to describe Lifshitz phase transitions in correlated electron systems.

We use an optimization procedure based on simulated bifurcation (SB) to solve the integer portfolio and trading trajectory problem with an unprecedented computational speed. The underlying algorithm is based on a classical description of quantum adiabatic evolutions of a network of non-linearly interacting oscillators. This formulation has already proven to beat state of the art computation times for other NP-hard problems and is expected to show similar performance for certain portfolio optimization problems. Inspired by such we apply the SB approach to the portfolio integer optimization problem with quantity constraints and trading activities. We show first numerical results for portfolios of up to 1000 assets, which already confirm the power of the SB algorithm for its novel use-case as a portfolio and trading trajectory optimizer.