Jin, Krajenbrink, and Bernard [Phys. Rev. Lett. 125, 040603 (2020)] defined a quantum asymmetric simple exclusion process (quantum ASEP) - a chain of spinless fermions whose hopping amplitudes are driven by operator-valued quantum Brownian noise - and showed that its height field (the time-integrated current) obeys a discrete quantum Kardar-Parisi-Zhang (KPZ) equation, which a quantum Cole-Hopf map linearizes to a quantum stochastic heat equation whose replica moments satisfy an operator-valued Lieb-Liniger problem. Whether this model is exactly solvable (e.g. by an operator Bethe ansatz), what its stationary measures and height-gradient correlation functions are, and whether a quantum KPZ universality class exists - a strong quantum KPZ fixed point with the classical 1/3 fluctuation exponent, or weak convergence under the (1:2:3) scaling - all remain open. The problem is to solve the quantum ASEP and its quantum KPZ / stochastic Burgers / stochastic heat trio exactly, derive its stationary structure and the role of off-diagonal quantum coherences, and determine its scaling limit and universality class.

Background

In one spatial dimension, the asymmetric simple exclusion process (ASEP) is a lattice gas of particles that hop stochastically to the right and left with rates favoring one direction, at most one particle per site. Its time-integrated current - equivalently a height field counting the net number of particles crossing a bond - is the discrete representative of the Kardar-Parisi-Zhang (KPZ) universality class, the generic class of fluctuating one-dimensional interfaces. In the continuum weak-asymmetry limit the height obeys the KPZ equation ∂th=ν∂x2h+λ(∂xh)2+η\partial_t h=\nu\partial_x^2 h+\lambda(\partial_x h)^2+\eta with additive white noise η\eta, and its exponential Z=e(λ/2ν)hZ=e^{(\lambda/2\nu)h} (the Cole-Hopf map) linearizes it to a stochastic heat equation (SHE) with multiplicative noise. The replica moments of ZZ then satisfy an imaginary-time Schr"odinger equation of Lieb-Liniger type with an attractive contact interaction, solvable by the Bethe ansatz, the exact-solution method that represents interacting one-dimensional many-body wavefunctions as superpositions of free plane waves with non-trivial scattering phases (Lieb and Liniger solved this model in 1963); this replica Bethe ansatz underlies the exact KPZ one-point distributions. The classical KPZ class is characterized by the 1:3:2 scaling exponents and two universality results: strong universality - a universal KPZ fixed point describing height fluctuations h∼v∞t+t1/3χh\sim v_\infty t+t^{1/3}\chi with χ\chi from a family depending only on large-scale initial data - and weak universality - convergence of weakly asymmetric growth models to the KPZ equation under the (1:2:3) KPZ scaling of height, space, and time. The derivative of the KPZ height, the stochastic Burgers equation, has a Gaussian stationary measure whose height-gradient correlation functions decay as t−2/3fKPZ(x/2t2/3)t^{-2/3}f_{\rm KPZ}(x/2t^{2/3}) with a universal scaling function fKPZf_{\rm KPZ}. Solving the classical SSEP and ASEP was the seed of macroscopic fluctuation theory (MFT), the effective theory of large deviations in diffusive out-of-equilibrium classical systems.

Jin, Krajenbrink, and Bernard asked what survives of this structure when the noise itself is quantum. Their precursor model, the quantum SSEP (QSSEP), is a chain of spinless fermions whose hopping amplitudes are independent Brownian operators obeying quantum Ito rules (Ito calculus being the stochastic calculus in which products of infinitesimal noise increments have deterministic values of order the time increment dtdt); its noise-averaged dynamics is the classical SSEP, and it is exactly solved: the invariant measure, entanglement statistics, and coherent fluctuations are described by free probability, the extension of probability theory to non-commuting operator-valued random variables. But QSSEP is diffusive - it has no asymmetry, no height field, and no KPZ scaling. The 2020 paper breaks the left-right symmetry: the fermion hopping amplitudes are driven by operator-valued quantum noises dWjdW^j attached to the lattice edges whose Ito contractions are dW‾jdWk=α dtd\overline{W}^j dW^k=\alpha\,dt and dWkdW‾j=(1+α) dtdW^k d\overline{W}^j=(1+\alpha)\,dt, with asymmetry parameter α\alpha (the mean number of excitations in the bath modeled as a bosonic reservoir; a Lindbladian is the standard Markovian generator of open-system dynamics). Averaging over the noise recovers the classical ASEP, but each quantum trajectory keeps operator-valued noise whose correlator depends on off-diagonal quantum coherences of the state. The paper derives the quantum analogue of the KPZ / stochastic Burgers / stochastic heat trio: a discrete quantum KPZ equation for the height operator h^k=∑j≤k(nj−12)\hat h_k=\sum_{j\le k}(n_j-\tfrac12) whose nonlinearity is a direct consequence of the non-commutativity of the noise (its discrete gradient is the quantum stochastic Burgers equation for the density); a quantum SHE with multiplicative operator-valued noise, to which the dynamics linearizes under the quantum Cole-Hopf (Gartner) operator Z^k=eδh^k\hat Z_k=e^{\delta\hat h_k} with δ=log⁡(α/(1+α))\delta=\log(\alpha/(1+\alpha)); and, for the replica moments of Z^\hat Z, an imaginary-time Schr"odinger equation on the space of operators with an attractive contact potential - an operator-valued analogue of the Lieb-Liniger model. In a weak-asymmetry (1:2:3) scaling with almost-commuting quantum noise, the Ito contraction of the quantum noise becomes classical at leading order while the KPZ nonlinearity survives as a remnant of non-commutativity, so the classical KPZ equation emerges as the formal leading limit - but no limit theorem, no stationarity result, and no universality statement is known. Because the noise is operator-valued and multiplicative, Wick-type tools - the Gaussian moment-factorization techniques underlying most exact results for stochastic dynamics - fail and the moment hierarchy does not close, so the standard exact-solution machinery of classical stochastic dynamics is unavailable.

Problem Statement

The problem is the exact solution and scaling-limit theory of the quantum ASEP of Jin-Krajenbrink-Bernard and its trio of quantum KPZ, quantum stochastic heat, and quantum stochastic Burgers equations, as one linked objective with three components. (1) Solvability and observables: construct an analytic solution method for the operator-valued replica Schr"odinger problem - the imaginary-time Schr"odinger equation on operator space with attractive contact interaction satisfied by the moments of the quantum Cole-Hopf operator Z^k=eδh^k\hat Z_k=e^{\delta \hat h_k}, δ=log⁡(α/(1+α))\delta=\log(\alpha/(1+\alpha)) - by any route, an adaptation of the Bethe ansatz to the operator contact-interaction Hamiltonian acting on the commuting replica operators being one candidate method rather than a requirement; use it to compute the height moments, the full counting statistics (the full probability distribution) of the time-integrated current, and the entanglement entropy dynamics; and thereby decide whether the model is integrable (integrability being used here in the sense of exact solvability - the model admits an exact analytic solution through an integrable structure such as a Bethe-ansatz solution or a family of commuting conserved charges, as opposed to the generic case in which the moment hierarchy fails to close and only a formal perturbative expansion is available). (2) Stationary structure and coherences: derive the stationary measure(s) of the quantum stochastic Burgers equation and the stationary correlation functions of the quantum height gradient, deciding whether they coincide with the classical KPZ stationary correlators t−2/3fKPZ(x/2t2/3)t^{-2/3}f_{\rm KPZ}(x/2t^{2/3}) or differ in systematic ways, and quantify how off-diagonal quantum coherences modify fluctuation and transport statistics. (3) Universality: determine whether a quantum KPZ universality class exists - whether weakly asymmetric quantum growth models with operator-valued noise converge under the (1:2:3) scaling (weak universality), and whether a quantum analogue of the KPZ fixed point with the 1/31/3 height fluctuation exponent and universal height distributions governs the strong-asymmetry regime (strong universality). All three components are required, and both sub-questions of (3) - weak (1:2:3) convergence and the strong KPZ-fixed-point statement - must be decided. An exact solution of (1) provides a natural common tool for (2) and (3), but it does not by itself resolve them, and a solution of any single component counts as partial progress rather than as a resolution of the task. A quantum extension of macroscopic fluctuation theory is motivation only and is not a required deliverable.

References

  1. T. Jin, A. Krajenbrink, D. Bernard, 'From stochastic spin chains to quantum Kardar-Parisi-Zhang dynamics', Phys. Rev. Lett. 125, 040603 (2020). DOI: 10.1103/PhysRevLett.125.040603, arXiv:2001.04278, https://arxiv.org/abs/2001.04278
  2. D. Bernard, T. Jin, 'Open Quantum Symmetric Simple Exclusion Process', Phys. Rev. Lett. 123, 080601 (2019). DOI: 10.1103/PhysRevLett.123.080601, arXiv:1904.01406, https://arxiv.org/abs/1904.01406
  3. D. Bernard, T. Jin, 'Solution to the Quantum Symmetric Simple Exclusion Process: The Continuous Case', Commun. Math. Phys. 384, 1141 (2021). DOI: 10.1007/s00220-021-04087-x, arXiv:2006.12222, https://arxiv.org/abs/2006.12222
  4. D. Bernard, L. Piroli, 'Entanglement distribution in the quantum symmetric simple exclusion process', Phys. Rev. E 104, 014146 (2021). DOI: 10.1103/PhysRevE.104.014146, arXiv:2102.04745, https://arxiv.org/abs/2102.04745
  5. L. Hruza, D. Bernard, 'Coherent Fluctuations in Noisy Mesoscopic Systems, the Open Quantum SSEP, and Free Probability', Phys. Rev. X 13, 011045 (2023). DOI: 10.1103/PhysRevX.13.011045, arXiv:2204.11680, https://arxiv.org/abs/2204.11680
  6. D. Bernard, F. H. L. Essler, L. Hruza, M. Medenjak, 'Dynamics of fluctuations in quantum simple exclusion processes', SciPost Phys. 12, 042 (2022). DOI: 10.21468/SciPostPhys.12.1.042, arXiv:2107.02662, https://arxiv.org/abs/2107.02662
  7. J. Robertson, F. H. L. Essler, 'Exact solution of a quantum asymmetric exclusion process with particle creation and annihilation', J. Stat. Mech. 2021, 103102. DOI: 10.1088/1742-5468/ac22f8, arXiv:2105.08828, https://arxiv.org/abs/2105.08828
  8. D. Bernard, 'Can the Macroscopic Fluctuation Theory be Quantized?', J. Phys. A 54, 433001 (2021). DOI: 10.1088/1751-8121/ac2597, arXiv:2107.04442, https://arxiv.org/abs/2107.04442
  9. G. Barraquand, D. Bernard, 'Introduction to quantum exclusion processes' (2025). arXiv:2507.01570, https://arxiv.org/abs/2507.01570
  10. M. Hairer, J. Quastel, 'A class of growth models rescaling to KPZ' (2015). arXiv:1512.07845, https://arxiv.org/abs/1512.07845
  11. K. Matetski, J. Quastel, D. Remenik, 'The KPZ fixed point', Acta Mathematica 227, 115 (2021). DOI: 10.4310/ACTA.2021.v227.n1.a3, arXiv:1701.00018, https://arxiv.org/abs/1701.00018
  12. K. Fujimoto, R. Hamazaki, Y. Kawaguchi, 'Dynamical Scaling of Surface Roughness and Entanglement Entropy in Disordered Fermion Models', Phys. Rev. Lett. 127, 090601 (2021). DOI: 10.1103/PhysRevLett.127.090601, https://doi.org/10.1103/PhysRevLett.127.090601
  13. J. De Nardis, S. Gopalakrishnan, R. Vasseur, 'Nonlinear Fluctuating Hydrodynamics for Kardar-Parisi-Zhang Scaling in Isotropic Spin Chains', Phys. Rev. Lett. 131, 197102 (2023). DOI: 10.1103/PhysRevLett.131.197102, https://doi.org/10.1103/PhysRevLett.131.197102
  14. T. Jin, D. G. Martin, 'Kardar-Parisi-Zhang Physics and Phase Transition in a Classical Single Random Walker under Continuous Measurement', Phys. Rev. Lett. 129, 260603 (2022). DOI: 10.1103/PhysRevLett.129.260603, https://doi.org/10.1103/PhysRevLett.129.260603
  15. G.-Q. Wang, C.-L. Zou, G.-C. Guo, X.-B. Zou, 'Kardar-Parisi-Zhang dynamics in an open integrable system: beyond the spontaneous-symmetry-breaking ansatz' (2026). arXiv:2607.02341, https://arxiv.org/abs/2607.02341
  16. K. Kwon, K. Fujimoto, J. Hur, B. Lee, S. Hwang, S. Kim, R. Hamazaki, Y. Kawaguchi, J. Choi, 'Universal Family-Vicsek scaling in quantum gases far from equilibrium' (2026). arXiv:2603.09060, https://arxiv.org/abs/2603.09060