Showing 1–30 of 30 results for author: Kos, P

Magnetic domains stabilized by symmetry-protected zero modes

Authors: Pavel Kos, Dominik S. Wild, Kristian Knakkergaard Nielsen

Abstract: Understanding mechanisms for the breakdown of thermalization in closed quantum systems is a central problem in quantum many-body physics. We demonstrate strong non-ergodic behavior in the XX model on coupled chains, where domain-wall initial states retain an inhomogeneous magnetization profile for arbitrarily long times. We find that this effect arises due to exponentially many zero modes protecte… ▽ More Understanding mechanisms for the breakdown of thermalization in closed quantum systems is a central problem in quantum many-body physics. We demonstrate strong non-ergodic behavior in the XX model on coupled chains, where domain-wall initial states retain an inhomogeneous magnetization profile for arbitrarily long times. We find that this effect arises due to exponentially many zero modes protected by chiral symmetry. Using an analysis based on the Lanczos algorithm, we identify a localization transition in the thermodynamic limit at a critical coupling between the chains. We further show that antiferromagnetic defects in the initial state and symmetry-breaking perturbations restore slow thermalization, whereas it remains robust for symmetry-conserving perturbations. These results establish that degenerate, symmetry-protected subspaces can give rise to thermodynamically stable non-ergodic dynamics in experimentally accessible quantum systems. △ Less

Submitted 16 April, 2026; originally announced April 2026.

Comments: 8 pages, 5 figures + Supplemental Material

Vanishing correlations in stochastic and bistochastic controlled circuits

Authors: Pavel Kos, Bruno Bertini, Tomaž Prosen

Abstract: We study the dynamics of circuits composed of stochastic and bistochastic controlled gates. This type of dynamics arises from quantum circuits with random controlled gates, as well as in stochastic circuits and deterministic classical cellular automata. We prove that stochastic and bistochastic controlled gates lead to two-point spatiotemporal correlation functions that vanish everywhere except wh… ▽ More We study the dynamics of circuits composed of stochastic and bistochastic controlled gates. This type of dynamics arises from quantum circuits with random controlled gates, as well as in stochastic circuits and deterministic classical cellular automata. We prove that stochastic and bistochastic controlled gates lead to two-point spatiotemporal correlation functions that vanish everywhere except when the two operators act on the same site. More generally, for multipoint correlations the two rightmost operators must act on the same site. We argue that autocorrelation, while hard to compute, typically decays exponentially toward a value that is exponentially small in the system size. Our results reveal a broad class of quantum systems that exhibit surprisingly simple correlation structures despite their complex microscopic dynamics. △ Less

Submitted 29 June, 2026; v1 submitted 20 January, 2026; originally announced January 2026.

Comments: 10 pages, 1 figure; v2: minor corrections

Journal ref: Phys. Rev. E 113, 064139 (2026)

Random Permutation Circuits Beyond Qubits are Quantum Chaotic

Authors: Bruno Bertini, Katja Klobas, Pavel Kos, Daniel Malz

Abstract: Random permutation circuits were recently introduced as minimal models for local many-body dynamics that can be interpreted both as classical and quantum. Standard dynamical complexity indicators such as damage spreading and out-of-time-order correlators (OTOCs), show that these systems exhibit sensitivity to initial conditions in the classical setting and operator scrambling in the quantum settin… ▽ More Random permutation circuits were recently introduced as minimal models for local many-body dynamics that can be interpreted both as classical and quantum. Standard dynamical complexity indicators such as damage spreading and out-of-time-order correlators (OTOCs), show that these systems exhibit sensitivity to initial conditions in the classical setting and operator scrambling in the quantum setting. Here, we address their quantum chaoticity - a stricter property - by studying the time evolution of local operator entanglement (LOE). We show that the behaviour of LOE in random permutation circuits depends on the dimension of the local configuration space q. When q = 2, i.e. the circuits act on qubits, random permutations are Clifford and the LOE of any local operator is bounded by a constant, indicating that they are not truly chaotic. On the other hand, when the dimension of the local configuration space exceeds two, the LOE grows linearly in time. We prove this in the limit of large q and present numerical evidence that a three-dimensional local configuration space is sufficient for a linear growth of LOE. Our findings highlight that quantum chaos can be produced by essentially classical dynamics. Moreover, we show that LOE can be defined also in the classical realm and put it forward as a universal indicator chaos, both quantum and classical. △ Less

Submitted 15 March, 2026; v1 submitted 14 August, 2025; originally announced August 2025.

Comments: 7+10 pages, 2 figures; v2 as appears in Phys. Rev. B

Journal ref: Phys. Rev. B 113, L100302 (2026)

Average-computation benchmarking for local expectation values in digital quantum devices

Authors: Flavio Baccari, Pavel Kos, Georgios Styliaris

Abstract: As quantum devices progress towards a quantum advantage regime, they become harder to benchmark. A particularly relevant challenge is to assess the quality of the whole computation, beyond testing the performance of each single operation. Here we introduce a scheme for this task that combines the target computation with variants of it, which, when averaged, allow for classically solvable correlati… ▽ More As quantum devices progress towards a quantum advantage regime, they become harder to benchmark. A particularly relevant challenge is to assess the quality of the whole computation, beyond testing the performance of each single operation. Here we introduce a scheme for this task that combines the target computation with variants of it, which, when averaged, allow for classically solvable correlation functions. Importantly, the variants exactly preserve the circuit architecture and depth, without simplifying the gates into a classically-simulable set. The method is based on replacing each gate by an ensemble of similar gates, which when averaged together form space-time channels [P. Kos and G. Styliaris, Quantum 7, 1020 (2023)]. We introduce explicit constructions for ensembles producing such channels, all applicable to arbitrary brickwork circuits, and provide a general recipe to find new ones through semidefinite programming. The resulting average computation retains important information about the original circuit and is able to detect noise beyond a Clifford benchmarking regime. Moreover, we provide evidence that estimating average-computation expectation values requires running only a limited number of different circuit realizations. △ Less

Submitted 2 April, 2026; v1 submitted 24 July, 2025; originally announced July 2025.

Comments: 15 pages, 2 figures. v2: improved presentation, added a part of the end matter about 2+1D. v3: minimal changes

Mixed state deep thermalization

Authors: Xie-Hang Yu, Wen Wei Ho, Pavel Kos

Abstract: We introduce the notion of the mixed state projected ensemble (MSPE), a collection of mixed states describing a local region of a quantum many-body system, conditioned upon measurements of the complementary region which are incomplete. This constitutes a generalization of the pure state projected ensemble in which measurements are assumed ideal and complete, and which has been shown to tend toward… ▽ More We introduce the notion of the mixed state projected ensemble (MSPE), a collection of mixed states describing a local region of a quantum many-body system, conditioned upon measurements of the complementary region which are incomplete. This constitutes a generalization of the pure state projected ensemble in which measurements are assumed ideal and complete, and which has been shown to tend towards limiting pure state distributions depending only on symmetries of the system, thus representing a new kind of universality in quantum equilibration dubbed deep thermalization. We study the MSPE generated by solvable (1+1)d dual-unitary quantum circuit evolution, and identify the limiting mixed state distributions which emerge at late times depending on the size of the incomplete measurement, which we assume to be lossy, finding that they correspond to certain random density matrix ensembles known in the literature. We also derive the rate of the emergence of such universality. Furthermore, we investigate the quantum information properties of the states composing the ensemble, specifically their capacity to teleport quantum information between the ends of the system. The teleportation fidelity is upper bounded by the quantum conditional entropy, which we find exhibits a sharp transition from zero to maximal when the number of measurements lost matches of that the number of degrees of freedom to be teleported. Our results initiate the first investigation of deep thermalization for mixed state ensembles, which are relevant for present-day quantum simulation experiments wherein measurements are typically not perfect, and also amount to a physical and natural way of sampling from hitherto abstract random density matrix ensembles. △ Less

Submitted 29 December, 2025; v1 submitted 12 May, 2025; originally announced May 2025.

Comments: 5+2+10 pages, 4 figures

Journal ref: Phys. Rev. Lett. 135, 260402 (2025)

Quantum and Classical Dynamics with Random Permutation Circuits

Authors: Bruno Bertini, Katja Klobas, Pavel Kos, Daniel Malz

Abstract: Understanding thermalisation in quantum many-body systems is among the most enduring problems in modern physics. A particularly interesting question concerns the role played by quantum mechanics in this process, i.e. whether thermalisation in quantum many-body systems is fundamentally different from that in classical many-body systems and, if so, which of its features are genuinely quantum. Here w… ▽ More Understanding thermalisation in quantum many-body systems is among the most enduring problems in modern physics. A particularly interesting question concerns the role played by quantum mechanics in this process, i.e. whether thermalisation in quantum many-body systems is fundamentally different from that in classical many-body systems and, if so, which of its features are genuinely quantum. Here we study this question in minimally structured many-body systems which are only constrained to have local interactions, i.e. local random circuits. We introduce a class of random permutation circuits (RPCs), where the gates locally permute basis states modelling generic microscopic classical dynamics, and compare them to random unitary circuits (RUCs), a standard toy model for generic quantum dynamics. We show that, like RUCs, RPCs permit the analytical computation of several key quantities such as out-of-time order correlators (OTOCs), or entanglement entropies. RPCs can be interpreted both as quantum or classical dynamics, which we use to find similarities and differences between the two. Performing the average over all random circuits, we discover a series of exact relations, connecting quantities in RUC and (quantum) RPCs. In the classical setting, we obtain similar exact results relating (quantum) purity to (classical) growth of mutual information and (quantum) OTOCs to (classical) decorrelators. Our results indicate that despite of the fundamental differences between quantum and classical systems, their dynamics exhibits qualitatively similar behaviours. △ Less

Submitted 9 April, 2025; v1 submitted 16 July, 2024; originally announced July 2024.

Comments: 26 (15+11) pages, 2 figures; v2 minor modifications

Journal ref: Phys. Rev. X 15, 011015 (2025)

Exact solution of long-range stabilizer Rényi entropy in the dual-unitary XXZ model

Authors: Jordi Arnau Montañà López, Pavel Kos

Abstract: Quantum systems can not be efficiently simulated classically due to the presence of entanglement and nonstabilizerness, also known as quantum magic. Here we study the generation of magic under evolution by a quantum circuit. To be able to provide exact solutions, we focus on the dual-unitary XXZ model and a measure of magic called stabilizer Rényi entropy (SRE). Moreover, we focus also on long-ran… ▽ More Quantum systems can not be efficiently simulated classically due to the presence of entanglement and nonstabilizerness, also known as quantum magic. Here we study the generation of magic under evolution by a quantum circuit. To be able to provide exact solutions, we focus on the dual-unitary XXZ model and a measure of magic called stabilizer Rényi entropy (SRE). Moreover, we focus also on long-range SRE, which cannot be removed by short-depth quantum circuits. To obtain exact solutions we use a ZX-calculus representation and graphical rules for the evaluation of the required expressions. We obtain exact results for SRE after short-time evolution in the thermodynamic limit and for long-range SRE for all times and all Rényi parameters for a particular partition of the state. Since the numerical evaluation of these quantities is exponentially costly in the Rényi parameter, we verify this numerically for low Rényi parameters and accessible system sizes and provide numerical results for the long-range SRE in other bipartitions. △ Less

Submitted 7 May, 2024; originally announced May 2024.

Comments: 43 pages, 11 figures

Journal ref: J. Phys. A: Math. Theor. 57 (2024) 475301

Dual-isometric Projected Entangled Pair States

Authors: Xie-Hang Yu, J. Ignacio Cirac, Pavel Kos, Georgios Styliaris

Abstract: Efficient characterization of higher dimensional many-body physical states presents significant challenges. In this paper, we propose a new class of Project Entangled Pair State (PEPS) that incorporates two isometric conditions. This new class facilitates the efficient calculation of general local observables and certain two-point correlation functions, which have been previously shown to be intra… ▽ More Efficient characterization of higher dimensional many-body physical states presents significant challenges. In this paper, we propose a new class of Project Entangled Pair State (PEPS) that incorporates two isometric conditions. This new class facilitates the efficient calculation of general local observables and certain two-point correlation functions, which have been previously shown to be intractable for general PEPS, or PEPS with only a single isometric constraint. Despite incorporating two isometric conditions, our class preserves the rich physical structure while enhancing the analytical capabilities. It features a large set of tunable parameters, with only a subleading correction compared to that of general PEPS. Furthermore, we analytically demonstrate that this class can encode universal quantum computations and can represent a transition from topological to trivial order. △ Less

Submitted 7 November, 2024; v1 submitted 25 April, 2024; originally announced April 2024.

Comments: 5+10 pages, 1 figure

Journal ref: Phys. Rev. Lett. 133, 190401 (2024)

Quantum information spreading in generalised dual-unitary circuits

Authors: Alessandro Foligno, Pavel Kos, Bruno Bertini

Abstract: We study the spreading of quantum information in a recently introduced family of brickwork quantum circuits that generalises the dual-unitary class. These circuits are unitary in time, while their spatial dynamics is unitary only in a restricted subspace. First, we show that local operators spread at the speed of light as in dual-unitary circuits, i.e., the butterfly velocity takes the maximal val… ▽ More We study the spreading of quantum information in a recently introduced family of brickwork quantum circuits that generalises the dual-unitary class. These circuits are unitary in time, while their spatial dynamics is unitary only in a restricted subspace. First, we show that local operators spread at the speed of light as in dual-unitary circuits, i.e., the butterfly velocity takes the maximal value allowed by the geometry of the circuit. Then, we prove that the entanglement spreading can still be characterised exactly for a family of compatible initial states (in fact, for an extension of the compatible family of dual-unitary circuits) and that the asymptotic entanglement slope is again independent on the Rényi index. Remarkably, however, we find that the entanglement velocity is generically smaller than one. We use these properties to find a closed-form expression for the entanglement membrane in these circuits. △ Less

Submitted 28 May, 2024; v1 submitted 5 December, 2023; originally announced December 2023.

Comments: 7 pages in the main text, 4 pages in the SM, 2 figures in the main tex

Journal ref: Phys. Rev. Lett. 132, 250402 (2024)

Hierarchical generalization of dual unitarity

Authors: Xie-Hang Yu, Zhiyuan Wang, Pavel Kos

Abstract: Quantum dynamics with local interactions in lattice models display rich physics, but is notoriously hard to study. Dual-unitary circuits allow for exact answers to interesting physical questions in clean or disordered one- and higher-dimensional quantum systems. However, this family of models shows some non-universal features, like vanishing correlations inside the light-cone and instantaneous the… ▽ More Quantum dynamics with local interactions in lattice models display rich physics, but is notoriously hard to study. Dual-unitary circuits allow for exact answers to interesting physical questions in clean or disordered one- and higher-dimensional quantum systems. However, this family of models shows some non-universal features, like vanishing correlations inside the light-cone and instantaneous thermalization of local observables. In this work we propose a generalization of dual-unitary circuits where the exactly calculable spatial-temporal correlation functions display richer behavior, and have non-trivial thermalization of local observables. This is achieved by generalizing the single-gate condition to a hierarchy of multi-gate conditions, where the first level recovers dual-unitary models, and the second level exhibits these new interesting features. We also extend the discussion and provide exact solutions to correlators with few-site observables and discuss higher-orders, including the ones after a quantum quench. In addition, we provide exhaustive parametrizations for qubit cases, and propose a new family of models for local dimensions larger than two, which also provides a new family of dual-unitary models. △ Less

Submitted 11 February, 2024; v1 submitted 6 July, 2023; originally announced July 2023.

Comments: 21 pages, 4 figures

Journal ref: Quantum 8, 1260 (2024)

Localised Dynamics in the Floquet Quantum East Model

Authors: Bruno Bertini, Pavel Kos, Tomaz Prosen

Abstract: We introduce and study the discrete-time version of the Quantum East model, an interacting quantum spin chain inspired by simple kinetically constrained models of classical glasses. Previous work has established that its continuous-time counterpart displays a disorder-free localisation transition signalled by the appearance of an exponentially large (in the volume) family of non-thermal, localised… ▽ More We introduce and study the discrete-time version of the Quantum East model, an interacting quantum spin chain inspired by simple kinetically constrained models of classical glasses. Previous work has established that its continuous-time counterpart displays a disorder-free localisation transition signalled by the appearance of an exponentially large (in the volume) family of non-thermal, localised eigenstates. Here we combine analytical and numerical approaches to show that: i) The transition persists for discrete times, in fact, it is present for any finite value of the time step apart from a zero measure set; ii) It is directly detected by following the non-equilibrium dynamics of the fully polarised state. Our findings imply that the transition is currently observable in state-of-the-art platforms for digital quantum simulation. △ Less

Submitted 24 February, 2024; v1 submitted 21 June, 2023; originally announced June 2023.

Comments: 6+7 pages; 4+6 figures; v2 as appears in PRL

Journal ref: Phys. Rev. Lett. 132, 080401 (2024)

Scrambling is Necessary but Not Sufficient for Chaos

Authors: Neil Dowling, Pavel Kos, Kavan Modi

Abstract: We show that out-of-time-order correlators (OTOCs) constitute a probe for Local-Operator Entanglement (LOE). There is strong evidence that a volumetric growth of LOE is a faithful dynamical indicator of quantum chaos, while OTOC decay corresponds to operator scrambling, often conflated with chaos. We show that rapid OTOC decay is a necessary but not sufficient condition for linear (chaotic) growth… ▽ More We show that out-of-time-order correlators (OTOCs) constitute a probe for Local-Operator Entanglement (LOE). There is strong evidence that a volumetric growth of LOE is a faithful dynamical indicator of quantum chaos, while OTOC decay corresponds to operator scrambling, often conflated with chaos. We show that rapid OTOC decay is a necessary but not sufficient condition for linear (chaotic) growth of the LOE entropy. We analytically support our results through wide classes of local-circuit models of many-body dynamics, including both integrable and non-integrable dual-unitary circuits. We show sufficient conditions under which local dynamics leads to an equivalence of scrambling and chaos. △ Less

Submitted 2 November, 2023; v1 submitted 14 April, 2023; originally announced April 2023.

Comments: 6+16 Pages. Comments welcome

Journal ref: Phys. Rev. Lett. 131, 180403 (2023)

Circuits of space and time quantum channels

Authors: Pavel Kos, Georgios Styliaris

Abstract: Exact solutions in interacting many-body systems are scarce but extremely valuable since they provide insights into the dynamics. Dual-unitary models are examples in one spatial dimension where this is possible. These brick-wall quantum circuits consist of local gates, which remain unitary not only in time, but also when interpreted as evolutions along the spatial directions. However, this setting… ▽ More Exact solutions in interacting many-body systems are scarce but extremely valuable since they provide insights into the dynamics. Dual-unitary models are examples in one spatial dimension where this is possible. These brick-wall quantum circuits consist of local gates, which remain unitary not only in time, but also when interpreted as evolutions along the spatial directions. However, this setting of unitary dynamics does not directly apply to real-world systems due to their imperfect isolation, and it is thus imperative to consider the impact of noise to dual-unitary dynamics and its exact solvability. In this work we generalise the ideas of dual-unitarity to obtain exact solutions in noisy quantum circuits, where each unitary gate is substituted by a local quantum channel. Exact solutions are obtained by demanding that the noisy gates yield a valid quantum channel not only in time, but also when interpreted as evolutions along one or both of the spatial directions and possibly backwards in time. This gives rise to new families of models that satisfy different combinations of unitality constraints along the space and time directions. We provide exact solutions for the spatio-temporal correlation functions, spatial correlations after a quantum quench, and the structure of steady states for these families of models. We show that noise unbiased around the dual-unitary family leads to exactly solvable models, even if dual-unitarity is strongly violated. We prove that any channel unital in both space and time directions can be written as an affine combination of a particular class of dual-unitary gates. Finally, we extend the definition of solvable initial states to matrix-product density operators. We completely classify them when their tensor admits a local purification. △ Less

Submitted 22 May, 2023; v1 submitted 24 June, 2022; originally announced June 2022.

Comments: v4: Accepted for publication in Quantum

Journal ref: Quantum 7, 1020 (2023)

Exact Spectral Statistics in Strongly Localised Circuits

Authors: Bruno Bertini, Pavel Kos, Tomaz Prosen

Abstract: Since the seminal work of Anderson, localisation has been recognised as a standard mechanism allowing quantum many-body systems to escape ergodicity. This idea acquired even more prominence in the last decade as it has been argued that localisation -- dubbed many-body localisation (MBL) in this context -- can sometimes survive local interactions in the presence of sufficiently strong disorder. A c… ▽ More Since the seminal work of Anderson, localisation has been recognised as a standard mechanism allowing quantum many-body systems to escape ergodicity. This idea acquired even more prominence in the last decade as it has been argued that localisation -- dubbed many-body localisation (MBL) in this context -- can sometimes survive local interactions in the presence of sufficiently strong disorder. A conventional signature of localisation is in the statistical properties of the spectrum -- spectral statistics -- which differ qualitatively from those in the ergodic phase. Although features of the spectral statistics are routinely used as numerical diagnostics for localisation, they have never been derived from first principles in the presence of non-trivial interactions. Here we fill this gap and provide the example of a simple class of quantum many-body systems -- which we dub strongly localised quantum circuits -- that are interacting, localised, and where the spectral statistics can be characterised exactly. Furthermore, we show that these systems exhibit a cascade of three different regimes for spectral correlations depending on the energy scale: at small, intermediate, and large scales they behave as disconnected patches of three decreasing sizes. We argue that these features appear in generic MBL systems, with the difference that only at the smallest scale they do become Poissonian. △ Less

Submitted 28 March, 2022; v1 submitted 29 October, 2021; originally announced October 2021.

Comments: 16 pages, 7 figures, 1 table; v2 added SM-I with explicit LIOMs; v3 as appears in PRB

Journal ref: Phys. Rev. B 105, 165142 (2022)

Thermalisation Dynamics and Spectral Statistics of Extended Systems with Thermalising Boundaries

Authors: Pavel Kos, Tomaz Prosen, Bruno Bertini

Abstract: We study thermalisation and spectral properties of extended systems connected, through their boundaries, to a thermalising Markovian bath. Specifically, we consider periodically driven systems modelled by brickwork quantum circuits where a finite section (block) of the circuit is constituted by arbitrary local unitary gates while its complement, which plays the role of the bath, is dual-unitary. W… ▽ More We study thermalisation and spectral properties of extended systems connected, through their boundaries, to a thermalising Markovian bath. Specifically, we consider periodically driven systems modelled by brickwork quantum circuits where a finite section (block) of the circuit is constituted by arbitrary local unitary gates while its complement, which plays the role of the bath, is dual-unitary. We show that the evolution of local observables and the spectral form factor are determined by the same quantum channel, which we use to characterise the system's dynamics and spectral properties. In particular, we identify a family of fine-tuned quantum circuits -- which we call strongly non-ergodic -- that fails to thermalise even in this controlled setting, and, accordingly, their spectral form factor does not follow the random matrix theory prediction. We provide a set of necessary conditions on the local quantum gates that lead to strong non-ergodicity, and in the case of qubits, we provide a complete classification of strongly non-ergodic circuits. We also study the opposite extreme case of circuits that are almost dual-unitary, i.e., where thermalisation occurs with the fastest possible rate. We show that, in these systems, local observables and spectral form factor approach respectively thermal values and random matrix theory prediction exponentially fast. We provide a perturbative characterisation of the dynamics and, in particular, of the time-scale for thermalisation. △ Less

Submitted 15 November, 2021; v1 submitted 17 August, 2021; originally announced August 2021.

Comments: 17 pages, 8 figures, 1 table; v2 18 pages, 8 figures, 1 table, improved exposition

Journal ref: Phys. Rev. B 104, 214303 (2021)

Random Matrix Spectral Form Factor of Dual-Unitary Quantum Circuits

Authors: Bruno Bertini, Pavel Kos, Tomaz Prosen

Abstract: We investigate a class of brickwork-like quantum circuits on chains of $d-$level systems (qudits) that share the so-called `dual unitarity' property. Namely, these systems generate unitary dynamics not only when propagating in the time direction, but also when propagating in the space direction. We consider space-time homogeneous (Floquet) circuits and perturb them with a quenched single-site diso… ▽ More We investigate a class of brickwork-like quantum circuits on chains of $d-$level systems (qudits) that share the so-called `dual unitarity' property. Namely, these systems generate unitary dynamics not only when propagating in the time direction, but also when propagating in the space direction. We consider space-time homogeneous (Floquet) circuits and perturb them with a quenched single-site disorder, i.e. by applying independent single site random unitaries drawn from arbitrary non-singular distribution over ${\rm SU}(d)$, e.g. one concentrated around the identity, after each layer of the circuit. We identify the spectral form factor at time $t$ in the limit of long chains as the dimension of the commutant of a finite set of operators on a qudit ring of $t$ sites. For general dual unitary circuits of qubits $(d=2)$ and a family of their extensions to higher $d>2$, we provide explicit construction of the commutant and prove that spectral form factor exactly matches the prediction of circular unitary ensemble for all $t$, if only the local 2-qubit gates are different from a SWAP (non-interacting gate). We discuss and partly prove possible extensions of our results to a weaker (more singular) forms of disorder averaging, as well as to quantum circuits with time-reversal symmetry, and to computing higher moments of the spectral form factor. △ Less

Submitted 16 July, 2021; v1 submitted 22 December, 2020; originally announced December 2020.

Comments: 30 pages; v2 rigorous results for spatially inhomogeneous interactions added; v3 extended version, it contains some unproven conjectures not published in CMP

Journal ref: Commun. Math. Phys. 387, 597-620 (2021)

Chaos and Ergodicity in Extended Quantum Systems with Noisy Driving

Authors: Pavel Kos, Bruno Bertini, Tomaž Prosen

Abstract: We study the time evolution operator in a family of local quantum circuits with random fields in a fixed direction. We argue that the presence of quantum chaos implies that at large times the time evolution operator becomes effectively a random matrix in the many-body Hilbert space. To quantify this phenomenon we compute analytically the squared magnitude of the trace of the evolution operator --… ▽ More We study the time evolution operator in a family of local quantum circuits with random fields in a fixed direction. We argue that the presence of quantum chaos implies that at large times the time evolution operator becomes effectively a random matrix in the many-body Hilbert space. To quantify this phenomenon we compute analytically the squared magnitude of the trace of the evolution operator -- the generalised spectral form factor -- and compare it with the prediction of Random Matrix Theory (RMT). We show that for the systems under consideration the generalised spectral form factor can be expressed in terms of dynamical correlation functions of local observables in the infinite temperature state, linking chaotic and ergodic properties of the systems. This also provides a connection between the many-body Thouless time $τ_{\rm th}$ -- the time at which the generalised spectral form factor starts following the random matrix theory prediction -- and the conservation laws of the system. Moreover, we explain different scalings of $τ_{\rm th}$ with the system size, observed for systems with and without the conservation laws. △ Less

Submitted 18 March, 2021; v1 submitted 23 October, 2020; originally announced October 2020.

Comments: 7+10 pages, 5 figures, 1 table; v2 improved exposition with the additional section I in SM

Journal ref: Phys. Rev. Lett. 126, 190601 (2021)

Correlations in Perturbed Dual-Unitary Circuits: Efficient Path-Integral Formula

Authors: Pavel Kos, Bruno Bertini, Tomaž Prosen

Abstract: Interacting many-body systems with explicitly accessible spatio-temporal correlation functions are extremely rare, especially in the absence of integrability. Recently, we identified a remarkable class of such systems and termed them dual-unitary quantum circuits. These are brick-wall type local quantum circuits whose dynamics are unitary in both time and space. For these systems the spatio-tempor… ▽ More Interacting many-body systems with explicitly accessible spatio-temporal correlation functions are extremely rare, especially in the absence of integrability. Recently, we identified a remarkable class of such systems and termed them dual-unitary quantum circuits. These are brick-wall type local quantum circuits whose dynamics are unitary in both time and space. For these systems the spatio-temporal correlation functions are non-trivial only at the edge of the causal light cone and can be computed in terms of one-dimensional transfer matrices. Dual-unitarity, however, requires fine-tuning and the degree of generality of the observed dynamical features remained unclear. Here we address this question by introducing arbitrary perturbations of the local gates. Considering fixed perturbations, we prove that for a particular class of unperturbed elementary dual-unitary gates the correlation functions are still expressed in terms of one-dimensional transfer matrices. These matrices, however, are now contracted over generic paths connecting the origin to a fixed endpoint inside the causal light cone. The correlation function is given as a sum over all such paths. Our statement is rigorous in the "dilute limit", where only a small fraction of the gates is perturbed, and in the presence of random longitudinal fields, but we provide theoretical arguments and stringent numerical checks supporting its validity even in the clean case and when all gates are perturbed. As a byproduct, in the case of random longitudinal fields -- which turns out to be equivalent to certain classical Markov chains -- we find four types of non-dual-unitary(and non-integrable) interacting many-body systems where the correlation functions are exactly given by the path-sum formula. △ Less

Submitted 26 October, 2020; v1 submitted 12 June, 2020; originally announced June 2020.

Comments: 28 pages, 9 figures, 2 tables; v3: 32 pages, 10 figures, 2 tables; presentation improved, Section 3 rewritten

Journal ref: Phys. Rev. X 11, 011022 (2021)

Low-entangled UHMWPE melt: analytical model and computer simulations

Authors: Artem Petrov, Vladimir Rudyak, Pavel Kos, Alexander Chertovich

Abstract: In this work we developed a theoretical model to describe the polymerization of very long macromolecules with simultaneous precipitation. As a reference system for our model we consider UHMWPE polymerization process, via homogeneous catalysis and with molecular weight above $10^6$ monomers. We derived time dependency of the entanglement length in the system, and assessed how polymerization rate an… ▽ More In this work we developed a theoretical model to describe the polymerization of very long macromolecules with simultaneous precipitation. As a reference system for our model we consider UHMWPE polymerization process, via homogeneous catalysis and with molecular weight above $10^6$ monomers. We derived time dependency of the entanglement length in the system, and assessed how polymerization rate and fraction of initiators affects this dependency. We show that decrease of the fraction of initiators leads to increase of the average entanglement length in the melt after complete polymerization. The results are supported by computer simulations and in principle could be applied to the wide set of polymerized materials, both crystallized and amorphous. △ Less

Submitted 15 January, 2020; v1 submitted 14 January, 2020; originally announced January 2020.

Operator Entanglement in Local Quantum Circuits II: Solitons in Chains of Qubits

Authors: Bruno Bertini, Pavel Kos, Tomaz Prosen

Abstract: We provide exact results for the dynamics of local-operator entanglement in quantum circuits with two-dimensional wires featuring ultralocal solitons, i.e. single-site operators which, up to a phase, are simply shifted by the time evolution. We classify all circuits allowing for ultralocal solitons and show that only dual-unitary circuits can feature moving ultralocal solitons. Then, we rigorously… ▽ More We provide exact results for the dynamics of local-operator entanglement in quantum circuits with two-dimensional wires featuring ultralocal solitons, i.e. single-site operators which, up to a phase, are simply shifted by the time evolution. We classify all circuits allowing for ultralocal solitons and show that only dual-unitary circuits can feature moving ultralocal solitons. Then, we rigorously prove that if a circuit has an ultralocal soliton moving to the left (right), the entanglement of local operators initially supported on even (odd) sites saturates to a constant value and its dynamics can be computed exactly. Importantly, this does not bound the growth of complexity in chiral circuits, where solitons move only in one direction, say to the left. Indeed, in this case we observe numerically that operators on the odd sublattice have unbounded entanglement. Finally, we present a closed-form expression for the local-operator entanglement entropies in circuits with ultralocal solitons moving in both directions. Our results hold irrespectively of integrability. △ Less

Submitted 11 March, 2020; v1 submitted 16 September, 2019; originally announced September 2019.

Comments: 22 pages, 1 figure; v2 24 pages, 2 figures, 1 table; v3 24 pages, 2 figures

Journal ref: SciPost Phys. 8, 068 (2020)

Operator Entanglement in Local Quantum Circuits I: Chaotic Dual-Unitary Circuits

Authors: Bruno Bertini, Pavel Kos, Tomaz Prosen

Abstract: The entanglement in operator space is a well established measure for the complexity of the quantum many-body dynamics. In particular, that of local operators has recently been proposed as dynamical chaos indicator, i.e. as a quantity able to discriminate between quantum systems with integrable and chaotic dynamics. For chaotic systems the local-operator entanglement is expected to grow linearly in… ▽ More The entanglement in operator space is a well established measure for the complexity of the quantum many-body dynamics. In particular, that of local operators has recently been proposed as dynamical chaos indicator, i.e. as a quantity able to discriminate between quantum systems with integrable and chaotic dynamics. For chaotic systems the local-operator entanglement is expected to grow linearly in time, while it is expected to grow at most logarithmically in the integrable case. Here we study local-operator entanglement in dual-unitary quantum circuits, a class of "statistically solvable" quantum circuits that we recently introduced. We identify a class of "completely chaotic" dual-unitary circuits where the local-operator entanglement grows linearly and we provide a conjecture for its asymptotic behaviour which is in excellent agreement with the numerical results. Interestingly, our conjecture also predicts a "phase transition" in the slope of the local-operator entanglement when varying the parameters of the circuits. △ Less

Submitted 11 March, 2020; v1 submitted 16 September, 2019; originally announced September 2019.

Comments: 23 pages, 6 figures; v2 29 pages, 6 figures, improved exposition

Journal ref: SciPost Phys. 8, 067 (2020)

Exact Correlation Functions for Dual-Unitary Lattice Models in 1+1 Dimensions

Authors: Bruno Bertini, Pavel Kos, Tomaz Prosen

Abstract: We consider a class of quantum lattice models in $1+1$ dimensions represented as local quantum circuits that enjoy a particular "dual-unitarity" property. In essence, this property ensures that both the evolution "in time" and that "in space" are given in terms of unitary transfer matrices. We show that for this class of circuits, generically non-integrable, one can compute explicitly all dynamica… ▽ More We consider a class of quantum lattice models in $1+1$ dimensions represented as local quantum circuits that enjoy a particular "dual-unitarity" property. In essence, this property ensures that both the evolution "in time" and that "in space" are given in terms of unitary transfer matrices. We show that for this class of circuits, generically non-integrable, one can compute explicitly all dynamical correlations of local observables. Our result is exact, non-pertubative, and holds for any dimension $d$ of the local Hilbert space. In the minimal case of qubits ($d = 2$) we also present a classification of all dual-unitary circuits which allows us to single out a number of distinct classes for the behaviour of the dynamical correlations. We find "non-interacting" classes, where all correlations are preserved, the ergodic and mixing one, where all correlations decay, and, interestingly, also classes that are are both interacting and non-ergodic. △ Less

Submitted 4 October, 2019; v1 submitted 3 April, 2019; originally announced April 2019.

Comments: 6+5 pages, no figures; v2 minor changes; v3 as appears in Phys. Rev. Lett

Journal ref: Phys. Rev. Lett. 123, 210601 (2019)

Modelling Polymerization-Induced Self Assembly (PISA)

Authors: Ruslan Shupanov, Pavel Kos, Alexei Gavrilov, Alexander Chertovich

Abstract: In this work we studied polymerization-induced self-assembly by means of computer simulations. Using this model, phase diagrams of the micelle states were constructed depending on the polymer concentration and the asymmetry of the composition for various reaction conditions. We found that if the reaction is ideal controlled radical polymerization (the initiation speed is much larger than the propa… ▽ More In this work we studied polymerization-induced self-assembly by means of computer simulations. Using this model, phase diagrams of the micelle states were constructed depending on the polymer concentration and the asymmetry of the composition for various reaction conditions. We found that if the reaction is ideal controlled radical polymerization (the initiation speed is much larger than the propagation speed and there are no side reactions such as termination or chain transfer), the phase diagram is no different from that obtained for pre-synthesized monodisperse diblock-copolymers with one insoluble block. Next, we studied two cases of slow initiation. We found that the phase diagram change dramatically: upon decreasing the initiation speed, the regions of spherical and cylindrical micelles shrink, while the region of vesicles/lamellae expands. This happens because at small initiation speed there is a significant amount of chains with a very short non-soluble block (and even without such block altogether), which do not participate in the formation of micelles. Therefore, decreasing the initiation speed essentially remaps the phase diagram coordinates by making the effective concentration lower (by decreasing the number of active chains) and the effective block length ratio higher (again, because the number of active chains gets higher, while the number of monomers remains the same). △ Less

Submitted 27 January, 2019; originally announced January 2019.

Comments: 12 pages, 10 figures

Journal ref: Polymers 2020, 12, 2599

Entanglement spreading in a minimal model of maximal many-body quantum chaos

Authors: Bruno Bertini, Pavel Kos, Tomaz Prosen

Abstract: The spreading of entanglement in out-of-equilibrium quantum systems is currently at the centre of intense interdisciplinary research efforts involving communities with interests ranging from holography to quantum information. Here we provide a constructive and mathematically rigorous method to compute the entanglement dynamics in a class of "maximally chaotic", periodically driven, quantum spin ch… ▽ More The spreading of entanglement in out-of-equilibrium quantum systems is currently at the centre of intense interdisciplinary research efforts involving communities with interests ranging from holography to quantum information. Here we provide a constructive and mathematically rigorous method to compute the entanglement dynamics in a class of "maximally chaotic", periodically driven, quantum spin chains. Specifically, we consider the so called "self-dual" kicked Ising chains initialised in a class of separable states and devise a method to compute exactly the time evolution of the entanglement entropies of finite blocks of spins in the thermodynamic limit. Remarkably, these exact results are obtained despite the models considered are maximally chaotic: their spectral correlations are described by the circular orthogonal ensemble of random matrices on all scales. Our results saturate the so called "minimal cut" bound and are in agreement with those found in the contexts of random unitary circuits with infinite-dimensional local Hilbert space and conformal field theory. In particular, they agree with the expectations from both the quasiparticle picture, which accounts for the entanglement spreading in integrable models, and the minimal membrane picture, recently proposed to describe the entanglement growth in generic systems. Based on a novel "duality-based" numerical method, we argue that our results describe the entanglement spreading from any product state at the leading order in time when the model is non-integrable. △ Less

Submitted 7 March, 2019; v1 submitted 12 December, 2018; originally announced December 2018.

Comments: 25 pages, 10 figures; v2 improved presentation; v3: 28 pages 11 figures, presentation improved, Section 7 rewritten

Journal ref: Phys. Rev. X 9, 021033 (2019)

Kinetic mechanisms of crumpled globule formation

Authors: Artem Petrov, Pavel Kos, Alexander Chertovich

Abstract: Homopolymer chain with beads forming pairwise reversible bonds is a well-known model in polymer physics. We studied kinetics of homopolymer chain collapse, which was induced by pairwise reversible bonds formation. We compared kinetic mechanism of this coil-globule transition with the mechanism of collapse in a poor solvent. We discovered, that coil-globule transition occurs sufficiently more homog… ▽ More Homopolymer chain with beads forming pairwise reversible bonds is a well-known model in polymer physics. We studied kinetics of homopolymer chain collapse, which was induced by pairwise reversible bonds formation. We compared kinetic mechanism of this coil-globule transition with the mechanism of collapse in a poor solvent. We discovered, that coil-globule transition occurs sufficiently more homogeneously on different scales, if collapse is induced by pairwise reversible bonds formation. This effect leads to formation of transient structures, which are not similar to the classical pearl-necklace conformations formed during collapse in a poor solvent. However, both types of collapse lead to formation of a metastable state of crumpled globule, which is one of the well-known models of interphase chromatin structure in different organisms. Moreover, we found out that stability and dynamics of this state can be controlled by fraction of reversible bonds and bond lifetime. △ Less

Submitted 1 October, 2019; v1 submitted 12 November, 2018; originally announced November 2018.

Exact Spectral Form Factor in a Minimal Model of Many-Body Quantum Chaos

Authors: Bruno Bertini, Pavel Kos, Tomaz Prosen

Abstract: The most general and versatile defining feature of quantum chaotic systems is that they possess an energy spectrum with correlations universally described by random matrix theory (RMT). This feature can be exhibited by systems with a well defined classical limit as well as by systems with no classical correspondence, such as locally interacting spins or fermions. Despite great phenomenological suc… ▽ More The most general and versatile defining feature of quantum chaotic systems is that they possess an energy spectrum with correlations universally described by random matrix theory (RMT). This feature can be exhibited by systems with a well defined classical limit as well as by systems with no classical correspondence, such as locally interacting spins or fermions. Despite great phenomenological success, a general mechanism explaining the emergence of RMT without reference to semiclassical concepts is still missing. Here we provide the example of a quantum many-body system with no semiclassical limit (no large parameter) where the emergence of RMT spectral correlations is proven exactly. Specifically, we consider a periodically driven Ising model and write the Fourier transform of spectral density's two-point function, the spectral form factor, in terms of a partition function of a two-dimensional classical Ising model featuring a space-time duality. We show that the self-dual cases provide a minimal model of many-body quantum chaos, where the spectral form factor is demonstrated to match RMT for all values of the integer time variable $t$ in the thermodynamic limit. In particular, we rigorously prove RMT form factor for odd $t$, while we formulate a precise conjecture for even $t$. The results imply ergodicity for any finite amount of disorder in the longitudinal field, rigorously excluding the possibility of many-body localization. Our method provides a novel route for obtaining exact nonperturbative results in non-integrable systems. △ Less

Submitted 9 January, 2019; v1 submitted 2 May, 2018; originally announced May 2018.

Comments: 6 + 22 pages, 3 figures; v2: improved presentation of the proofs in the appendices; v3: as appears in Physical Review Letters

Journal ref: Phys. Rev. Lett. 121, 264101 (2018)

Many-body quantum chaos: Analytic connection to random matrix theory

Authors: Pavel Kos, Marko Ljubotina, Tomaz Prosen

Abstract: A key goal of quantum chaos is to establish a relationship between widely observed universal spectral fluctuations of clean quantum systems and random matrix theory (RMT). For single particle systems with fully chaotic classical counterparts, the problem has been partly solved by Berry (1985) within the so-called diagonal approximation of semiclassical periodic-orbit sums. Derivation of the full R… ▽ More A key goal of quantum chaos is to establish a relationship between widely observed universal spectral fluctuations of clean quantum systems and random matrix theory (RMT). For single particle systems with fully chaotic classical counterparts, the problem has been partly solved by Berry (1985) within the so-called diagonal approximation of semiclassical periodic-orbit sums. Derivation of the full RMT spectral form factor $K(t)$ from semiclassics has been completed only much later in a tour de force by Mueller et al (2004). In recent years, the questions of long-time dynamics at high energies, for which the full many-body energy spectrum becomes relevant, are coming at the forefront even for simple many-body quantum systems, such as locally interacting spin chains. Such systems display two universal types of behaviour which are termed as `many-body localized phase' and `ergodic phase'. In the ergodic phase, the spectral fluctuations are excellently described by RMT, even for very simple interactions and in the absence of any external source of disorder. Here we provide the first theoretical explanation for these observations. We compute $K(t)$ explicitly in the leading two orders in $t$ and show its agreement with RMT for non-integrable, time-reversal invariant many-body systems without classical counterparts, a generic example of which are Ising spin 1/2 models in a periodically kicking transverse field. △ Less

Submitted 16 May, 2018; v1 submitted 7 December, 2017; originally announced December 2017.

Comments: 10 pages in RevTex with 4 figures and a few diagrams; v3: version accepted by PRX

Journal ref: Phys. Rev. X 8, 021062 (2018)

Crystallization in melts and poor-solvent solutions of semiflexible polymers: extensive DPD study

Authors: P. I. Kos, V. A. Ivanov, A. V. Chertovich

Abstract: In the present work, crystallization in melts and poor-solvent solutions of semiflexible polymers with different concentration was studied by means of dissipative particle dynamics simulation technique. We use a coarse-grained polymer model trying to catch general principles of crystallization in such systems on large time and length scales. We observe the crystallization process starting from an… ▽ More In the present work, crystallization in melts and poor-solvent solutions of semiflexible polymers with different concentration was studied by means of dissipative particle dynamics simulation technique. We use a coarse-grained polymer model trying to catch general principles of crystallization in such systems on large time and length scales. We observe the crystallization process starting from an initial randomly prepared system with different polymer volume fractions in a poor solvent. Because the solvent is very poor, the macrophase polymer-solvent separation takes place very fast and is accompanied by partial polymer crystallization. We have found that the overall crystalline fraction at the end of crystallization process decreases upon increasing the polymer volume fraction in the initial randomly prepared system, while the steady-state crystallization speed is almost the same at polymer volume fractions larger than 50\%. At the same time, the average crystallite size differs considerably and has a maximum value in the systems with 90\% polymer volume fraction. We assume that this polymer concentration is an optimal value in a sense of a balance between the amount of polymer material available for increasing crystallite size and chain entanglements preventing crystallites growth and merging. △ Less

Submitted 28 January, 2020; v1 submitted 30 August, 2017; originally announced August 2017.

Comments: 44 pages, 9 figures

Time-dependent Correlation Functions in Open Quadratic Fermionic Systems

Authors: Pavel Kos, Tomaz Prosen

Abstract: We formulate and discuss explicit computation of dynamic correlation functions in open quadradic fermionic systems which are driven and dissipated by the Lindblad jump processes that are linear in canonical fermionic operators. Dynamic correlators are interpreted in terms of local quantum quench where the pre-quench state is the non-equilibrium steady state, i.e. a fixed point of the Liouvillian.… ▽ More We formulate and discuss explicit computation of dynamic correlation functions in open quadradic fermionic systems which are driven and dissipated by the Lindblad jump processes that are linear in canonical fermionic operators. Dynamic correlators are interpreted in terms of local quantum quench where the pre-quench state is the non-equilibrium steady state, i.e. a fixed point of the Liouvillian. As an example we study the XY spin 1/2 chain and the Kitaev Majorana chains with boundary Lindblad driving, whose dynamics exhibits asymmetric (skewed) light cone behaviour. We also numerically treat the two dimensional XY model and the XY spin chain with additional Dzyaloshinskii-Moriya interactions. The latter exhibits a new non-equilibrium phase transition which can be understood in terms of bifurcations of the quasi-particle dispersion relation. Finally, considering in some detail the periodic Kitaev chain (fermionic ring) with dissipation at a single (arbitrary) site, we present analytical expressions for the first order corrections (in the strength of dissipation) to the spectrum and the non-equilibrium steady state (NESS) correlation functions. △ Less

Submitted 23 August, 2017; originally announced August 2017.

Comments: 25 pages, 10 figures

Quantum spin liquid ground states of the Heisenberg-Kitaev model on the triangular lattice

Authors: Pavel Kos, Matthias Punk

Abstract: We study quantum disordered ground states of the two dimensional Heisenberg-Kitaev model on the triangular lattice using a Schwinger boson approach. Our aim is to identify and characterize potential gapped quantum spin liquid phases that are stabilized by anisotropic Kitaev interactions. For antiferromagnetic Heisenberg- and Kitaev couplings and sufficiently small spin $S$ we find three different… ▽ More We study quantum disordered ground states of the two dimensional Heisenberg-Kitaev model on the triangular lattice using a Schwinger boson approach. Our aim is to identify and characterize potential gapped quantum spin liquid phases that are stabilized by anisotropic Kitaev interactions. For antiferromagnetic Heisenberg- and Kitaev couplings and sufficiently small spin $S$ we find three different symmetric $Z_2$ spin liquid phases, separated by two continuous quantum phase transitions. Interestingly, the gap of elementary excitations remains finite throughout the transitions. The first spin liquid phase corresponds to the well known zero-flux state in the Heisenberg limit, which is stable with respect to small Kitaev couplings and develops $120^\circ$ order in the semi-classical limit at large $S$. In the opposite Kitaev limit we find a different spin liquid ground-state, which is a quantum disordered version of a magnetically ordered state with antiferromagnetic chains, in accordance with results in the classical limit. Finally, at intermediate couplings we find a spin liquid state with unconventional spin correlations. Upon spinon condensation this state develops Bragg peaks at incommensurate momenta in close analogy to the magnetically ordered Z2 vortex crystal phase, which has been analyzed in recent theoretical works. △ Less

Submitted 16 November, 2016; originally announced November 2016.

Comments: 14 pages, 13 figures

Journal ref: Phys. Rev. B 95, 024421 (2017)