### Algorithmic Phases in Variational Quantum Ground-State Preparation

The fidelity of a variational quantum circuit state prepared within stochastic gradient descent depends on, in addition to the circuit architecture, the number $N_s$ of measurements performed to estimate the gradient components. Simulating the variational quantum eigensolver (VQE) approach applied to two-dimensional frustrated quantum magnets, we observe that this dependence has systematic features. First, the algorithm manifests pronouncedly separated regimes on the $N_s$ axis with state fidelity $\mathcal{F}$ vanishing at $N_s < N_s^c$ and rapidly growing at $N_s > N_s^c$. {The point of transition $N_s^c$ is marked by a peak of energy variance, resembling the behaviour of specific heat in second-order phase transitions.} The extrapolation of the system-dependent threshold value $N_s^c$ to the thermodynamic limit suggests the possibility of obtaining sizable state fidelities with an affordable shots budget, even for large-scale spin clusters. Second, above $N_s^c$, the state infidelity $\mathcal{I} = 1 - \mathcal{F}$ satisfies $\mathcal{I} - \mathcal{I}_0 \propto 1/(\Delta^2 N_s)$, with $\mathcal{I}_0$ representing the circuit's inability to express the exact state, $\mathcal{F}$ is the achieved state fidelity, and $\Delta$ represents the system energy gap. This $1 / \Delta^2$ empirical law implies optimization resources increase inversely proportional to the squared gap of the system. We provide a symmetry-enhanced simulation protocol, which, in case of a closing gap, can significantly reduce the frustrated magnets simulation costs in quantum computers.

### Power and temperature dependent model for High Q superconductors

Measuring the internal quality factor of coplanar waveguide superconducting resonators is an established method of determining small losses in superconducting devices. Traditionally, the resonator losses are only attributed to two-level system (TLS) defects using a power dependent model for the quality factor. However, excess non-equilibrium quasiparticles can also limit the quality factor of the planar superconducting resonators used in circuit quantum electrodynamics. At millikelvin temperatures, quasiparticles can be generated by breaking Cooper pairs via a single high-energy or multiple sub-gap photons. Here a two-temperature, power and temperature dependent model is proposed to evaluate resonator losses for isolating TLS and quasiparticle loss simultaneously. The model combines the conventional TLS power and temperature dependence with an effective temperature non-equilibrium quasiparticle description of the superconducting loss. The quasiparticle description is based on the quasiparticle number density calculated using rate equations for an external quasiparticle generation source, recombination, and trapping. The number density is translated to an effective temperature using a thermal distribution that may be different from the bath. Experimental measurements of high-quality factor resonators fabricated from single crystal aluminum and titanium nitride thin films on silicon are interpreted with the presented model. This approach enables identification of quasiparticle and TLS loss, resulting in the determination that the TiN resonator has comparable TLS and quasiparticle loss at low power and low-temperature, while the low-temperature Al resonator behavior is dominated by non-equilibrium quasiparticle loss.

### A version of de Broglie's double solution theory reproducing Landau's quantization in a uniform magnetic field

In this work we develop a model based on the double solution theory of de Broglie in order to reproduce the famous Landau levels splitting in a constant magnetic field.

### Qubit-efficient simulation of thermal states with quantum tensor networks

We present a holographic quantum simulation algorithm to approximately prepare thermal states of $d$-dimensional interacting quantum many-body systems, using only enough hardware qubits to represent a ($d$-1)-dimensional cross-section. This technique implements the thermal state by approximately unraveling the quantum matrix-product density operator (qMPDO) into a stochastic mixture of quantum matrix product states (sto-qMPS), and variationally optimizing the quantum circuits that generate the sto-qMPS tensors, and parameters of the probability distribution generating the stochastic mixture. We demonstrate this technique on Quantinuum's trapped-ion quantum processor to simulate thermal properties of correlated spin-chains over a wide temperature range using only a single pair of hardware qubits. We then explore the representational power of two versions of sto-qMPS ansatzes for larger and deeper circuits through classical simulations and establish empirical relationships between the circuit resources and the accuracy of the variational free-energy.

### Scheduling Quantum Teleportation with Noisy Memories

Quantum teleportation channels can overcome the effects of photonic loss, a major challenge in the implementation of a quantum network over fiber. Teleportation channels are created by distributing an entangled state between two nodes which is a probabilistic process requiring classical communication. This causes critical delays that can cause information loss as quantum data suffers from decoherence when stored in memory. In this work, we quantify the effect of decoherence on fidelity at a node in a quantum network due to the storage of qubits in noisy memory platforms. We model the memory platform as a buffer that stores incoming qubits waiting for the creation of a teleportation channel. Memory platforms are parameterized with decoherence rate and buffer size, in addition to the order in which the incoming qubits are served. We show that fidelity at a node is a linear sum of terms, exponentially decaying with time, where the decay rate depends on the decoherence rate of the memory platform. This allows us to utilize Laplace Transforms to derive efficiently computable functions of average fidelity with respect to the load, buffer size, and decoherence rate of the memory platform. We prove that serving qubits in a Last In First Out order with pushout for buffer overflow management is optimal in terms of average fidelity. Lastly, we apply this framework to model a single repeater node to calculate the average fidelity of the teleportation channels created by this repeater assuming perfect gate operations.

### Engineered Josephson Parametric Amplifier in quantum two-modes squeezed radar

Josephson parametric amplifier (JPA) engineering is a significant component in the quantum two-mode squeezed radar (QTMS), to enhance, for instance, radar performance and the detection range or bandwidth. In this study, we apply quantum theory to a research domain focusing the design of QTMS radar. We apply engineered JPA (EJPA) to enhance the performance of a quantum radar (QR). We investigate the correlation between the signal and idler using and we propose strategies for maintaining entanglement at room temperature. We define the quantum signal-to-noise ratio (SNR) and detection range equations of the QTMS radar. The engineering JPA, leads to a remarkable improvement of the quantum radar performance, i.e. a large enhancement in quantum SNR of about 6 dB, a substantial improvement in the probability of detection through far fewer channels, and a huge increase in QTMS radar range, from half a meter in the conventional JPA to 482 m in the current study.

### Retrodictive Quantum Computing

Quantum models of computation are widely believed to be more powerful than classical ones. Efforts center on proving that, for a given problem, quantum algorithms are more resource efficient than any classical one. All this, however, assumes a standard predictive paradigm of reasoning where, given initial conditions, the future holds the answer. How about bringing information from the future to the present and exploit it to one's advantage? This is a radical new approach for reasoning, so-called Retrodictive Computation, that benefits from the specific form of the computed functions. We demonstrate how to use tools of symbolic computation to realize retrodictive quantum computing at scale and exploit it to efficiently, and classically, solve instances of the quantum Deutsch-Jozsa, Bernstein-Vazirani, Simon, Grover, and Shor's algorithms.

### Distributed quantum sensing with a mode-entangled network of spin-squeezed atomic states

Quantum sensors are used for precision timekeeping, field sensing, and quantum communication. Comparisons among a distributed network of these sensors are capable of, for example, synchronizing clocks at different locations. The performance of a sensor network is limited by technical challenges as well as the inherent noise associated with the quantum states used to realize the network. For networks with only local entanglement at each node, the noise performance of the network improves at best with square root of the number of nodes. Here, we demonstrate that nonlocal entanglement between network nodes offers better scaling with network size. A shared quantum nondemolition measurement entangles a clock network with up to four nodes. This network provides up to 4.5 dB better precision than one without nonlocal entanglement, and 11.6 dB improvement as compared to a network of sensors operating at the quantum projection noise limit. We demonstrate the generality of the approach with atomic clock and atomic interferometer protocols, in scientific and technologically relevant configurations optimized for intrinsically differential comparisons of sensor outputs.

### Dynamics and entanglement in quantum and quantum-classical systems: lessons for gravity

Motivated by quantum gravity, semiclassical theory, and quantum theory on curved spacetime, we study the system of an oscillator coupled to two spin-1/2 particles. This simple model provides a prototype for comparing three types of dynamics: the full quantum theory, the classical oscillator with spin backreaction, and spins propagating on a fixed oscillator background. From nonperturbative calculations of oscillator and entanglement entropy dynamics, we find that (i) entangled tripartite states produce novel oscillator trajectories, (ii) the three systems give equivalent dynamics for sufficiently weak oscillator-spin couplings, and (iii) spins driven by a classical oscillator, with or without backreaction, can produce entangled spin states. The latter result suggests a counterpoint to claims that gravity must be quantized to produce entangled matter states.

### Efficient Quantum State Tracking in Noisy Environments

Quantum state tomography, which aims to find the best description of a quantum state -- the density matrix, is an essential building block in quantum computation and communication. Standard techniques for state tomography are incapable of tracking changing states and often perform poorly in the presence of environmental noise. Although there are different approaches to solve these problems theoretically, experimental demonstrations have so far been sparse. Our approach, matrix-exponentiated gradient tomography, is an online tomography method that allows for state tracking, updates the estimated density matrix dynamically from the very first measurements, is computationally efficient, and converges to a good estimate quickly even with noisy data. The algorithm is controlled via a single parameter, its learning rate, which determines the performance and can be tailored in simulations to the individual experiment. We present an experimental implementation of matrix-exponentiated gradient tomography on a qutrit system encoded in the transverse spatial mode of photons. We investigate the performance of our method on stationary and evolving states, as well as significant environmental noise, and find fidelities of around 95% in all cases.

### Tightening monogamy and polygamy relations of unified entanglement in multipartite systems

We study the monogamy and polygamy inequalities of unified entanglement in multipartite quantum systems. We first derive the monogamy inequality of unified-$(q, s)$ entanglement for multi-qubit states under arbitrary bipartition, and then obtain the monogamy inequalities of the $\alpha$th ($0\leq\alpha\leq\frac{r}{2}, r\geq\sqrt{2}$) power of entanglement of formation for tripartite states and their generalizations in multi-qubit quantum states. We also generalize the polygamy inequalities of unified-$(q, s)$ entanglement for multi-qubit states under arbitrary bipartition. Moreover, we investigate polygamy inequalities of the $\beta$th ($\beta\geq \max\{1, s\}, 0\leq s\leq s_0, 0\leq s_0\leq\sqrt{2}$) power of the entanglement of formation for $2\otimes2\otimes2$ and $n$-qubit quantum systems. Finally, using detailed examples, we show that the results are tighter than previous studies.

### Multi-party quantum private comparison of size relation with d-level single-particle states

In this paper, by using d-level single-particle states, two novel multi-party quantum private comparison protocols for size relation comparison with two semi-honest third parties and one semi-honest third party are constructed, respectively. Here, each protocol can compare the size relation of secret integers from n parties rather than just the equality within one time execution. In each protocol, every third party is assumed to be semi-honest in the sense that she may misbehave on her own but is not allowed to collude with anyone else; and each party employs the qudit shifting operation to encode her secret integer. Each protocol can resist both the outside attack and the participant attack. Specially, each party's secret integer can be kept unknown to other parties and the third parties. The proposed protocol with two third parties is workable in a stranger environment, as there are no communication and no pre-shared key between each pair of party. The proposed protocol with one third party is workable in an acquaintance environment, as all parties need to share a common private key beforehand.

### Measure-resend semi-quantum private comparison without entanglement

In this paper, we successfully design the semi-quantum private comparison (SQPC) protocol with the measure-resend characteristic by using two-particle product states as the initial prepared quantum resource which allows two classical users to compare the equality of their private secrets under the help of a quantum third party (TP). The quantum TP is semi-honest in the sense that he is allowed to misbehave on his own but cannot conspire with either of users. Both the output correctness and the security against the outside attack and the participant attack can be guaranteed. Compared with the previous SQPC protocols, the advantage of our protocol lies in that it only employs two-particle product states as the initial prepared quantum resource, only requires TP to perform single-photon measurements and does not need quantum entanglement swapping. Our protocol can be realized with current quantum technologies.

### Quantum discords of tripartite quantum systems

The quantum discord of bipartite systems is one of the best-known measures of non-classical correlations and an important quantum resource. In the recent work appeared in [Phys. Rev. Lett 2020, 124:110401], the quantum discord has been generalized to multipartite systems. In this paper, we give analytic solutions of the quantum discord for tripartite states with fourteen parameters.

### Local unitary equivalence of generic multi-qubits based on the CP decomposition

The CANDECOMP/PARAFAC (CP) decomposition is a generalization of the spectral decomposition of matrices to higher-order tensors. In this paper we use the CP decomposition to study unitary equivalence of higher order tensors and construct several invariants of local unitary equivalence for general higher order tensors. Based on this new method, we study the coefficient tensors of $3$-qubit states and obtain a necessary and sufficient criterion for local unitary equivalence of general tripartite states in terms of the CP decomposition. We also generalize this method to obtain some invariants of local unitary equivalence for general multi-partite qudits.

### Learning quantum graph states with product measurements

We consider the problem of learning $N$ identical copies of an unknown $n$-qubit quantum graph state with product measurements. These graph states have corresponding graphs where every vertex has exactly $d$ neighboring vertices. Here, we detail an explicit algorithm that uses product measurements on multiple identical copies of such graph states to learn them. When $n \gg d$ and $N = O(d \log(1/\epsilon) + d^2 \log n ),$ this algorithm correctly learns the graph state with probability at least $1- \epsilon$. From channel coding theory, we find that for arbitrary joint measurements on graph states, any learning algorithm achieving this accuracy requires at least $\Omega(\log (1/\epsilon) + d \log n)$ copies when $d=o(\sqrt n)$. We also supply bounds on $N$ when every graph state encounters identical and independent depolarizing errors on each qubit.

### Extraction of ergotropy: free energy bound and application to open cycle engines

The second law of thermodynamics uses change in free energy of macroscopic systems to set a bound on performed work. Ergotropy plays a similar role in microscopic scenarios, and is defined as the maximum amount of energy that can be extracted from a system by a unitary operation. In this analysis, we quantify how much ergotropy can be induced on a system as a result of system's interaction with a thermal bath, with a perspective of using it as a source of work performed by microscopic machines. We provide the fundamental bound on the amount of ergotropy which can be extracted from environment in this way. The bound is expressed in terms of the non-equilibrium free energy difference and can be saturated in the limit of infinite dimension of the system's Hamiltonian. The ergotropy extraction process leading to this saturation is numerically analyzed for finite dimensional systems. Furthermore, we apply the idea of extraction of ergotropy from environment in a design of a new class of stroke heat engines, which we label open-cycle engines. Efficiency and work production of these machines can be completely optimized for systems of dimensions 2 and 3, and numerical analysis is provided for higher dimensions.

### Entropic Accord: A new measure in the quantum correlation hierarchy

Quantum correlation often refers to correlations exhibited by two or more local subsystems under a suitable measurement. These correlations are beyond the framework of classical statistics and the associated classical probability distribution. Quantum entanglement is the most well known of such correlations and plays an important role in quantum information theory. However, there exist non-entangled states that still possess quantum correlations which cannot be described by classical statistics. One such measure that captures these non-classical correlations is discord. Here we introduce a new measure of quantum correlations which we call entropic accord that fits between entanglement and discord. It is defined as the optimised minimax mutual information of the outcome of the projective measurements between two parties. We show a strict hierarchy exists between entanglement, entropic accord and discord. We study two-qubit states which shows the relationship between the three entropic quantities. In addition to revealing a class of correlations that are distinct from discord and entanglement, the entropic accord measure can be inherently more intuitive in certain contexts.

### Reconstructing Non-Markovian Open Quantum Evolution From Multi-time Measurements

For a quantum system undergoing non-Markovian open quantum dynamics, we demonstrate a tomography algorithm based on multi-time measurements of the system, which reconstructs a minimal environment coupled to the system, such that the system plus environment undergoes unitary evolution and that the reduced dynamics of the system is identical to the observed dynamics of it. The reconstructed open quantum evolution model can be used to predict any future dynamics of the system when it is further assumed to be time-independent. We define the memory size and memory complexity for the non-Markovian open quantum dynamics which characterize the complexity of the reconstruction.

### Security Proof for Circular Mediated Semi-quantum Key Distribution

Mediated semi-quantum key distribution (M-SQKD) allows two limited semiquantum or "classical" users to share a secret key with the help of a quantum third party (TP), who may be untrusted. Several such protocols have been investigated recently, but no one has yet considered the security proof of M-SQKD based on circular transport structure. In this paper, we design a circular M-SQKD protocol based on Bell states and prove the protocol is unconditional security in the asymptotic scenario. By utilizing the parameters observed in the channel, we can compute a lower bound of our protocol's key rate and derive a threshold value of noise tolerance. Besides, due to the scalability of the circular transport structure, our M-SQKD protocol can be easily extended to multi-party scenarios, which offers an approach of realizing multiple "classical" users' key distribution. Compared with the previous M-SQKD protocols, our protocol has comparable noise tolerance and can realize key distribution between multiple "classical" users. Thus, the proposed protocol may have potential applications in future quantum communications based on these characteristics.

### Quantum control of tunable-coupling transmons using dynamical invariants of motion

We analyse the implementation of a fast nonadiabatic CZ gate between two transmon qubits with tuneable coupling. The gate control method is based on a theory of dynamical invariants which leads to reduced leakage and robustness against decoherence. The gate is based on a description of the resonance between the $|11\rangle$ and $|20\rangle$ using an effective Hamiltonian with the 6 lowest energy states. A modification of the invariants method allows us to take into account the higher-order perturbative corrections of this effective model. This enables a gate fidelity several orders of magnitude higher than other quasiadiabatic protocols, with gate times that approach the theoretical limit.

### Fast scanning nitrogen-vacancy magnetometry by spectrum demodulation

We demonstrate a spectrum demodulation technique for greatly speeding up the data acquisition rate in scanning nitrogen-vacancy center magnetometry. Our method relies on a periodic excitation of the electron spin resonance by fast, wide-band frequency sweeps combined with a phase-locked detection of the photo-luminescence signal. The method can be extended by a frequency feedback to realize real-time tracking of the spin resonance. Fast scanning magnetometry is especially useful for samples where the signal dynamic range is large, of order millitesla, like for ferro- or ferrimagnets. We demonstrate our method by mapping stray fields above the model antiferromagnet $\alpha$-Fe$_2$O$_3$ (hematite) at pixel rates of up to 100\,Hz and an image resolution exceeding one megapixel.

### Inverse design in nuclear quantum optics: From artificial x-ray multi-level schemes to spectral observables

Ensembles of M\"ossbauer nuclei embedded in thin-film cavities form a promising platform for x-ray quantum optics. A key feature is that the joint nuclei-cavity system can be considered as an artificial x-ray multi-level scheme in the low-excitation regime. Using the cavity environment, the structure and parameters of such level schemes can be tailored beyond those offered by the bare nuclei. However, so far, the direct determination of a cavity structure providing a desired quantum optical functionality has remained an open challenge. Here, we address this challenge using an inverse design methodology. As a first qualitative result, we show that the established fitting approach based on scattering observables in general is not unique, since the analysis may lead to different multi-level systems for the same cavity if based on observables in different scattering channels. Motivated by this, we distinguish between scattering signatures and the microscopic level scheme as separate design objectives, with the latter being uniquely determined by an \textit{ab initio} approach. We find that both design objectives are of practical relevance and that they complement each other regarding potential applications. We demonstrate the inverse design for both objectives using example tasks, such as realising electromagnetically induced transparency. Our results pave the way for new applications in nuclear quantum optics involving more complex x-ray cavity designs.

### The geometric link between Hardy and Clauser-Horne-Shimony-Holt

We show that the Hardy nonlocality condition is equivalent to the violation of the CHSH inequality with additional constraints. We adapt the geometrical optimization of the violation of the CHSH inequality to these additional constraints and show that the Hardy condition is equivalent to optimizing the length difference of two sides in a triangle. Furthermore, we discuss the effects of the different constraints.

### Barren plateaus from learning scramblers with local cost functions

The existence of barren plateaus has recently revealed new training challenges in quantum machine learning (QML). Uncovering the mechanisms behind barren plateaus is essential in understanding the scope of problems that QML can efficiently tackle. Barren plateaus have recently been shown to exist when learning global properties of random unitaries. Establishing whether local cost functions can circumvent these barren plateaus is pertinent if we hope to apply QML to quantum many-body systems. We prove a no-go theorem showing that local cost functions encounter barren plateaus in learning random unitary properties.

### Provably Optimal Control for Multiplicative Amplitude Control Noise

We provide a technique to obtain provably optimal control sequences for quantum systems under the influence of time-correlated multiplicative control noise. Utilizing the circuit-level noise model introduced in [Phys. Rev. Research 3, 033229(2021)], we show that we can map the problem of finding such a sequence to a convex optimization problem with guaranteed optimality that follows from the convexity. We also show that this technique is compatible with more general off-axis time-correlated dephasing noise. In spite of losing provable optimality, numerically optimized control sequences under this scenario can still achieve nearly optimal performance when the control noise is strong relative to the dephasing contribution. This approach will enable the development of optimal quantum logic gates in systems where noise due to amplitude drifts in the control is strong relative to dephasing such as in ion-trap based quantum computers or in the limit of fast control.

### Squeezed light generated with hyperradiance without nonlinearity

We propose that the squeezed light accompanied by hyperradiance is induced by quantum interference in a linear system consisting of a high quality optical cavity and two coherently driven two-level qubits. When two qubits are placed at the crest and trough of the standing wave in the cavity respectively (i.e., they have the opposite coupling coefficient to the cavity), we show that squeezed light is generated in the hyperradiance regime under the conditions of strong coupling and weak driving. Simultaneously, the Klyshko's criterion alternates up and down at unity when the photon number is even or odd. Moreover, the orthogonal angles of the squeezed light can be controlled by adjusting the frequency detuning pressure between the driving field and the qubits. It can be implemented in a variety of quantum systems, including but not limited to two-level systems such as atoms, quantum dots in single-mode cavities.

### Time Series Quantum Reservoir Computing with Weak and Projective Measurements

Quantum machine learning represents a promising avenue for data processing, also for purposes of sequential temporal data analysis, as recently proposed in quantum reservoir computing (QRC). The possibility to operate on several platforms and noise intermediate-scale quantum devices makes QRC a timely topic. A challenge that has not been addressed yet, however, is how to efficiently include quantum measurement in realistic protocols, while retaining the reservoir memory needed for sequential time series processing and preserving the quantum advantage offered by large Hilbert spaces. In this work, we propose different measurement protocols and assess their efficiency in terms of resources, through theoretical predictions and numerical analysis. We show that it is possible to exploit the quantumness of the reservoir and to obtain ideal performance both for memory and forecasting tasks with two successful measurement protocols. One consists in rewinding part of the dynamics determined by the fading memory of the reservoir and storing the corresponding data of the input sequence, while the other employs weak measurements operating online at the trade-off where information can be extracted accurately and without hindering the needed memory. Our work establishes the conditions for efficient protocols, being the fading memory time a key factor, and demonstrates the possibility of performing genuine online time-series processing with quantum systems.

### Black Holes Decohere Quantum Superpositions

We show that if a massive body is put in a quantum superposition of spatially separated states, the mere presence of a black hole in the vicinity of the body will eventually destroy the coherence of the superposition. This occurs because, in effect, the gravitational field of the body radiates soft gravitons into the black hole, allowing the black hole to acquire "which path" information about the superposition. A similar effect occurs for quantum superpositions of electrically charged bodies. We provide estimates of the decoherence time for such quantum superpositions. We believe that the fact that a black hole will eventually decohere any quantum superposition may be of fundamental significance for our understanding of the nature of black holes in a quantum theory of gravity.

### Transport and entanglement growth in long-range random Clifford circuits

Conservation laws and hydrodynamic transport can constrain entanglement dynamics in isolated quantum systems, manifest in a slowdown of higher R\'enyi entropies. Here, we introduce a class of long-range random Clifford circuits with U$(1)$ symmetry, which act as minimal models for more generic quantum systems and provide an ideal framework toexplore this phenomenon. Depending on the exponent $\alpha$ controlling the probability $\propto r^{-\alpha}$ of gates spanning a distance $r$, transport in such circuits varies from diffusive to superdiffusive and then to superballistic. We unveil that the different hydrodynamic regimes reflect themselves in the asymptotic entanglement growth according to $S(t) \propto t^{1/z}$, where $z$ is the $\alpha$-dependent dynamical transport exponent. We explain this finding in terms of the inhibited operator spreading in U$(1)$-symmetric Clifford circuits, where the emerging light cones are intimately related to the transport behavior and are significantly narrower compared to circuits without conservation law. For sufficiently small $\alpha$, we show that the presence of hydrodynamic modes becomes irrelevant such that $S(t)$ behaves similarly in circuits with and without conservation law. Our work sheds light on the interplay of transport and entanglement and emphasizes the usefulness of constrained Clifford circuits to explore questions in quantum many-body dynamics.

### Average capacity of quantum entanglement

As an alternative to entanglement entropies, the capacity of entanglement becomes a promising candidate to probe and estimate the degree of entanglement of quantum bipartite systems. In this work, we study the typical behavior of entanglement capacity over major models of random states. In particular, the exact and asymptotic formulas of average capacity have been derived under the Hilbert-Schmidt and Bures-Hall ensembles. The obtained formulas generalize some partial results of average capacity computed recently in the literature. As a key ingredient in deriving the results, we make use of recent advances in random matrix theory pertaining to the underlying orthogonal polynomials and special functions. Numerical study has been performed to illustrate the usefulness of average capacity as an entanglement indicator.

### Design and Implementation of a Quantum Kernel for Natural Language Processing

Natural language processing (NLP) is the field that attempts to make human language accessible to computers, and it relies on applying a mathematical model to express the meaning of symbolic language. One such model, DisCoCat, defines how to express both the meaning of individual words as well as their compositional nature. This model can be naturally implemented on quantum computers, leading to the field quantum NLP (QNLP). Recent experimental work used quantum machine learning techniques to map from text to class label using the expectation value of the quantum encoded sentence. Theoretical work has been done on computing the similarity of sentences but relies on an unrealized quantum memory store. The main goal of this thesis is to leverage the DisCoCat model to design a quantum-based kernel function that can be used by a support vector machine (SVM) for NLP tasks. Two similarity measures were studied: (i) the transition amplitude approach and (ii) the SWAP test. A simple NLP meaning classification task from previous work was used to train the word embeddings and evaluate the performance of both models. The Python module lambeq and its related software stack was used for implementation. The explicit model from previous work was used to train word embeddings and achieved a testing accuracy of $93.09 \pm 0.01$%. It was shown that both the SVM variants achieved a higher testing accuracy of $95.72 \pm 0.01$% for approach (i) and $97.14 \pm 0.01$% for (ii). The SWAP test was then simulated under a noise model defined by the real quantum device, ibmq_guadalupe. The explicit model achieved an accuracy of $91.94 \pm 0.01$% while the SWAP test SVM achieved 96.7% on the testing dataset, suggesting that the kernelized classifiers are resilient to noise. These are encouraging results and motivate further investigations of our proposed kernelized QNLP paradigm.

### Correlated driving-and-dissipation equation for non-Condon spectroscopy with the Herzberg-Teller vibronic coupling

Correlated driving-and-dissipation equation (CODDE) is an optimized complete second-order quantum dissipation approach, which is originally concerned with the reduced system dynamics only. However, one can actually extract the hybridized bath dynamics from CODDE with the aid of dissipaton-equation-of-motion theory, a statistical quasi-particle quantum dissipation formalism. Treated as an one{dissipaton theory, CODDE is successfully extended to deal with the Herzberg-Teller vibronic couplings in dipole-field interactions. Demonstrations will be carried out on the non-Condon spectroscopies of a model dimer system.

### Angular-time evolution for the Affleck-Kennedy-Lieb-Tasaki chain and its edge-state dynamics

We study the angular-time evolution that is a parameter-time evolution defined by the entanglement Hamiltonian for the bipartitioned ground state of the Affleck-Kennedy-Lieb-Tasaki (AKLT) chain with the open boundary. In particular, we analytically calculate angular-time spin correlation functions $\langle S_n^\alpha(\tau)S_n^\alpha(0)\rangle$ with $\alpha = x,y,z$, using the matrix-product-state (MPS) representation of the valence-bond-solid state with edges. We also investigate how the angular-time evolution operator can be represented in the physical spin space with the use of gauge transformation for the MPS. We then discuss the physical interpretation of the angular-time evolution in the AKLT chain.

### Emergence and Dynamical Stability of Charge Time-Crystal in a Current-Carrying Quantum Dot Simulator

Periodically-driven open quantum systems that never thermalize exhibit a discrete time-crystal behavior, a non-equilibrium quantum phenomenon that has shown promise in quantum information processing applications. Measurements of time-crystallinity are currently limited to (magneto-) optical experiments in atom-cavity systems and spin-systems making it an indirect measurement. We theoretically show that time-crystallinity can be measured directly in the charge-current from a spin-less Hubbard ladder, which can be simulated on a quantum-dot array. We demonstrate that one can dynamically tune the system out and then back into the time-crystal phase, proving its robustness against external forcings. These findings motivate further theoretical and experimental efforts to simulate the time-crystal phenomena in current-carrying nano-scale systems.

### Exact Factorization Adventures: A Promising Approach for Non-bound States

Modeling the dynamics of non-bound states in molecules requires an accurate description of how electronic motion affects nuclear motion and vice-versa. The exact factorization (XF) approach offers a unique perspective, in that it provides potentials that act on the nuclear subsystem or electronic subsystem, which contain the effects of the coupling to the other subsystem in an exact way. We briefly review the various applications of the XF idea in different realms, and how features of these potentials aid in the interpretation of two different laser-driven dissociation mechanisms. We present a detailed study of the different ways the coupling terms in recently-developed XF-based mixed quantum-classical approximations are evaluated, where either truly coupled trajectories, or auxiliary trajectories that mimic the coupling are used, and discuss their effect in both a surface-hopping framework as well as the rigorously-derived coupled-trajectory mixed quantum-classical approach.

### High-order harmonic generations in tilted Weyl semimetals

We investigate high-order harmonic generations (HHGs) under the comparison of Weyl cones in two types. Due to the hyperboloidal electron pocket structure, strong noncentrosymmetrical generations in high orders are observed around a single type-II Weyl point, especially at frequency zero. Such remarkable DC signal is proved to have attributions from the intraband transition after spectral decomposition. Under weak pulse electric field , the linear optical response of a non-tilted Weyl cone is consistent with the Kubo theory. With more numerical simulations, we conclude the non-zero chemical potential can enhance the even-order generations, from the slightly tilted system to the over-tilted systems. In consideration of dynamical symmetries, type-I and -II Weyl cones also show different selective responses under the circularly polarized light. Finally, using a more realistic model containing two pairs of Weyl points, we demonstrate the paired Weyl points with opposite chirality could suppress the overall even-order generations.

### Photoinduced prethermal order parameter dynamics in the two-dimensional large-$N$ Hubbard-Heisenberg model

A central topic in current research in non-equilibrium physics is the design of pathways to control and induce order in correlated electron materials with time-dependent electromagnetic fields. The theoretical description of such processes, in particular in two spatial dimensions, is very challenging and often relies on phenomenological modelling in terms of free energy landscapes. Here, we present a semiclassical scheme that describes dephasing dynamics beyond mean-field and allows to simulate the light-induced manipulation of prethermal order in a two-dimensional model with competing phases microscopically. We calculate the time-evolution of the relevant order parameters under pulsed driving. We find that the induced prethermal order does not depend on the amount of absorbed energy alone but also explicitly on the driving frequency and amplitude. While this dependency is pronounced in the low-frequency regime, it is suppressed at high driving frequencies.

### Driving Force and Nonequilibrium Vibronic Dynamics in Charge Separation of Strongly Bound Electron-Hole Pairs

Electron-hole pairs in organic photovoltaics dissociate efficiently despite their Coulomb-binding energy exceeding thermal energy at room temperature. The electronic states involved in charge separation couple to structured vibrational environments containing multiple underdamped modes. The non-perturbative simulations of such large, spatially extended electronic-vibrational (vibronic) systems remains an outstanding challenge. Current methods bypass this difficulty by considering effective one-dimensional Coulomb potentials or unstructured environments. Here we extend and apply a recently developed method for the non-perturbative simulation of open quantum systems to the dynamics of charge separation in one, two and three-dimensional donor-acceptor networks. This allows us to identify the precise conditions in which underdamped vibrational motion induces efficient long-range charge separation. Our analysis provides a comprehensive picture of ultrafast charge separation by showing how different mechanisms driven either by electronic or vibronic couplings are well differentiated for a wide range of driving forces and how entropic effects become apparent in large vibronic systems. These results allow us to quantify the relative importance of electronic and vibronic contributions in organic photovoltaics and provide a toolbox for the design of efficient charge separation pathways in artificial nanostructures.

### Entanglement distillation towards minimal bond cut surface in tensor networks

We propose that a minimal bond cut surface is characterized by entanglement distillation in tensor networks. Our proposal is not only consistent with the holographic models of perfect or tree tensor networks, but also can be applied for several different classes of tensor networks including matrix product states and multi-scale entanglement renormalization ansatz. We confirmed our proposal by a numerical simulation based on the random tensor network. The result sheds new light on a deeper understanding of the Ryu-Takayanagi formula for entanglement entropy in holography.

### Unveiling the mixed nature of polaritonic transport: From enhanced diffusion to ballistic motion approaching the speed of light

In recent years it has become clear that the transport of excitons and charge carriers in molecular systems can be enhanced by coherent coupling with photons, giving rise to the formation of hybrid excitations known as polaritons. Such enhancement has far-reaching technological implications, however, the enhancement mechanism and the transport nature of these composite light-matter excitations in such systems still remain elusive. Here we map the ultrafast spatiotemporal dynamics of surface-bound optical waves strongly coupled to a self-assembled molecular layer and fully resolve them in energy/momentum space. Our studies reveal intricate behavior which stems from the hybrid nature of polaritons. We find that the balance between the molecular disorder and long-range correlations induced by the coherent mixing between light and matter leads to a mobility transition between diffusive and ballistic transport, which can be controlled by varying the light-matter composition of the polaritons. Furthermore, we directly demonstrate that the coupling with light can enhance the diffusion coefficient of molecular excitons by six orders of magnitude and even lead to ballistic flow at two-thirds the speed of light.

### Spatial entanglement in two dimensional QCD: Renyi and Ryu-Takayanagi entropies

We derive a general formula for the replica partition function in the vacuum state, for a large class of interacting theories with fermions, with or without gauge fields, using the equal-time formulation on the light front. The result is used to analyze the spatial entanglement of interacting Dirac fermions in two-dimensional QCD. A particular attention is paid to the issues of infrared cut-off dependence and gauge invariance. The Renyi entropy for a single interval, is given by the rainbow dressed quark propagator to order ${\cal O}(N_c)$. The contributions to order ${\cal O}(1)$, are shown to follow from the off-diagonal and off mass-shell mesonic T-matrix, with no contribution to the central charge. The construction is then extended to mesonic states on the light front, and shown to probe the moments of the partonic PDFs for large light-front separations. In the vacuum and for small and large intervals, the spatial entanglement entropy following from the Renyi entropy, is shown to be in agreement with the Ryu-Takayanagi geometrical entropy, using a soft-wall AdS$_3$ model of two-dimensional QCD.

### Dynamical maps and symmetroids

Starting from the canonical symmetroid $\mathcal{S}(G)$ associated with a groupoid $G$, the issue of describing dynamical maps in the groupoidal approach to Quantum Mechanics is addressed. After inducing a Haar measure on the canonical symmetroid $\mathcal{S}(G)$, the associated von-Neumann groupoid algebra is constructed. It is shown that the left-regular representation allows to define linear maps on the groupoid-algebra of the groupoid $G$ and given subsets of functions are associated with completely positive maps. Some simple examples are also presented.