Stochastic Quantization Research Articles (Page 1)

Abstract We propose the use of the ‘spin-opstring’, derived from Stochastic Series Expansion quantum Monte Carlo (QMC) simulations as machine learning (ML) input data. It offers a compact, memory-efficient representation of QMC simulation cells, combining the initial state with an operator string that encodes the state’s evolution through imaginary time. Using supervised ML, we demonstrate the input’s effectiveness in capturing both conventional and topological phase transitions, and in a regression task to predict non-local observables. We also demonstrate the capability of spin-opstring data in transfer learning by training models on one quantum system and successfully predicting on another, as well as showing that models trained on smaller system sizes generalize well to larger ones. Importantly, we illustrate a clear advantage of spin-opstring over conventional spin configurations in the accurate prediction of a quantum phase transition. Finally, we show how the inherent structure of spin-opstring provides an elegant framework for the interpretability of ML predictions. Using two state-of-the-art interpretability techniques, Layer-wise Relevance Propagation and SHapley Additive exPlanations, we show that the ML models learn and rely on physically meaningful features from the input data. Together, these findings establish the spin-opstring as a broadly-applicable and interpretable input format for ML in quantum many-body physics.

Abstract We present here the first lattice simulation of symplectic quantization, a new functional approach to quantum field theory which allows to define an algorithm to numerically sample the quantum fluctuations of fields directly in Minkowski space-time, at variance with all other present approaches. Symplectic quantization is characterized by a Hamiltonian deterministic dynamics evolving with respect to an additional time parameter τ analogous to the fictious time of Parisi-Wu stochastic quantization. The difference between stochastic quantization and the present approach is that the former is well defined only for Euclidean field theories, while the latter allows to sample the causal structure of space-time. In this work we present the numerical study of a real scalar field theory on a 1+1 space-time lattice with a λϕ 4 interaction. We find that for λ ≪ 1 the two-point correlation function obtained numerically reproduces qualitatively well the shape of the free Feynman propagator. Within symplectic quantization the expectation values over quantum fluctuations are computed as dynamical averages along the dynamics in τ, in force of a natural ergodic hypothesis connecting Hamiltonian dynamics with a generalized microcanonical ensemble. Analytically, we prove that this microcanonical ensemble, in the continuum limit, is equivalent to a canonical-like one where the probability density of field configurations is P [ϕ] ∝ exp(zS[ϕ]/ℏ). The results from our simulations correspond to the value z = 1 of the parameter in the canonical weight, which in this case is a well-defined probability density for field configurations in causal space-time, provided that a lower bounded interaction potential is considered. The form proposed for P [ϕ] suggests that our theory can be connected to ordinary quantum field theory by analytic continuation in the complex-z plane.

On the Mathematical Theory of Quantum Stochastic Filtering Equations for Mixed States

Feynman diagrams are an essential tool for simulating strongly correlated electron systems. However, stochastic quantum MonteCarlo sampling suffers from the sign problem, particularly when solving a multiorbital quantum impurity model. Recently, two approaches have been proposed for efficient numerical treatment of Feynman diagrams: tensor cross interpolation (TCI) to replace stochastic sampling and the quantics tensor train (QTT) representation for compressing space-time dependence. One of the remaining challenges is the nontrivial task of identifying low-rank structures in weak-coupling Feynman diagrams for multiorbital electron-phonon systems. In particular, the traditional TCI algorithm faces an ergodicity problem, which prevents it from fully exploring the multiorbital space. To address this, we incorporate a new algorithm called global search, which resolves this issue. By combining this approach with QTT, we uncover low-rank structures and achieve efficient numerical integration with exponential resolution in time and faster-than-power-law convergence of error relative to computational cost. Additionally, our approach does not require the division of discontinuous regions necessary in nonquantics TCI.

Elliptic stochastic quantization of Sinh-Gordon QFT

Abstract We present a simple PDE construction of the sine‐Gordon measure below the first threshold (), in both the finite and infinite volume settings, by studying the corresponding parabolic sine‐Gordon model. We also establish pathwise global well‐posedness of the hyperbolic sine‐Gordon model in finite volume for .

This review explores the Complex Langevin Method (CLM), a stochastic quantization technique designed to address the sign problem in quantum field theories with complex actions. Beginning with foundational principles, the review examines the application of CLM across a range of models, including zero- and two-dimensional systems, supersymmetric quantum mechanics and the IKKT matrix model, a candidate for nonperturbative string theory. Key advancements, such as stabilization techniques and mass deformations, are highlighted as solutions to challenges like numerical instability and singular drift terms. The review emphasizes the capacity of CLM to simulate complex systems and reveal nonperturbative phenomena, positioning it as a powerful tool for exploring quantum field theory and string theory. Future directions, including higher-dimensional applications and benchmarking against quantum simulations, underscore the potential of CLM to advance both theoretical understanding and computational methodologies.

Abstract We quote a definitive simple proof that neither classical stochastic dynamics nor quantum dynamics can be nonlinear if we stick to their standard statistical interpretations. A recently proposed optomechanical test of gravity’s classicality versus quantumness is based on the nonlinear Schrödinger—Newton equation (SNE) which is the nonrelativistic limit of standard semiclassical gravity. While in typical cosmological applications of semiclassical gravity the predicted violation of causality is ignored, it cannot be disregarded in applications of the SNE in high sensitive laboratory tests hoped for the coming years. We reveal that, in a recently designed experiment, quantum optical monitoring of massive probes predicts fake action-at-a-distance (acausality) on a single probe already. The proposed experiment might first include the direct test of this acausality.

Abstract The quantum phase transition between Z 2 plaquette valence bound solid (PVBS) and superfluid (SF) phases on the planar pyrochlore lattice (square ice) is under debate. To gain further insight, here, we focus on the dynamical features of the hard-core Bose–Hubbard model on this lattice and study the excitation spectra by combining stochastic analytic continuation and quantum Monte Carlo simulation. In both PVBS and SF phases, a flat band with bow-tie structure is observed and can be explained by certain symmetries. At the transition point, the spectra turn to be continuous and gapless. A (2+1)-dimensional Abelian–Higgs model with mixed ’t Hooft anomaly is proposed to describe the transition, where the anomaly matching predicts that the deconfinement can exist on the domain walls. From the snapshot of the spin configuration in real space, we found the existence of the domain wall. We also found that the spectrum along a specific path in momentum space from PVBS phase to the transition point can be well described by an XXZ spin chain, and the critical theory of XXZ spin chain matches the anomaly. The two-spinon continuum along this specific path implies additional domain walls (point defect) can emerge in the domain walls (line defect) and take the role of deconfinement at the transition point.

The study of information revivals, witnessing the violation of certain data-processing inequalities, has provided an important paradigm in the study of non-Markovian quantum stochastic processes. Although often used interchangeably, we argue here that the notions of “revivals” and “backflows,” i.e., flows of information from the environment back into the system, are distinct: an information revival can occur without any backflow ever taking place. In this paper, we examine in detail the phenomenon of noncausal revivals and relate them to the theory of short Markov chains and squashed non-Markovianity. We also provide an operational condition, in terms of system-only degrees of freedom, to witness the presence of genuine backflow that cannot be explained by noncausal revivals. As a byproduct, we demonstrate that focusing on processes with genuine backflows, while excluding those with only noncausal revivals, resolves the issue of nonconvexity of Markovianity, thus enabling the construction of a convex resource theory of genuine quantum non-Markovianity.

A new construction of non-Gaussian, rotation-invariant and reflection positive probability measures μ \mu associated with the φ 3 4 \varphi ^4_3 -model of quantum field theory is presented. Our construction uses a combination of semigroup methods, and methods of stochastic partial differential equations (SPDEs) for finding solutions and stationary measures of the natural stochastic quantization associated with the φ 3 4 \varphi ^4_3 -model. Our starting point is a suitable approximation μ M , N \mu _{M,N} of the measure μ \mu , which we intend to construct. μ M , N \mu _{M,N} is parametrized by an M M -dependent space cut-off function ρ M : R 3 → R \rho _M: {\mathbb R}^3\rightarrow {\mathbb R} and an N N -dependent momentum cut-off function ψ N : R ^ 3 ≅ R 3 → R \psi _N: \widehat {\mathbb R}^3 \cong {\mathbb R}^3 \rightarrow {\mathbb R} , that act on the interaction term (nonlinear term and counterterms). The corresponding family of stochastic quantization equations yields solutions ( X t M , N , t ≥ 0 ) (X_t^{M,N}, t\geq 0) that have μ M , N \mu _{M,N} as an invariant probability measure. By a combination of probabilistic and functional analytic methods for singular stochastic differential equations on negative-indices weighted Besov spaces (with rotation invariant weights) we prove the tightness of the family of continuous processes ( X t M , N , t ≥ 0 ) M , N (X_t^{M,N},t \geq 0)_{M,N} . Limit points in the sense of convergence in law exist, when both M M and N N diverge to + ∞ +\infty . The limit processes ( X t ; t ≥ 0 ) (X_t; t\geq 0) are continuous on the intersection of suitable Besov spaces and any limit point μ \mu of the μ M , N \mu _{M,N} is a stationary measure of X X . μ \mu is shown to be a rotation-invariant and non-Gaussian probability measure and we provide results on its support. It is also proven that μ \mu satisfies a further important property belonging to the family of axioms for Euclidean quantum fields, namely it is reflection positive.

Stochastic quantum power flow for risk assessment in power systems

Abstract We develop a theory of random non-Hermitian action that, after quantization, describes the stochastic nonlinear dynamics of quantum states in Hilbert space. Focusing on fermionic fields, we propose both canonical quantization and path integral quantization, demonstrating that these two approaches are equivalent. Using this formalism, we investigate the evolution of a single-particle Gaussian wave packet under the influence of non-Hermiticity and randomness. Our results show that specific types of non-Hermiticity lead to wave packet localization, while randomness affects the central position of the wave packet, causing the variance of its distribution to increase with the strength of the randomness.

We explore the mathematical relationship between the holographic Wilsonian renormalization group (HWRG) and stochastic quantization(SQ) motivated by the similarity of the monotonicity in RG flow with Langevin dynamics of non-equilibrium thermodynamics. We look at scalar field theory in AdS space with its generic mass, self-interaction, and marginal boundary deformation in the momentum space. Identifying the stochastic time t with radial coordinate r in AdS, we establish maps between the fictitious time evolution of stochastic multi-point correlation function and the radial evolution of multi-trace deformation, which respectively, express the relaxation process of Langevin dynamics and holographic RG flow. We show that the multi-trace deformations in the HWRG are successfully captured by the Langevin dynamics of SQ.

Abstract We develop a system of non-linear stochastic evolution equations that describes the continuous measurements of quantum systems with mixed initial state. We address quantum systems with unbounded Hamiltonians and unbounded interaction operators. Using arguments of the theory of quantum measurements we derive a system of stochastic interacting wave functions (SIWFs for short) that models the continuous monitoring of quantum systems. We prove the existence and uniqueness of the solution to this system under conditions general enough for the applications. We obtain that the mixed state generated by the SIWF at any time does not depend on the initial state, and satisfies the diffusive stochastic quantum master equation, which is also known as Belavkin equation. We present two physical examples. In one, the SIWF becomes a system of non-linear stochastic partial differential equations. In the other, we deal with a model of a circuit quantum electrodynamics.

Deriving propagators with stochastic quantum evolution

We define quantum chaos and integrability in open quantum many-body systems as a dynamical property of single stochastic realizations, referred to as quantum trajectories. This definition relies on the predictions of random matrix theory applied to the subset of the Liouvillian eigenspectrum involved in each quantum trajectory. Our approach, which we name (SSQT), enables a natural distinction between transient and steady-state quantum chaos as general phenomena in open setups. We test the generality and reliability of the SSQT criterion on several dissipative systems, further showing that an open system with a chaotic structure can evolve towards either a chaotic or integrable steady state. We apply our theoretical framework to two driven-dissipative bosonic systems. First, we study the driven-dissipative Bose-Hubbard model, a paradigmatic example of a quantum simulator, clarifying the interplay of integrability, transient, and steady-state chaos across its phase diagram. Our analysis shows the existence of an emergent dissipative quantum chaotic phase, whereas the classical and semiclassical limits display an integrable behavior. In this regime, chaos arises from the quantum and classical fluctuations associated with the dissipation mechanisms. Second, we investigate dissipative quantum chaos in the dispersive readout of a transmon qubit: a measurement technique ubiquitous in superconducting-based quantum hardware. Through the SSQT, we distinguish several regimes where the performance of the measurement instrument can be connected to the integrable or chaotic nature of the underlying driven-dissipative bosonic system. Our work offers a general understanding of the integrable and chaotic dynamics of open quantum systems and paves the way for the investigation of dissipative quantum chaos and its consequences on state-of-the-art noisy intermediate-scale quantum devices.

Quantum signal processing (QSP) provides a systematic framework for implementing a polynomial transformation of a linear operator, and unifies nearly all known quantum algorithms. In parallel, recent works have developed randomized compiling, a technique that promotes a unitary gate to a quantum channel and enables a quadratic suppression of error (i.e., ϵ → O(ϵ2)) at little to no overhead. Here we integrate randomized compiling into QSP through Stochastic Quantum Signal Processing. Our algorithm implements a probabilistic mixture of polynomials, strategically chosen so that the average evolution converges to that of a target function, with an error quadratically smaller than that of an equivalent individual polynomial. Because nearly all QSP-based algorithms exhibit query complexities scaling as O(log(1/ϵ))—stemming from a result in functional analysis—this error suppression reduces their query complexity by a factor that asymptotically approaches 1/2. By the unifying capabilities of QSP, this reduction extends broadly to quantum algorithms, which we demonstrate on algorithms for real and imaginary time evolution, phase estimation, ground state preparation, and matrix inversion.

We consider an infinite-range interacting quantum spin-1/2 model, undergoing periodic kicking and dissipatively coupled with an environment. In the thermodynamic limit, it is described by classical mean-field equations that can show regular and chaotic regimes. At finite size, we describe the system dynamics using stochastic quantum trajectories. We find that the asymptotic nonstabilizerness (alias the magic, a measure of quantum complexity), averaged over trajectories, mirrors to some extent the classical chaotic behavior, while the entanglement entropy has no relation with chaos in the thermodynamic limit.

ABSTRACTThis Conference Proceedings volume contains the written versions of most of the contributions presented during the 11th International Workshop on Astronomy and Relativistic Astrophysics (IWARA 2024). The Workshop took place in Machupicchu Pueblo (Aguas Calientes), Peru, from September 2–6, 2024. The Workshop provided a setting for discussing recent developments in a wide variety of topics, including Archaeoastronomy and Cognition, the Kerr metric and the Hawking temperature, Damour‐Solodukhin‐type Wormholes, Stellar Physics and General Relativity, Stochastic Quantum Mechanics in Curved Spaces, Gauss‐Bonnet Theory, Quantum Gravity, Black Holes, Fast Radio Bursts, Neutron Stars, Strange Stars, X‐ray Binaries, Pulsars, Gravitational Waves, and Dark Matter, among others. Most of these contributions are included in this volume.