© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

arXiv:2305.15319 · Scientific Reports

Proposal of a quantum version of active particles via a nonunitary quantum walk

Manami Yamagishi, Naomichi Hatano, Hideaki Obuse

Published Nov 19, 2024• 11 citations

Abstract

Abstract The main aim of the present paper is to define an active particle in a quantum framework as a minimal model of quantum active matter and investigate difference and commonalities of quantum and classical active matters. Although the research field of active matter has been expanding wider and wider, most research is conducted in classical systems. We here propose a truly deterministic quantum active-particle model with a non-unitary quantum walk as minimal models of quantum active matter. We aim to reproduce similar results obtained previously with classical active Brownian particle; that is, a Brownian particle, with a finite energy take-up, becomes active and climbs up a potential wall. We realize such a system with non-unitary quantum walks. We introduce new internal states, the ground state |G⟩ and the excited state |E⟩, and a new non-unitary operator N(g) for an asymmetric transition between |G⟩ and |E⟩. The non-Hermiticity parameter g promotes transition to the excited state and hence the particle takes up energy from the environment. With our quantum active particle, we successfully observe that the movement of the quantum walker becomes more active in a non-trivial way as we increase the non-Hermiticity parameter g, which is similar to the classical active Brownian particle. Meanwhile, we also observe three unique features of quantum walks, namely, ballistic propagation of peaks in one dimension, the walker staying on the constant energy plane in two dimensions, and oscillations originating from the resonant transition between the ground state |G⟩ and excited state |E⟩ both in one and two dimensions.

DOI: 10.1038/s41598-024-78986-z

Related papers

Modeling dissipation in quantum active matter

Alexander P. Antonov, Sangyun Lee, Benno Liebchen +5 more

Active matter is characterized by a constant influx and dissipation of energy that gives rise to directed motion. Dissipation requires interactions with an external environment, such that extending the paradigm of active matter to a quantum framework requires an appropriate description of this environment. In this work, we consider a driven quantum particle undergoing noise and dissipation, with external driving exhibiting characteristics of classical activity. We model the non-unitary dynamics with time-local master equations and analyze the particle motion at different time scales for different forms of the master equations, satisfying different criteria. We systematically compare predictions on the dynamics of particle trajectories and thereby we uncover how the particle motion evolves under the interplay of quantum effects, dissipation, and active-like dynamics. These results are essential for guiding possible experiments aimed at realizing quantum analogues of classical active systems.

quant-pharXiv:2511.21502•Physical Review A•Nov 27, 2025•3 citations

Emergent Quantum Walk Dynamics from Classical Interacting Particles

Surajit Saha

The dynamics of a discrete-time quantum walk (DTQW) can be realized within a purely classical interacting particle system composed of some boxes and a large but finite number of balls, and can, in principle, be implemented in a tabletop experimental setting. The distribution of the balls evolves under stochastic, occupation-dependent update rules at each lattice site, producing quantum-walk dynamics without invoking a wavefunction. The update parameters are fixed by the parameters of coin and shift operations of the DTQW. This framework naturally yields a generalized active spin model and provides a minimal lattice-based microscopic understanding of the emergence of quantum-like dynamics in active matter systems. This interdisciplinary approach connects the classical models to the broad range of applications where DTQWs are successfully employed.

arXiv:2506.06896•Jun 7, 2025•0 citations

A novel construction of lackadaisical quantum walk

Yunguo Lin, Yiwen Ye, Jie Yao +1 more

Abstract This paper constructs a new model for the lackadaisical quantum walk and analyzes its features. Traditionally, the particle state is represented by a three-dimensional complex vector, with the Grover operator as the evolution unitary. Analytical methods calculate position probability distributions, but computational complexity and error grow exponentially with steps. To address this, the paper introduces an evolution unitary operator composed of three operators for left movement, right movement, and staying. The staying state corresponds to multiple self-loops at position vertices. A path analysis method is employed to calculate position probabilities and characteristics in polynomial time. Lackadaisical quantum walks differ from classical walks, increasing self-loops enhances diffusion, following a Gaussian mixture model. Numerical experiments reveal lackadaisical characteristics such as slower diffusion, localization, and ballistic diffusion.

Physica Scripta•Apr 29, 2025•0 citations

Quantum features derived from the classical model of a bouncer-walker coupled to a zero-point field

H. Schwabl, J Mesa Pascasio, Siegfried Fussy +1 more

In our bouncer-walker model a quantum is a nonequilibrium steady-state maintained by a permanent throughput of energy. Specifically, we consider a "particle" as a bouncer whose oscillations are phase-locked with those of the energy-momentum reservoir of the zero-point field (ZPF), and we combine this with the random-walk model of the walker, again driven by the ZPF. Starting with this classical toy model of the bouncer-walker we were able to derive fundamental elements of quantum theory [1]. Here this toy model is revisited with special emphasis on the mechanism of emergence. Especially the derivation of the total energy ℏω o and the coupling to the ZPF are clarified. For this we make use of a sub-quantum equipartition theorem. It can further be shown that the couplings of both bouncer and walker to the ZPF are identical. Then we follow this path in accordance with Ref. [2], expanding the view from the particle in its rest frame to a particle in motion. The basic features of ballistic diffusion are derived, especially the diffusion constant D, thus providing a missing link between the different approaches of our previous works [1, 2].

arXiv:1205.4519•Journal of Physics Conference Series•May 10, 2012•4 citations

Directional quantum random walk induced by coherence

Jin-Fu Chen, Yu-Han Ma, Chang-Pu Sun

Quantum walk (QW), which is considered as the quantum counterpart of the classical random walk (CRW), is actually the quantum extension of CRW from the single-coin interpretation. The sequential unitary evolution engenders correlation between different steps in QW and leads to a non-binomial position distribution. In this paper, we propose an alternative quantum extension of CRW from the ensemble interpretation, named quantum random walk (QRW), where the walker has many unrelated coins, modeled as two-level systems, initially prepared in the same state. We calculate the walker’s position distribution in QRW for different initial coin states with the coin operator chosen as Hadamard matrix. In one-dimensional case, the walker’s position is the asymmetric binomial distribution. We further demonstrate that in QRW, coherence leads the walker to perform directional movement. For an initially decoherenced coin state, the walker’s position distribution is exactly the same as that of CRW. Moreover, we study QRW in 2D lattice, where the coherence plays a more diversified role in the walker’s position distribution.

arXiv:1907.12072•Frontiers of Physics•Jan 3, 2020•1 citation

Active Quantum Particles from Engineered Dissipation

Jeanne Gipouloux, Matteo Brunelli, Leticia Cugliandolo +2 more

We introduce and characterize different models for an active quantum particle where activity arises from engineered dissipation-- specifically, from a suitably coupled nonequilibrium environment. These include a model of a particle moving on a lattice with coherent and dissipative hopping, as well as quantum generalizations of well-studied models of active behavior, such as the active Ornstein-Uhlenbeck process, run-and-tumble dynamics, and the active Brownian particle. Despite the different microscopic mechanisms at play, we show that all these models display key features of active motion. Notably, we observe a crossover from diffusive to active-diffusive behavior at long times, leading to an effective Péclet number, as well as a strong sensitivity to boundary conditions which, in our open quantum system context, arises from the Liouville skin effect. We discuss the role of quantum fluctuations and experimental realizations with superconducting circuits or cold gases, closing with perspectives for many-body effects in quantum active matter.

quant-pharXiv:2603.19094•Mar 20, 2026•0 citations

Abstract The main aim of the present paper is to define an active particle in a quantum framework as a minimal model of quantum active matter and investigate difference and commonalities of quantum and classical active matters. Although the research field of active matter has been expanding wider and wider, most research is conducted in classical systems. We here propose a truly deterministic quantum active-particle model with a non-unitary quantum walk as minimal models of quantum active matter. We aim to reproduce similar results obtained previously with classical active Brownian particle; that is, a Brownian particle, with a finite energy take-up, becomes active and climbs up a potential wall. We realize such a system with non-unitary quantum walks. We introduce new internal states, the ground state |G⟩ and the excited state |E⟩, and a new non-unitary operator N(g) for an asymmetric transition between |G⟩ and |E⟩. The non-Hermiticity parameter g promotes transition to the excited state and hence the particle takes up energy from the environment. With our quantum active particle, we successfully observe that the movement of the quantum walker becomes more active in a non-trivial way as we increase the non-Hermiticity parameter g, which is similar to the classical active Brownian particle. Meanwhile, we also observe three unique features of quantum walks, namely, ballistic propagation of peaks in one dimension, the walker staying on the constant energy plane in two dimensions, and oscillations originating from the resonant transition between the ground state |G⟩ and excited state |E⟩ both in one and two dimensions.

Active matter is characterized by a constant influx and dissipation of energy that gives rise to directed motion. Dissipation requires interactions with an external environment, such that extending the paradigm of active matter to a quantum framework requires an appropriate description of this environment. In this work, we consider a driven quantum particle undergoing noise and dissipation, with external driving exhibiting characteristics of classical activity. We model the non-unitary dynamics with time-local master equations and analyze the particle motion at different time scales for different forms of the master equations, satisfying different criteria. We systematically compare predictions on the dynamics of particle trajectories and thereby we uncover how the particle motion evolves under the interplay of quantum effects, dissipation, and active-like dynamics. These results are essential for guiding possible experiments aimed at realizing quantum analogues of classical active systems.

The dynamics of a discrete-time quantum walk (DTQW) can be realized within a purely classical interacting particle system composed of some boxes and a large but finite number of balls, and can, in principle, be implemented in a tabletop experimental setting. The distribution of the balls evolves under stochastic, occupation-dependent update rules at each lattice site, producing quantum-walk dynamics without invoking a wavefunction. The update parameters are fixed by the parameters of coin and shift operations of the DTQW. This framework naturally yields a generalized active spin model and provides a minimal lattice-based microscopic understanding of the emergence of quantum-like dynamics in active matter systems. This interdisciplinary approach connects the classical models to the broad range of applications where DTQWs are successfully employed.

Abstract This paper constructs a new model for the lackadaisical quantum walk and analyzes its features. Traditionally, the particle state is represented by a three-dimensional complex vector, with the Grover operator as the evolution unitary. Analytical methods calculate position probability distributions, but computational complexity and error grow exponentially with steps. To address this, the paper introduces an evolution unitary operator composed of three operators for left movement, right movement, and staying. The staying state corresponds to multiple self-loops at position vertices. A path analysis method is employed to calculate position probabilities and characteristics in polynomial time. Lackadaisical quantum walks differ from classical walks, increasing self-loops enhances diffusion, following a Gaussian mixture model. Numerical experiments reveal lackadaisical characteristics such as slower diffusion, localization, and ballistic diffusion.

In our bouncer-walker model a quantum is a nonequilibrium steady-state maintained by a permanent throughput of energy. Specifically, we consider a "particle" as a bouncer whose oscillations are phase-locked with those of the energy-momentum reservoir of the zero-point field (ZPF), and we combine this with the random-walk model of the walker, again driven by the ZPF. Starting with this classical toy model of the bouncer-walker we were able to derive fundamental elements of quantum theory [1]. Here this toy model is revisited with special emphasis on the mechanism of emergence. Especially the derivation of the total energy ℏω o and the coupling to the ZPF are clarified. For this we make use of a sub-quantum equipartition theorem. It can further be shown that the couplings of both bouncer and walker to the ZPF are identical. Then we follow this path in accordance with Ref. [2], expanding the view from the particle in its rest frame to a particle in motion. The basic features of ballistic diffusion are derived, especially the diffusion constant D, thus providing a missing link between the different approaches of our previous works [1, 2].

Quantum walk (QW), which is considered as the quantum counterpart of the classical random walk (CRW), is actually the quantum extension of CRW from the single-coin interpretation. The sequential unitary evolution engenders correlation between different steps in QW and leads to a non-binomial position distribution. In this paper, we propose an alternative quantum extension of CRW from the ensemble interpretation, named quantum random walk (QRW), where the walker has many unrelated coins, modeled as two-level systems, initially prepared in the same state. We calculate the walker’s position distribution in QRW for different initial coin states with the coin operator chosen as Hadamard matrix. In one-dimensional case, the walker’s position is the asymmetric binomial distribution. We further demonstrate that in QRW, coherence leads the walker to perform directional movement. For an initially decoherenced coin state, the walker’s position distribution is exactly the same as that of CRW. Moreover, we study QRW in 2D lattice, where the coherence plays a more diversified role in the walker’s position distribution.

We introduce and characterize different models for an active quantum particle where activity arises from engineered dissipation-- specifically, from a suitably coupled nonequilibrium environment. These include a model of a particle moving on a lattice with coherent and dissipative hopping, as well as quantum generalizations of well-studied models of active behavior, such as the active Ornstein-Uhlenbeck process, run-and-tumble dynamics, and the active Brownian particle. Despite the different microscopic mechanisms at play, we show that all these models display key features of active motion. Notably, we observe a crossover from diffusive to active-diffusive behavior at long times, leading to an effective Péclet number, as well as a strong sensitivity to boundary conditions which, in our open quantum system context, arises from the Liouville skin effect. We discuss the role of quantum fluctuations and experimental realizations with superconducting circuits or cold gases, closing with perspectives for many-body effects in quantum active matter.