Papers

Harnessing active matter to power functional devices has attracted growing interest. We propose a minimal battery model for storing energy extracted from active matter and derive its charging and discharging characteristics analytically. When the energetic cost of sustaining activity is included, the charging power-efficiency trade-off becomes loop-like and the efficiency attains a finite-time maximum, in sharp contrast to passive energy devices.

A majority of measurable dynamic quantities of a physical system is described as an equilibrium correlation function of the form $G_{AB}(t-t') = \left<A(t)B(t')\right>$, where $t'$ represents the impact time and $t$ represents the response time. Conventional quantum algorithms to compute such quantities via Hamiltonian simulation such as the Hadamard test, variational methods, or linear-response-based algorithms share one feature in common: each time point $t-t'$ is calculated by a different quantum circuit, leading to at least $O(N_t)$ total shots, and at least $O(N_t^2)$ total quantum gates. Here, we provide a different approach: utilizing $O(\log N_t)$ ancilla qubits to store time as a quantum variable, and effectively parallelize the computation of an equilibrium correlation function by requiring only a single quantum circuit for all time points. We show that this circuit needs to be run only $O(\log N_t)$ times, which in total requires $O(N_t \log N_t)$ quantum resources in the large system size limits. We prove that this scaling is optimal, and demonstrate our algorithm calculating the Green's function of a one-dimensional spinless Hubbard model.

We develop a spectral theory for dynamical large deviations in non-Markov jump processes and non-Markov chains, whose dynamics depends on the past through state- and jump-dependent empirical observables. We demonstrate that a multiscale Wentzel-Kramers-Brillouin-Jeffreys (WKBJ) Ansatz separates fast configurational relaxation from slow memory evolution, reducing the Feynman–Kac equation for occupation and flux statistics to an eigenvalue problem for a new tilted operator coupled to Hamilton–Jacobi characteristics. This provides a computationally efficient framework for quantifying fluctuations in a broad class of non-Markovian systems. We illustrate our general results with a bistable self-induced East model.

Simulating open quantum systems reveals how environmental coupling shapes relaxation, excitation transport, and the dynamics of quantum correlations. On quantum hardware, dissipative channels add operations and might seem to increase cost. However, we show that a broad class of Pauli noise, including depolarization, can ease quantum and classical simulation by exponentially suppressing high-weight components of Heisenberg-evolved observables. For bounded-degree Lindblad dynamics, this permits system-size-independent Trotter steps when estimating local observables. Combining this compression with Richardson extrapolation, we provide classical and quantum error bounds with respect to the 2-norm of the propagated observable. These give high-probability expectation-value guarantees for random input states or random Hamiltonians acting on a fixed state and control (out-of-time-order) correlators. Concretely, we show that expectation values of local observables at time $t$ can be estimated to accuracy $\varepsilon$ using circuits of maximum depth $O\!\left(\left[1+\left(t/\gamma\right)^{3/2}\right]\log^2(1/\varepsilon)\right)$, independent of the system size $n$. Classically, we show that sparse Pauli propagation runs in time polynomial in $n$, $t$ and $1/\varepsilon$ for every fixed $\gamma>0$, with polynomial degree $O(1/\gamma)$ in the weak dissipation limit. The gap between our quantum and classical upper bounds leaves room for a substantial polynomial quantum advantage as dissipation weakens, an observation further supported by our numerical results.

In this article, we explore how transport at a solid–electrolyte interface is affected by the internal fluctuations of the solid and quantify the role of solid excitations on ionic friction. Combining molecular dynamics simulations with theoretical modeling, we show that the presence of ions enhances the dynamical response of the electrolyte compared to pure water. This leads to an ionic contribution to the fluctuation-induced interfacial friction. We show that collective ionic modes dominate the dissipation, such that ionic friction arises from many-body effects rather than from individual drag forces. Furthermore, high-frequency molecular modes are found to be ion-specific, suggesting possible routes toward spectral separation of ions, with implications for desalination, filtration, and energy conversion.

Measures of non-stabilizerness (“magic”) of a quantum evolution quantify how hard it is to simulate on a classical computer. A natural question is how to extract information about magic from physical correlators, for example at finite temperature or in the ground state. In this paper we show how to measure magic using out-of-time-order correlators evaluated in an arbitrary state. For every full-rank state, and in particular for every Gibbs state, the resulting quantity vanishes if and only if the evolution is Clifford. We establish its basic properties, relate it to the stabilizer entropy of the Choi state, and prove that, even with a single fixed probe operator, it gives a lower bound on the number of $T$ gates needed to implement the dynamics in any circuit architecture.

Planar parity-preserving tensor networks (ppTNs) admit a fermionic formulation in which the Gaussian, efficiently contractible limit is deformed by local interactions. We investigate how many-body methods can be used to analyze such contractions in a minimal two-parameter ppTN. Its contraction is simultaneously an interacting fermionic partition function, a loop gas, a deformed toric-code norm, and a quartic Ising model, allowing the same phase diagram to be approached with complementary tools. At its center lies a `topological island', characterized beyond the Gaussian limit by a boundary-twist $\mathbb Z_2$ indicator that reduces to Chern-number parity and admits a loop-winding interpretation under duality. Fermionic perturbation theory predicts the interacting phase boundaries, tensor-network numerics establish their critical behavior, and the loop and spin descriptions reveal a self-dual line with a $c=1$ four-state-Potts multicritical fixed point.

The Scrooge ensemble associated with a density matrix $\rho$ is the maximally random ensemble whose average density matrix is $\rho$. It has attracted growing interest for its applications to quantum many-body dynamics and quantum information. The intrinsic statistical properties of the Scrooge ensembles are encoded in their higher-order moments, yet they are nontrivial to analyze and prepare for they feature a non-polynomial Haar integrand, blocking direct use of Haar integration machinery. Existing physical mechanisms to sample multiple copies of states from the Scrooge ensembles, meanwhile, typically require costly postselection, rendering them inefficient for large system size. In this work, we develop efficient and accurate methods for analyzing and implementing Scrooge moments. We first introduce a polynomial approximation to the Scrooge $k$-th moments whose error is tunable down to a threshold that improves from inverse polynomial to exponential in $1/||\rho||_\infty$ over previous results. Building on our approximation, we then give the first efficient quantum algorithm that estimates observable expectation values using $\widetilde{\mathcal{O}}(k^3/\epsilon^5)$ copies of $\rho$ for any target error $\epsilon$ above this threshold. We further provide a construction of a block encoding of the Scrooge $k$-th moment when the preparation circuit of $\rho$ is known. Of independent interest could be a key subroutine in our algorithms, which efficiently implements projection onto the $k$-fold symmetric subspace for $\rho^{\otimes k}$ using $\mathcal{O}(k^3/\epsilon^2)$ samples of $\rho$, avoiding the naive $k!$ postselection cost of such a projection. These results provide new tools for both the theoretical analysis and practical prediction of properties of Scrooge ensembles, with potential applications to quantum many-body dynamics and quantum information processing.

We study the problem of optimally placing a large number of small absorbing traps to minimize the mean first-passage time (MFPT) of particles diffusing in a heterogeneous medium with space-dependent diffusivity $D(x)$ and prescribed initial particle distribution $\omega(x)$. In two and three dimensions we identify two distinct regimes, depending on how strongly the trap ensemble depletes the particles. In the weak trapping regime (fewer traps) the optimal trap density $\rho$ is the arithmetic average $\rho = \frac12(\mu+\omega)$ of the normalized inverse diffusivity $\mu \propto 1/D$ and the initial particle distribution $\omega$. In the strong trapping regime (more traps), it is proportional to their geometric average, $\rho \propto \sqrt{\mu \omega}$. We revisit the classical Green's function approach and show that it applies to weak trapping only; in 2D this requires the trap size to be exponentially small in the number of traps, so that most applications of interest fall in the strong trapping regime instead. We therefore develop a homogenization approach that captures both regimes. A special choice of initial particle distribution is $\omega \propto 1/D$, which corresponds to the equilibrium particle distribution in the absence of traps according to the Itô interpretation. In this case we show that the optimal trap distribution is the stationary particle distribution itself, regardless of the trapping strength, and this makes the MFPT constant throughout the domain; the same trap distribution also minimizes the worst-case capture time. Finally, in one dimension we find instead $\rho\propto\left(\mu \omega\right) ^{1/3}$, and a similar analysis covers space-dependent drift and thin domains of variable cross-section. Direct numerical optimization confirms the analytical results.

We introduce a simple reaction-network model of a quorum-sensing population that couples the cell density $x$ to the signal density $w$. In the four-channel cell-signal network studied here, scaling signal production and removal by the same factor $r$ leaves all deterministic equilibria, and their stability types, unchanged. Nevertheless, we show that $r$ shifts the quasipotential barrier $\Delta V(r)$ for rare transitions towards extinction, and thus, under metastable exit assumptions, the mean time to reach a fixed neighbourhood of the extinction state on the exponential scale $e^{N\Delta V(r)}$. We compute the barrier by minimization of the path action with the signal retained as a fluctuating coordinate, and we compare it with exact stochastic simulation of population-threshold crossing times regressed in $N$. Over a range of $r$, the minimum-action barriers satisfy $\Delta V(r)=\Delta V_\infty+O(1/r)$, where $\Delta V_\infty$ is obtained by eliminating the signal first. At $r=1$, the barrier is 55% larger than $\Delta V_\infty$. The saddle barrier $\Delta V(r)$ is also the least action needed to enter the basin of extinction, but the density threshold can be crossed more cheaply: at $r=0.5$ the cheapest crossing costs 7.7% less action, keeps the signal high, and is usually followed by recovery. Simulated arrival times in this neighbourhood, which include failed attempts, grow with slopes within two fitted standard errors of the saddle barrier at every tested rate, and within $0.004$ of it if the logarithmic prefactor term is omitted. Under either regression model they exclude $\Delta V_\infty$ at $r\le2$ by at least $4.4$ fitted standard errors, without using the action solver.

We introduce a classification of quantum phases of mixed states based on the ability to preserve and fault tolerantly transfer information. We formulate phase equivalence between sets of states that encode logical quantum information and are connected by shallow local channel circuits assisted by local measurements and global classical communication of measurement outcomes, where all quantum operations, including measurements, must be robust to noise. We show that the existence of a recovery threshold for local noise is a property of an entire phase. We prove that any two sets of states with nonzero thresholds that are related by finite-depth local channel circuits belong to the same fault tolerant phase. Our approach leads to a classification of subsystem codes without favoring any state of the gauge subsystem, and we demonstrate phase equivalence between 2D and 3D color codes using dimensional jumps within the 3D gauge color code. Lastly, we study how syndrome structure affects fault tolerance in circuits with measurements. We show that qudit stabilizer codes with extended syndromes permit single-shot state preparation, whereas a broad class of shallow circuits measuring point-like syndromes, including state preparation for 2D topological codes and non-Abelian topological orders with solvable anyon theories, is not fault tolerant.

Large-scale events such as political scandals or misinformation campaigns can abruptly shift the collective opinion of a population and reshape the dynamics in ways that standard local-interaction models of sociophysics do not capture. Here, we investigate the dynamics of opinion formation when a group of interacting agents is subjected to sudden, macroscopic exogenous shocks. Our framework is general and applies to arbitrary opinion dynamics models; to illustrate its scope, we consider the voter model and variations thereof. We verify that the mathematical description is excellent in describing the first-passage properties to consensus of the voter model on the complete graph and on complex networks. Remarkably, we identify the conditions under which shocks can be advantageous or detrimental to reach consensus. For the case of the all-to-all noisy voter model under exogenous events, we unveil a rich phase diagram characterized by novel trimodal and asymmetric bimodal states. Ultimately, this work provides a rigorous yet flexible stochastic-process framework for understanding the out-of-equilibrium behavior of opinion dynamics under recurrent exogenous stressors.

Melting temperature calculation is one of the important topics in computational materials science. In high-throughput in silico screening and artificial intelligence assisted design of materials, it usually requires a rapid and autonomous assessment of the melting temperature of the target. Unfortunately, molecular dynamics (MD) simulations of the melting point require many cumbersome and manual operations, making large-scale calculation of the melting point challenging. In this work, we introduce MyTm, a toolkit that employs MD to automatically determine the melting point. The method is fully modularized, and by combining these modules, the program enables fully automated melting calculations by using commonly adopted approaches, including the direct-heating method, the void method, the modified void method, the solid-liquid coexistence method, and the Z method. Moreover, a machine learning (ML) method is proposed and employed to recognize and classify the solid like and liquid like atoms, which effectively resolve the low accuracy issue in conventional classification approaches, thus making the automated high throughput pipeline of melting-point calculation possible. The robustness and efficacy of MyTm have been demonstrated by several well studied systems.

Inferring the entropy production rate $\sigma$ from recorded trajectories is a key challenge in stochastic thermodynamics. In experiments, the forces governing the dynamics are usually unknown and often only some of the system's degrees of freedom can be observed. Building on a recently introduced method [I. Di Terlizzi, Phys. Rev. Lett. 135, 237101 (2025)], we develop and validate CYNAR (computation yields nonequilibrium analysis and reconstruction), a pipeline that estimates $\sigma$ from trajectories. It exploits a kinetic decomposition of $\sigma$ into a traffic term ${\cal T}$, inferred from the short-time curvature of correlation functions, and an inflow rate ${\cal G}$, computed from the steady-state score, i.e., the gradient of the log-density. We show that the same decomposition, evaluated on any subset of observed coordinates, always bounds $\sigma$ from below, for any diffusion tensor and without knowledge of the hidden degrees of freedom. We then compare CYNAR with a more standard approach that estimates $\sigma$ from steady-state probability currents, on four systems of increasing complexity. CYNAR proves to be more reliable, with no detectable bias close to equilibrium, weak dependence on spatial discretization, and marked robustness to measurement noise. It is also readily applicable to high-dimensional systems, where the traffic alone provides a lower bound close to $\sigma$, and remains informative under partial observation, including cases where the flux-based estimate vanishes identically. CYNAR is provided as an open-source Python package.

Maximum-likelihood decoding provides an optimal decoding strategy for quantum error correction under stochastic Pauli noise. However, computing logical-class probabilities is challenging, and the leading exact tensor-network contraction requires time exponential in treewidth. In this work, we introduce a new decoding algorithm based on rank-decomposition dynamic programming (Rank DP). We express decoding partition functions with independent local fault factors as quadratic sums of powers and apply Rank DP. The resulting exact evaluator has arithmetic complexity polynomial in the input size and exponential in the Tanner graph's rank-width with respect to the fault partition, including decomposition construction. After Gaussian elimination, it gives polynomial-time decoding for punctured quantum Reed-Muller codes and a Steane-concatenated family with growing distance under independent single-qubit Pauli noise. Standard contraction of the corresponding tensor networks requires superpolynomial time. A nonnegative realization gives relative floating-point error bounds under explicit arithmetic assumptions. Numerical experiments demonstrate runtime advantages over the tested tensor-network implementations on selected code-capacity and circuit-level instances, with full likelihood evaluation for Reed-Muller codes up to $1{,}023$ qubits. We further use these likelihoods to learn circuit noise parameters from syndromes, evaluate rare postselection probabilities, and quantify decoder optimality gaps. Our work opens new avenues for exploiting algebraic structure in quantum decoding and noise characterization.

We prove a local amplitude-ratio bound for positive ground states and an explicit reverse comparison for fidelity under finite-region erasure. An exact lattice calculation for a connected critical Ising chain gives amplitude exponent $1/8$, preserved by arbitrary optimal control on any fixed spin set and, at the level of logarithmic exponents, on $o(\log N)$ spins. We derive the finite Cauchy product and its uniform strip asymptotics, and document exact two-spin optimization and an interferometric protocol for measuring the recovery. A separate finite-mediator theorem quantifies local compression and a decoder while retaining the rest of the environment exactly. We also give a ground-preparation Ramsey bound, counterexamples to stronger projected-state inferences, and independent full-Hamiltonian calculations.

Topology and geometry are inseparable in physical networks. Here we show that, after factoring out the topology-dependent number of eligible edge pairs, exchangeability makes the mean crossing propensity of those pairs exactly graph independent while preserving topology in fluctuations. We derive a microscopic, parameter-free theory for the response when topology and geometry are coupled. Synthetic and real three-dimensional networks confirm that structure hidden from the normalized mean survives in correlations that determine response.

Understanding randomness formation under conservation laws is important both for characterizing chaotic phenomena in isolated quantum systems and for symmetry-constrained quantum information processing. $U(1)$-symmetric random circuits on an $N$-qubit system provide a minimal tractable setting for this problem, in which the formation rate of unitary designs was conjectured to be governed by charge transport in the underlying circuit geometry. However, whether charge transport indeed sets the slowest relaxation mode has remained an open problem, due to the difficulty in deriving a tight lower bound on the symmetric design formation rate. In this work, we resolve this problem by determining the optimal scaling of the unitary $k$-design formation rate for $k\leq O(\sqrt{\log N})$ across a wide range of circuit geometries, including lattices, expander graphs, and all-to-all interactions. In particular, for $k\geq 2$, we find that the formation rate is governed by a two-particle encounter rate—the rate at which two random walkers encounter each other in a given circuit geometry. This encounter rate is parametrically smaller than the single-particle transport rate in many geometries, thereby disproving the previous conjecture. We further establish that this encounter mechanism indeed sets the design formation rate by deriving its matching lower bound with a novel proof technique based on an auxiliary subgroup ensemble. Although this encounter mode imposes a robust bottleneck on design formation in $U(1)$-symmetric random circuits, we also show that this bottleneck can be circumvented by using symmetry-breaking local gates and provide an explicit protocol that achieves faster design formation. These results establish a new picture of randomness formation under conservation laws and provide guiding principles for characterizing and efficiently generating symmetry-constrained randomness.

It is widely expected that the von Neumann entanglement of a local operator grows at most logarithmically in integrable many-body systems in infinite volume. We give a counterexample in a four-state brickwork circuit whose gate is a permutation matrix solving the constant Yang–Baxter equation. The von Neumann operator entropy grows as $(\log2)\sqrt{t/\pi}+O(\log t)$, whereas fixed-index Rényi entropies grow linearly below index one and logarithmically above it. At fixed relative Hilbert–Schmidt error below one, matrix product operator simulations require a bond dimension growing at least as $\exp(c\sqrt t)$ for some $c>0$.

Despite the widespread use of Low-Rank Adaptation (LoRA), little is known about its dynamics in continual learning and the mechanisms by which low-rank updates affect catastrophic forgetting. We provide an asymptotically exact dynamical characterization of LoRA in a solvable two-task teacher-student model. In the high-dimensional online-learning limit, we derive a closed system of ordinary differential equations for a finite set of macroscopic order parameters, yielding exact expressions for the generalization errors throughout both the initial Task 1 learning phase and the subsequent LoRA fine-tuning on Task 2. The theory quantitatively matches finite-dimensional simulations and exposes two characteristic effects of LoRA: low-rank adaptation reduces interference with features learned on the first task, but its initialization slows adaptation to the second task. Building on this mechanistic picture, we analyze a state-dependent masking strategy that freezes hidden units carrying the strongest first-task representations and restricts adaptation to the complementary subspace. This structural partitioning markedly reduces forgetting, while preserving plasticity on the new task. Our framework further clarifies the role of adapter rank: transfer improves only up to the intrinsic dimensionality of the target task and saturates beyond it, while forgetting continues to grow with rank. These results provide a dynamical and geometric account of how low-rank adaptation organizes information across sequential tasks and are qualitatively reproduced on a sequential MNIST benchmark.

We develop a theory of phase separation in fluids conserving higher-order multipole moments of a scalar density, and investigate the resulting coarsening dynamics, nucleation, and interfacial fluctuations. We find that higher-order conservation laws slow the relaxation of all phenomena investigated, whereas nonequilibrium violations of the fluctuation–dissipation theorem suppress density and interfacial fluctuations while hindering nucleation. Simulations support these predictions.

We investigate the phase diagram and critical properties of the two-dimensional classical XY model with interactions decaying as $1/r^{2+\sigma}$. At the marginal case $\sigma=2$, we show that the low-temperature phase is characterized by logarithmic quasi-long-range order (log-QLRO), where spin correlations decay as a power of the logarithm of distance, $C(r)\sim(\ln r)^{-\eta_\ell}$ with $\eta_\ell\propto T$, and that the transition into this phase is a continuous transition beyond both the Ginzburg-Landau-Wilson paradigm and the Berezinskii-Kosterlitz-Thouless mechanism. Our analysis combines a near-exact Gaussian spin-wave theory, an adiabatic renormalization-group analysis, and large-scale Monte Carlo simulations. We find that the $1/r^4$ interaction logarithmically modifies the spin-wave stiffness kernel as $\gamma(k)\simeq\kappa k^2\ln(1/k)$, which suppresses vortex proliferation and gives rise to a nonuniversal correlation-length exponent $\nu\propto1/\sqrt{2\pi\kappa}$. Crucially, our simulations reveal that the short-range algebraic QLRO phase is unstable against a weak long-range perturbation for $\sigma\le2$. These results establish $\sigma=2$ as the crossover boundary between the long-range and short-range universality classes and provide a critical evaluation of recent theoretical proposals.

We show that periodically modulating the self-propulsion of active particles in an asymmetric potential can both enhance rectified transport and reverse its direction, even when the propulsion speed itself never changes sign. Current enhancement results from the nonlinear response of active transport to propulsion strength, whereas current reversal emerges from the interplay between activity-induced spreading and passive relaxation. Our theoretical analysis identifies the regimes governed by these distinct rectification mechanisms and predicts the finite range of driving periods over which current reversal occurs. These results establish cyclic activity as a means of dynamically controlling rectified active transport, with potential applications to the selective sorting of active particles.

Experiments often detect only selected transitions, leaving the underlying state dynamics and dissipation hidden. We show that waiting-time distributions between directed visible events can reconstruct the boundary dynamics without observing the reverse of every detected transition. Under a simple source-target closure condition, they determine visible rates and the hidden excursions connecting observed states, yielding a rigorous lower bound on total entropy production that improves the standard waiting-time estimate when reverse events are available. Remarkably, observing only a spanning directed cycle is sufficient to identify the full Markov generator, even when all reverse cycle edges and other microscopic transitions remain unseen.

When driven by a large enough electric field or current, the insulating state of vanadium dioxide ($VO_2$) typically collapses abruptly into the metallic state (the insulator-metal transition, IMT) following a brief incubation delay. These incubation delays, typically lasting microseconds or less, have been analyzed predominantly through macroscopic electro-thermal modeling, providing little statistical insight into the transition dynamics. Here, we investigate the electrically driven IMT in a two-dimensional $VO_2$ lattice and demonstrate that the material behaves as a complex, interdependent network despite being a single structural entity. Because metallic domains dissipate significantly more Joule heat than insulating regions, each switching event raises the Joule heating of the network, resulting in network dependency and triggering a cascading chain reaction. Combining time-resolved resistance measurements with a Joule-coupled resistor-network model and interdependent-network theory, we observe an abrupt (first-order) transition preceded by an anomalously long incubation plateau lasting thousands of seconds. During this metastable plateau, an effective branching factor of the switching activity approaches unity at criticality, and the plateau lifetime diverges with exponent $\zeta=1/2$; the order parameter scales with exponent $\beta=1/2$. These signatures coincide with those predicted for interdependent networks, indicating that dissipative Joule coupling alone can drive mixed-order catastrophic cascades in a single-network correlated-oxide system.

Engineering dissipation has emerged as a powerful strategy for controlling quantum many-body dynamics, complementing purely coherent control for state preparation. However, finite-time driving across a quantum critical point generates nonadiabatic excitations and coherences, and existing dissipation-engineering approaches have largely characterized how a fixed environment modifies critical dynamics rather than actively steering a system toward a prescribed thermal trajectory. Here we develop a Markovian dissipative framework for finite-time thermal-state tracking based on engineered population currents with non-negative transition rates, and apply it to the transverse-field Ising model driven across its quantum critical point. Compared with unitary dynamics and a natural thermal bath, the engineered dynamics strongly suppresses thermal excitation deviations and provides additional control over coherence damping through a free common current $Q_k$, while exhibiting a substantially steeper residual scaling, $\delta n_{\mathrm{ex}}^{\mathrm{th}}\sim(v/v_{\mathrm{ch}})^2$, than the Kibble–Zurek behavior recovered for unitary evolution. These results point toward engineered dissipation as a route for controlling thermal-state preparation in driven quantum critical systems.

The effective Hamiltonian (EH), also referred to as the potential of mean force or free-energy landscape, is widely used to describe equilibrium properties and slow dynamics in chemical and soft-matter systems. However, the conventional EH has been argued to be ill-defined because it does not transform as a scalar under coordinate transformations. We identify an implicit change of reference measure (gauge), rather than the coordinate transformation itself, as the origin of this apparent problem. We formulate the EH as a scalar relative to a chosen gauge and derive the intrinsic, gauge-independent Fokker–Planck equation (FPE) on the manifold of slow variables. The conventional EH and the diffusion-dependent EH introduced in a Riemannian formulation, together with their corresponding FPEs, are thus different gauge representations of the same equilibrium measure and intrinsic stochastic dynamics. We further demonstrate that the gauge choice has an operational meaning: different restraint protocols select different reference measures. In the macroscopic limit, the leading EH becomes gauge independent within a class of subleading gauges and is determined by the large-deviation rate function. In the same limit, the stochastic dynamics reduces to gauge-independent deterministic dynamics.

It is well-known that thermalization breaks down in presence of an extensive number of symmetries or conservation laws that are mutually compatible or commuting with each other. We ask here how an extensive number of mutually incompatible local symmetries govern eigenstate thermalization (ETH). We will investigate this through models within the class of bond-dependent $\mathbb{Z}_2$-symmetric quantum spin-$\frac{1}{2}$ Hamiltonians that naturally admit this structure. This leads to exponentially large degeneracies in the spectrum governed by the incompatible symmetry structure. Here, using spectral diagnostics, we show that non-integrability can survive despite extensively many conserved quantities when they are incompatible. Furthermore, we show that ETH behavior persists deep into the many-body spectral edge due to this structure. In other words, incompatible local symmetries enable a mechanism to obtain enough thermodynamic entropy at low excitation energy densities to make ETH operational or predictive near the ground state.

We introduce a quantum-echo Markov process for combinatorial optimization. Quantum dynamics based on quantum annealing (QA) or the quantum approximate optimization algorithm (QAOA) is used to engineer the transition kernel. Increasing the annealing time in QA or the number of layers in QAOA enables transitions to explore distant configurations while suppressing large energy changes. The Hamming-space delocalization originates from operator spreading, whereas the energy-space localization arises from a dynamically generated correlation. For optimization, we propose quantum-echo local optimization with and without greedy descent. We find that its performance is governed by the interplay between Hamming-space nonlocality and energy-space locality, and that excessive energy-space localization can degrade optimization performance. Incorporating greedy descent substantially improves the performance, highlighting the complementary roles of quantum dynamics for exploration and greedy descent for exploitation. These results establish quantum-echo dynamics as a framework for engineering structured transition kernels and provide a route to using finite-resource quantum many-body dynamics as a computational primitive for iterative optimization.

We study the ergotropy and the thermodynamic potentials of a qubit battery charged by repeated interactions with clusters of thermal ancillas, qubits or oscillators, through a composite system-bath coupling that generates steady-state coherence in the energy basis. A single parameter encoding the bath quantum statistics yields the coherence and ergotropy in closed form for any cluster size. For large clusters, the ergotropy becomes independent of the bath statistics and, at optimal couplings, reaches a universal fraction of the equilibrium internal energy of the qubit at the bath temperature. Since the charged state has no population inversion, all extractable work comes from coherence. The ergotropy is flat at low temperatures. We trace this plateau to the third law, and show that the ergotropy is proportional to the equilibrium enthalpy of the qubit, whose slope vanishes with the heat capacity. For a battery composed of multiple qubits, the coherence grows exponentially with the number of qubits while the ergotropy grows linearly, both remaining a nearly constant fraction of their respective maxima. The charged state is a non-equilibrium steady state with finite entropy at zero temperature, sustained by the work of switching the collisional interaction on and off. The cost is nonzero whenever the parallel coupling is nonzero, and, unlike the ergotropy, it depends on the bath statistics.