Statistical Mechanical Model Research Articles

Efficient Classical Simulation of Random Shallow 2D Quantum Circuits

A central question of quantum computing is determining the source of the advantage of quantum computation over classical computation. Even though simulating quantum dynamics on a classical computer is thought to require exponential overhead in the worst case, efficient simulations are known to exist in several special cases. It was widely assumed that these easy-to-simulate cases as well as any yet-undiscovered ones could be avoided by choosing a quantum circuit at random. We prove that this intuition is false by showing that certain families of constant-depth, 2D random circuits can be approximately simulated on a classical computer in time only linear in the number of qubits and gates, even though the same families are capable of universal quantum computation and are hard to exactly simulate in the worst case (under standard hardness assumptions). While our proof applies to specific random circuit families, we demonstrate numerically that typical instances of more general families of sufficiently shallow constant-depth 2D random circuits are also efficiently simulable. We propose two classical simulation algorithms. One is based on first simulating spatially local regions which are then “stitched” together via recovery maps. The other reduces the 2D simulation problem to a problem of simulating a form of 1D dynamics consisting of alternating rounds of random local unitaries and weak measurements. Similar processes have recently been the subject of an intensive research focus, which has observed that the dynamics generally undergo a phase transition from a low-entanglement (and efficient-to-simulate) regime to a high-entanglement (and inefficient-to-simulate) regime as measurement strength is varied. Via a mapping from random quantum circuits to classical statistical mechanical models, we give analytical evidence that a similar computational phase transition occurs for both of our algorithms as parameters of the circuit architecture like the local Hilbert space dimension and circuit depth are varied and, additionally, that the effective 1D dynamics corresponding to sufficiently shallow random quantum circuits falls within the efficient-to-simulate regime. Implementing the latter algorithm for the depth-3 “brickwork” architecture, for which exact simulation is hard, we find that a laptop could simulate typical instances on a 409×409 grid with a total variation distance error less than 0.01 in approximately one minute per sample, a task intractable for previously known circuit simulation algorithms. Numerical results support our analytic evidence that the algorithm is asymptotically efficient.4 MoreReceived 4 March 2021Revised 15 January 2022Accepted 14 February 2022DOI:https://doi.org/10.1103/PhysRevX.12.021021Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasEntanglement productionPhase transitionsQuantum computationQuantum entanglementTechniquesComputational complexityInformation theoryIsing modelMatrix product statesTensor network methodsQuantum InformationStatistical Physics

Molecular dynamics simulations of the evaporation of hydrated ions from aqueous solution

Although important for atmospheric processes and gas-phase catalysis, very little is known about the hydration state of ions in the vapor phase. Here we study the evaporation energetics and kinetics of a chloride ion from liquid water by molecular dynamics simulations. As chloride permeates the interface, a water finger forms and breaks at a chloride separation of ≈ 2.8 nm from the Gibbs dividing surface. For larger separations from the interface, about 7 water molecules are estimated to stay bound to chloride in saturated water vapor, as corroborated by continuum dielectrics and statistical mechanics models. This ion hydration significantly reduces the free-energy barrier for evaporation. The effective chloride diffusivity in the transition state is found to be about 6 times higher than in bulk, which reflects the highly mobile hydration dynamics as the water finger breaks. Both effects significantly increase the chloride evaporation flux from the quiescent interface of an electrolyte solution, which is predicted from reaction kinetic theory.

High-Force Application by a Nanoscale DNA Force Spectrometer.

The ability to apply and measure high forces (>10 pN) on the nanometer scale is critical to the development of nanomedicine, molecular robotics, and the understanding of biological processes such as chromatin condensation, membrane deformation, and viral packaging. Established force spectroscopy techniques including optical traps, magnetic tweezers, and atomic force microscopy rely on micron-sized or larger handles to apply forces, limiting their applications within constrained geometries including cellular environments and nanofluidic devices. A promising alternative to these approaches is DNA-based molecular calipers. However, this approach is currently limited to forces on the scale of a few piconewtons. To study the force application capabilities of DNA devices, we implemented DNA origami nanocalipers with tunable mechanical properties in a geometry that allows application of force to rupture a DNA duplex. We integrated static and dynamic single-molecule characterization methods and statistical mechanical modeling to quantify the device properties including force output and dynamic range. We found that the thermally driven dynamics of the device are capable of applying forces of at least 20 piconewtons with a nanometer-scale dynamic range. These characteristics could eventually be used to study other biomolecular processes such as protein unfolding or to control high-affinity interactions in nanomechanical devices or molecular robots.

In most 6-regular toroidal graphs all 5-colorings are Kempe equivalent

A Kempe swap in a proper coloring interchanges the colors on some maximal connected 2-colored subgraph. Two k-colorings are k-equivalent if we can transform one into the other using Kempe swaps. The triangulated toroidal grid, T[m×n], is formed from (a toroidal embedding of) the Cartesian product of Cm and Cn by adding parallel diagonals inside all 4-faces. Mohar and Salas showed that not all 4-colorings of T[m×n] are 4-equivalent. In contrast, Bonamy, Bousquet, Feghali, and Johnson showed that all 6-colorings of T[m×n] are 6-equivalent. They asked whether the same is true for 5-colorings. We answer their question affirmatively when m,n≥6. Further, we show that if G is 6-regular with a toroidal embedding where every non-contractible cycle has length at least 7, then all 5-colorings of G are 5-equivalent. Our results relate to the antiferromagnetic Pott’s model in statistical mechanics.

Echo chambers in the Ising model and implications on the mean magnetization

The echo-chamber effect is a common term in opinion dynamic modeling to describe how a person’s opinion might be artificially enhanced as it is reflected back at her through social interactions. Here, we study the existence of this effect in statistical mechanics models, which are commonly used to study opinion dynamics. We show that the Ising model does not exhibit echo-chambers, but this result is a consequence of a special symmetry. We then distinguish between three types of models: (i) those with a strong echo-chamber symmetry, that have no echo-chambers at all; (ii) those with a weak echo-chamber symmetry that can exhibit echo-chambers but only if there are external fields in the system, and (iii) models without echo-chamber symmetry that generically have echo-chambers. We use these results to construct an efficient algorithm to efficiently and precisely calculate magnetization in arbitrary tree networks. Finally, we apply this algorithm to study two systems: phase transitions in the random field Ising model on a Bethe lattice and the influence optimization problem in social networks.

Packaging contests between viral RNA molecules and kinetic selectivity.

The paper presents a statistical-mechanics model for the kinetic selection of viral RNA molecules by packaging signals during the nucleation stage of the assembly of small RNA viruses. The effects of the RNA secondary structure and folding geometry of the packaging signals on the assembly activation energy barrier are encoded by a pair of characteristics: the wrapping number and the maximum ladder distance. Kinetic selection is found to be optimal when assembly takes place under conditions of supersaturation and also when the concentration ratio of capsid protein and viral RNA concentrations equals the stoichiometric ratio of assembled viral particles. As a function of the height of the activation energy barrier, there is a form of order-disorder transition such that for sufficiently low activation energy barriers, kinetic selectivity is erased by entropic effects associated with the number of assembly pathways.

Quantification of Entropic Excluded Volume Effects Driving Crowding-Induced Collapse and Folding of a Disordered Protein.

We investigate the conformational properties of the intrinsically disordered DNA-binding domain of CytR in the presence of the polymeric crowder polyethylene glycol (PEG). Integrating circular dichroism, nuclear magnetic resonance, and single-molecule Förster resonance energy transfer measurements, we demonstrate that disordered CytR populates a well-folded minor conformation in its native ensemble, while the unfolded ensemble collapses and folds with an increase in crowder density independent of the crowder size. Employing a statistical-mechanical model, the effective reduction in the accessible conformational space of a residue in the unfolded state is estimated to be 10% at 300 mg/mL PEG8000, relative to dilute conditions. The experimentally consistent PEG-temperature phase diagram thus constructed reveals that entropic effects can stabilize disordered CytR by 10 kJ mol-1, driving the equilibrium toward folded conformations under physiological conditions. Our work highlights the malleable conformational landscape of CytR, the presence of a folded conformation in the disordered ensemble, and proposes a scaling relation for quantifying excluded volume effects on protein stability.

Statistical mechanics of beam self-cleaning in GRIN multimode optical fibers.

Since its first demonstration in graded-index multimode fibers, spatial beam self-cleaning has attracted a growing research interest. It allows for the propagation of beams with a bell-shaped spatial profile, thus enabling the use of multimode fibers for several applications, from biomedical imaging to high-power beam delivery. So far, beam self-cleaning has been experimentally studied under several different experimental conditions. Whereas it has been theoretically described as the irreversible energy transfer from high-order modes towards the fundamental mode, in analogy with a beam condensation mechanism. Here, we provide a comprehensive theoretical description of beam self-cleaning, by means of a semi-classical statistical mechanics model of wave thermalization. This approach is confirmed by an extensive experimental characterization, based on a holographic mode decomposition technique, employing laser pulses with temporal durations ranging from femtoseconds up to nanoseconds. An excellent agreement between theory and experiments is found, which demonstrates that beam self-cleaning can be fully described in terms of the basic conservation laws of statistical mechanics.

CDFTPY: A python package for performing classical density functional theory calculations for molecular liquids

Classical density functional theory (CDFT) provides a rigorous theoretical framework for the statistical mechanics based analysis of many-body systems. This approach has proven to be successful in simulations of mono-atomic, i.e. simple liquids, and there is an ongoing theoretical effort in extending it to more complex polyatomic, molecular liquid systems. Sharing these developments in the form of open-source and easily accessible codes could greatly benefit these efforts. In this work, we present python-based CDFT code that contains both conventional Reference Interaction Site Model (RISM) and recently developed renormalized site density theory (RSDFT) approach. The current implementation is focused on ion solvation - the problem of both fundamental and practical importance. It allows the calculation of individual ions as well as comparative analysis across a range of interaction parameters. Program summaryProgram Title: cdftpyCPC Library link to program files:https://doi.org/10.17632/p8dsgz5n4g.1Developer's repository link:https://github.com/opencdftLicensing provisions: GPLv3Programming language: python 3.9+Nature of problem: Computational modeling of molecular liquids at the atomistic level of resolution is an important capability across many scientific areas. Classical density functional theory (CDFT) approaches this problem by building statistical mechanics model of the system in terms of its average atomic (site) density. Such an approach can provide orders of magnitudes improvements in efficiency compared to conventional molecular dynamics simulations, but requires special treatment of multi-scale interactions in a molecular liquid. A practical utility of our open-source python based implementation of CDFT is the study the problem solvation of ions or Lennard-Jones particles.Solution method: Python package developed in this work provides two CDFT implementations for molecular liquids - renormalized site density functional theory and reference interaction site model. It enables calculations of thermodynamic and structural properties related to solvation of spherical solutes. The nonlinear integral equations associated with the two methods are solved iteratively, utilizing Fast Fourier Transform (FFT) for the calculation of the numerically intensive convolution integrals. The resulting code provides near instantaneous evaluation of the solvated properties of individual solutes and high-throughput screening across the range of different solute parameters.

An Algebraic Approach to Product-form Stationary Distributions for Some Reaction Networks

Exact results for product-form stationary distributions of Markov chains are of interest in different fields. In stochastic reaction networks (CRNs), stationary distributions are mostly known in special cases where they are of product-form. However, there is no full characterization of the classes of networks whose stationary distributions have product-form. We develop an algebraic approach to product-form stationary distributions in the framework of CRNs. Under certain hypotheses on linearity and decomposition of the state space for conservative CRNs, this gives sufficient and necessary algebraic conditions for product-form stationary distributions. Correspondingly, we obtain a semialgebraic subset of the parameter space that captures rates where, under the corresponding hypotheses, CRNs have product-form. We employ the developed theory to CRNs and some models of statistical mechanics, besides sketching the pertinence in other models from applied probability.

What are the limits of universality?

It is a central prediction of renormalization group theory that the critical behaviours of many statistical mechanics models on Euclidean lattices depend only on the dimension and not on the specific choice of lattice. We investigate the extent to which this universality continues to hold beyond the Euclidean setting, taking as case studies Bernoulli bond percolation and lattice trees. We present strong numerical evidence that the critical exponents governing these models on transitive graphs of polynomial volume growth depend only on the volume-growth dimension of the graph and not on any other large-scale features of the geometry. For example, our results strongly suggest that percolation, which has upper-critical dimension 6, has the same critical exponents onZ4and the Heisenberg group despite the distinct large-scale geometries of these two lattices preventing the relevant percolation models from sharing a common scaling limit. On the other hand, we also show that no such universality should be expected to hold on fractals, even if one allows the exponents to depend on a large number of standard fractal dimensions. Indeed, we give natural examples of two fractals which share Hausdorff, spectral, topological and topological Hausdorff dimensions but exhibit distinct numerical values of the percolation Fisher exponentτ. This gives strong evidence against a conjecture of Balankinet al.(2018Phys. Lett. A382, 12–19 (doi:10.1016/j.physleta.2017.10.035)).

Random Quantum Circuits Anticoncentrate in Log Depth

We consider quantum circuits consisting of randomly chosen two-local gates and study the number of gates needed for the distribution over measurement outcomes for typical circuit instances to be anti-concentrated, roughly meaning that the probability mass is not too concentrated on a small number of measurement outcomes. Understanding the conditions for anti-concentration is important for determining which quantum circuits are difficult to simulate classically, as anti-concentration has been in some cases an ingredient of mathematical arguments that simulation is hard and in other cases a necessary condition for easy simulation. Our definition of anti-concentration is that the expected collision probability, that is, the probability that two independently drawn outcomes will agree, is only a constant factor larger than if the distribution were uniform. We show that when the 2-local gates are each drawn from the Haar measure (or any two-design), at least $\Omega(n \log(n))$ gates (and thus $\Omega(\log(n))$ circuit depth) are needed for this condition to be met on an $n$ qudit circuit. In both the case where the gates are nearest-neighbor on a 1D ring and the case where gates are long-range, we show $O(n \log(n))$ gates are also sufficient, and we precisely compute the optimal constant prefactor for the $n \log(n)$. The technique we employ relies upon a mapping from the expected collision probability to the partition function of an Ising-like classical statistical mechanical model, which we manage to bound using stochastic and combinatorial techniques.

Nonlinear one-dimensional constitutive model for magnetostrictive materials

This paper presents an analytic model of one dimensional magnetostriction. We show how specific assumptions regarding the symmetry of key micromagnetic energies (magnetocrystalline, magnetoelastic, and Zeeman) reduce a general three-dimensional statistical mechanics model to a one-dimensional form with an exact solution. We additionally provide a useful form of the analytic equations to help ensure numerical accuracy. Numerical results show that the model maintains accuracy over a large range of applied magnetic fields and stress conditions extending well outside those produced in standard laboratory conditions. A comparison to experimental data is performed for several magnetostrictive materials. The model is shown to accurately predict the behavior of Terfenol-D, while two compositions of Galfenol are modeled with varying accuracy. To conclude we discuss what conditions facilitate the description of materials with cubic crystalline anisotropy as transversely isotropic, to achieve peak model performance.

Topological Quantum Statistical Mechanics and Topological Quantum Field Theories

The Ising model describes a many-body interacting spin (or particle) system, which can be utilized to imitate the fundamental forces of nature. Although it is the simplest many-body interacting system of spins (or particles) with Z2 symmetry, the phenomena revealed in Ising systems may afford us lessons for other types of interactions in nature. In this work, we first focus on the mathematical structure of the three-dimensional (3D) Ising model. In the Clifford algebraic representation, many internal factors exist in the transfer matrices of the 3D Ising model, which are ascribed to the topology of the 3D space and the many-body interactions of spins. They result in the nonlocality, the nontrivial topological structure, as well as the long-range entanglement between spins in the 3D Ising model. We review briefly the exact solution of the ferromagnetic 3D Ising model at the zero magnetic field, which was derived in our previous work. Then, the framework of topological quantum statistical mechanics is established, with respect to the mathematical aspects (topology, algebra, and geometry) and physical features (the contribution of topology to physics, Jordan–von Neumann–Wigner framework, time average, ensemble average, and quantum mechanical average). This is accomplished by generalizations of our findings and observations in the 3D Ising models. Finally, the results are generalized to topological quantum field theories, in consideration of relationships between quantum statistical mechanics and quantum field theories. It is found that these theories must be set up within the Jordan–von Neumann–Wigner framework, and the ergodic hypothesis is violated at the finite temperature. It is necessary to account the time average of the ensemble average and the quantum mechanical average in the topological quantum statistical mechanics and to introduce the parameter space of complex time (and complex temperature) in the topological quantum field theories. We find that a topological phase transition occurs near the infinite temperature (or the zero temperature) in models in the topological quantum statistical mechanics and the topological quantum field theories, which visualizes a symmetrical breaking of time inverse symmetry.

Vertex electrical model: Lagrangian and nonnegativity properties

This note is a modest addition to the work arXiv:2109.13952. Here we construct an embedding of the space of electrical networks to the totally non-negative Lagrangian Grassmannian in a generic situation with the help of the technique of vertex integrable models of statistical mechanics.

Thermodynamics and surface properties of Fe–V and Fe–Ti liquid alloys

Abstract A simple statistical mechanical model, based on a quasi-lattice approximation in which one assumes the formation of complexes, has been used to study bulk properties, such as free energy of mixing, the thermodynamic activity and enthalpy of mixing, in liquid Fe –Vand Fe –Ti alloys. The energetics and its effect on the alloying behavior of the liquid alloys has been investigated with the aim of correlating bulk phenomena with surface effects. The analysis shows that, assuming the formation of intermetallic complexes of the form Fe2V and FeTi in the liquid alloys, one can explain the energetics of the bulk alloys. Our results for the bulk calculations indicate that Fe –V and Fe –Ti both exhibit a significant tendency for compound formation. From a perusal of the diffusion coefficient D, we observe the same trend towards compound formation, as demonstrated by the chemical short-range order parameter (CSRO) close to the assumed stoichiometric composition. Furthermore, using the model calculations in the bulk, we study some surface properties. Our calculations indicate that Fe segregates to the surface at all bulk compositions in Fe –Vand Fe –Ti liquid, though the segregation effect is more pronounced in the former alloy. The reason for this is that Fe –Ti is a more ordered system than Fe –V and, thus, the driving force for surface segregation in these alloys is their energetics.

Structural-Energetic Basis for Coupling between Equilibrium Fluctuations and Phosphorylation in a Protein Native Ensemble.

The functioning of proteins is intimately tied to their fluctuations in the native ensemble. The structural–energetic features that determine fluctuation amplitudes and hence the shape of the underlying landscape, which in turn determine the magnitude of the functional output, are often confounded by multiple variables. Here, we employ the FF1 domain from human p190A RhoGAP protein as a model system to uncover the molecular basis for phosphorylation of a buried tyrosine, which is crucial to the transcriptional activity associated with transcription factor TFII-I. Combining spectroscopy, calorimetry, statistical–mechanical modeling, molecular simulations, and in vitro phosphorylation assays, we show that the FF1 domain samples a diverse array of conformations in its native ensemble, some of which are phosphorylation-competent. Upon eliminating unfavorable charge–charge interactions through a single charge-reversal (K53E) or charge-neutralizing (K53Q) mutation, we observe proportionately lower phosphorylation extents due to the altered structural coupling, damped equilibrium fluctuations, and a more compact native ensemble. We thus establish a conformational selection mechanism for phosphorylation in the FF1 domain with K53 acting as a “gatekeeper”, modulating the solvent exposure of the buried tyrosine. Our work demonstrates the role of unfavorable charge–charge interactions in governing functional events through the modulation of native ensemble characteristics, a feature that could be prevalent in ordered protein domains.

A general statistical mechanical model for fluid system thermodynamics: Application to sub- and super-critical water.

We propose in this paper a theoretical model for fluid state thermodynamics based on modeling the fluctuation distributions and, hence, the corresponding moment generating functions providing the free energy of the system. Using the relatively simple and physically coherent gamma model for the fluctuation distributions, we obtain a complete theoretical equation of state, also giving insight into the statistical/molecular organization and phase or pseudo-phase transitions occurring under the sub- and super-critical conditions, respectively. Application to sub- and super-critical fluid water and a comparison with the experimental data show that this model provides an accurate description of fluid water thermodynamics, except close to the critical point region where limited but significant deviations from the experimental data occur. We obtain quantitative evidence of the correspondence between the sub- and super-critical thermodynamic behaviors, with the super-critical water pseudo-liquid and pseudo-gas phases being the evolution of the sub-critical water liquid and gas phases, respectively. Remarkably, according to our model, we find that for fluid water the minimal subsystem corresponding to either the liquid-like or the gas-like condition includes an infinite number of molecules in the sub-critical regime (providing the expected singularities due to macroscopic phase transitions) but only five molecules in the super-critical regime (coinciding with the minimal possible hydrogen-bonding cluster), thus suggesting that the super-critical regime be characterized by the coexistence of nanoscopic subsystems in either the pseudo-liquid or the pseudo-gas phase with each subsystem fluctuating between forming and disrupting the minimal hydrogen-bonding network.

Fundamental thresholds of realistic quantum error correction circuits from classical spin models

Mapping the decoding of quantum error correcting (QEC) codes to classical disordered statistical mechanics models allows one to determine critical error thresholds of QEC codes under phenomenological noise models. Here, we extend this mapping to admit realistic, multi-parameter noise models of faulty QEC circuits, derive the associated strongly correlated classical spin models, and illustrate this approach for a quantum repetition code with faulty stabilizer readout circuits. We use Monte-Carlo simulations to study the resulting phase diagram and benchmark our results against a minimum-weight perfect matching decoder. The presented method provides an avenue to assess fundamental thresholds of QEC circuits, independent of specific decoding strategies, and can thereby help guiding the development of near-term QEC hardware.

Cubatic structural transformation of the packing of granular cylinders.

Packing structures of granular cylinders with the aspect ratio close to one have been reconstructed with the help of magnetic resonance imaging techniques. By controlling the container boundary conditions and preparation protocols, a structural transformation from a disordered liquid-like state to an orientationally ordered state with cubatic symmetry at a high packing fraction is observed. This ordering process is accompanied by the formation of more faceted contacts, which lower the elastic energy between jammed granular particles to drive the transformation. With the help of Edwards' volume ensemble theory, this granular structural transformation is explained using a phenomenological thermodynamic model and a self-consistent mean-field statistical mechanical model. Both models predict a sharp but continuous change of order parameter when the effective granular temperature is lowered. The intrinsic difference and connection between this granular structural transformation and the entropy-driven phase transition of conventional thermal hard-particle systems are discussed.