Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2411.02546 · Journal of High Energy Physics

On complexity and duality

Jeff Murugan, Zayd Pandit, Hendrik J. R. Van Zyl

Published Mar 11, 2025• 0 citations
arXivHTMLPDF

Abstract

A bstract We explore the relationship between complexity and duality in quantum systems, focusing on how local and non-local operators evolve under time evolution. We find that non-local operators, which are dual to local operators under specific mappings, exhibit behaviour that mimics the growth of their local counterparts, particularly when considering state complexity. For the open transverse Ising model this leads to a neat organisation of the operator dynamics on either side of the duality, both consistent with growth expected in a quadratic fermion model like the Kitaev chain. When examining periodic chains, however, the mapping of boundary terms provides access to multiple branches of highly complex operators. These give rise to much larger saturation values of complexity for parity-mixing operators and are in contrast to what one would expect for a quadratic Hamiltonian. Our results shed light on the intricate relationship between non-locality, complexity growth, and duality in quantum systems.

DOI
10.1007/jhep03(2025)062

Related papers

K-complexity and the Jordan-Wigner transformation

Zayd Pandit

Krylov complexity is a measure of operator growth that demonstrates universal properties and bounds a large class of complexities. One such measure from this bounded class is operator size. The relationship between operator size and operator growth has been conjectured to be non-trivial due to the existence of duality transformations such as the Jordan-Wigner (JW) transformation which map small operators to large, non-local operators. We investigate this claim directly in the case of the JW transformation which maps the XY Heisenberg chain to the Kitaev chain. We numerically calculate the complexity of dual operators, and analyse the early and late time behaviour and symmetries. We find that for Open Boundary Conditions (OBC) the early time behaviour of the K-Complexity correlates with operator size, but that large operators can have very low K-Complexity if dual to a small operator. We find that for Periodic Boundary Conditions (PBC) larger operators produce larger early time growth, but do not correlate to larger late-time complexity regardless of the size of the dual operator. The difference between the OBC and PBC results arise from an often overlooked break in translational symmetry across the PBC Jordan-Wigner transformation. We also find that state complexity is not sensitive to the break in translational symmetry.

•Jan 1, 2025•0 citations

Investigation of quantum chaos in local and non-local Ising models

Reza Pirmoradian, Elham Sadoogh, Maryam Teymouri +3 more

We investigate signatures of quantum chaos within Ising spin chains subjected to transverse and longitudinal fields, incorporating both local (nearest-neighbor) and non-local (long-range) couplings. While local Ising models may exhibit integrable or chaotic dynamics contingent on interaction strengths and field parameters, systems with non-local interactions generally display a stronger propensity toward chaos, even when the non-local couplings are weak. By examining the distribution of energy level spacings through the level spacing ratio, we delineate the transition from integrable to chaotic regimes and characterize the emergence of quantum chaos in these systems. Our analysis demonstrates that non-local couplings facilitate faster operator spreading and more intricate dynamical behavior, enabling these systems to approach maximal chaos more readily than their local counterparts. Additionally, we analyze Krylov complexity as a dynamical probe of chaos, observing a characteristic peak followed by a plateau at late times in chaotic regimes. This behavior provides a quantitative means to distinguish between integrable and chaotic phases, with the growth rate and saturation level of the complexity serving as effective indicators. Our findings underscore the role of non-local interactions in accelerating the onset of chaos and modifying dynamical complexity in quantum spin chains.

quant-pharXiv:2512.21713•Dec 26, 2025•0 citations

Investigation of Quantum Chaos in Local and Non-local Ising Models

Reza Pirmoradian, Maryam Teymouri, Elham Sadoogh +3 more

Time New Roman 10 We investigate signatures of quantum chaos within Ising spin chains subjected to transverse and longitudinal fields, incorporating both local (nearest-neighbor) and non-local (long-range) couplings. While local Ising models may exhibit integrable or chaotic dynamics contingent on interaction strengths and field parameters, systems with non-local interactions generally display a stronger propensity toward chaos, even when the non-local couplings are weak. By examining the distribution of energy level spacings through the level spacing ratio, we delineate the transition from integrable to chaotic regimes and characterize the emergence of quantum chaos in these systems. Our analysis demonstrates that non-local couplings facilitate faster operator spreading and more intricate dynamical behavior, enabling these systems to approach maximal chaos more readily than their local counterparts. Additionally, we analyze Krylov complexity as a dynamical probe of chaos, observing a characteristic peak followed by a plateau at late times in chaotic regimes. This behavior provides a quantitative means to distinguish between integrable and chaotic phases, with the growth rate and saturation level of the complexity serving as effective indicators. Our findings underscore the role of non-local interactions in accelerating the onset of chaos and modifying dynamical complexity in quantum spin chains.

•Jul 24, 2025•0 citations

Isospectrality and Operator Complexity

Pradip Kattel, Yicheng Tang, Natan Andrei

We study a pair of exactly solvable, isospectral fermion chains, one strongly interacting and one quadratic, that nevertheless display remarkably different phase structures and operator dynamics. A nonlocal nonlinear unitary transformation maps one onto the other while preserving the entire many-body spectrum and converting local fermion operators into extended many-body strings. Thus, operators that evolve within a closed linear subspace in the quadratic model become interacting operators that generate increasingly higher-body terms and exhibit asymptotic Lanczos growth $b_n\propto\sqrt n$. Despite their identical spectra, the two models realize distinct phases and sharply different notions of operator complexity. Our results demonstrate that free many-body spectra and interacting operator dynamics are fundamentally compatible.

quant-pharXiv:2606.05294•Jun 5, 2026•0 citations

Complexity growth of operators in the SYK model and in JT gravity

Shao-Kai Jian, Brian G. Swingle, Zhuo-Yu Xian

Abstract The concepts of operator size and computational complexity play important roles in the study of quantum chaos and holographic duality because they help characterize the structure of time-evolving Heisenberg operators. It is particularly important to understand how these microscopically defined measures of complexity are related to notions of complexity defined in terms of a dual holographic geometry, such as complexity-volume (CV) duality. Here we study partially entangled thermal states in the Sachdev-Ye-Kitaev (SYK) model and their dual description in terms of operators inserted in the interior of a black hole in Jackiw-Teitelboim (JT) gravity. We compare a microscopic definition of complexity in the SYK model known as K-complexity to calculations using CV duality in JT gravity and find that both quantities show an exponential-to-linear growth behavior. We also calculate the growth of operator size under time evolution and find connections between size and complexity. While the notion of operator size saturates at the scrambling time, our study suggests that complexity, which is well defined in both quantum systems and gravity theories, can serve as a useful measure of operator evolution at both early and late times.

arXiv:2008.12274•Journal of High Energy Physics•Mar 1, 2021•116 citations

Krylov complexity in open quantum systems

Chang Liu, Tang Haifeng, Hui Zhai

Krylov complexity is a measure of operator complexity that exhibits universal behavior and bounds a large class of other measures. In this paper, we generalize Krylov complexity from a closed system to an open system coupled to a Markovian bath, where Lindbladian evolution replaces Hamiltonian evolution. We show that Krylov complexity in open systems can be mapped to a non-Hermitian tight-binding model in a half-infinite chain. We discuss the properties of the non-Hermitian terms and show that the strengths of the non-Hermitian terms increase linearly with the increase of the Krylov basis index $n$. Such a non-Hermitian tight-binding model can exhibit localized edge modes that determine the long-time behavior of Krylov complexity. Hence, the growth of Krylov complexity is suppressed by dissipation, and at long times, Krylov complexity saturates at a finite value much smaller than that of a closed system with the same Hamiltonian. Our conclusions are supported by numerical results on several models, such as the Sachdev-Ye-Kitaev model and the interacting fermion model. Our work provides insights for discussing complexity, chaos, and holography for open quantum systems.

arXiv:2207.13603•Physical Review Research•Aug 7, 2023•84 citations
A bstract We explore the relationship between complexity and duality in quantum systems, focusing on how local and non-local operators evolve under time evolution. We find that non-local operators, which are dual to local operators under specific mappings, exhibit behaviour that mimics the growth of their local counterparts, particularly when considering state complexity. For the open transverse Ising model this leads to a neat organisation of the operator dynamics on either side of the duality, both consistent with growth expected in a quadratic fermion model like the Kitaev chain. When examining periodic chains, however, the mapping of boundary terms provides access to multiple branches of highly complex operators. These give rise to much larger saturation values of complexity for parity-mixing operators and are in contrast to what one would expect for a quadratic Hamiltonian. Our results shed light on the intricate relationship between non-locality, complexity growth, and duality in quantum systems.
Krylov complexity is a measure of operator growth that demonstrates universal properties and bounds a large class of complexities. One such measure from this bounded class is operator size. The relationship between operator size and operator growth has been conjectured to be non-trivial due to the existence of duality transformations such as the Jordan-Wigner (JW) transformation which map small operators to large, non-local operators. We investigate this claim directly in the case of the JW transformation which maps the XY Heisenberg chain to the Kitaev chain. We numerically calculate the complexity of dual operators, and analyse the early and late time behaviour and symmetries. We find that for Open Boundary Conditions (OBC) the early time behaviour of the K-Complexity correlates with operator size, but that large operators can have very low K-Complexity if dual to a small operator. We find that for Periodic Boundary Conditions (PBC) larger operators produce larger early time growth, but do not correlate to larger late-time complexity regardless of the size of the dual operator. The difference between the OBC and PBC results arise from an often overlooked break in translational symmetry across the PBC Jordan-Wigner transformation. We also find that state complexity is not sensitive to the break in translational symmetry.
We investigate signatures of quantum chaos within Ising spin chains subjected to transverse and longitudinal fields, incorporating both local (nearest-neighbor) and non-local (long-range) couplings. While local Ising models may exhibit integrable or chaotic dynamics contingent on interaction strengths and field parameters, systems with non-local interactions generally display a stronger propensity toward chaos, even when the non-local couplings are weak. By examining the distribution of energy level spacings through the level spacing ratio, we delineate the transition from integrable to chaotic regimes and characterize the emergence of quantum chaos in these systems. Our analysis demonstrates that non-local couplings facilitate faster operator spreading and more intricate dynamical behavior, enabling these systems to approach maximal chaos more readily than their local counterparts. Additionally, we analyze Krylov complexity as a dynamical probe of chaos, observing a characteristic peak followed by a plateau at late times in chaotic regimes. This behavior provides a quantitative means to distinguish between integrable and chaotic phases, with the growth rate and saturation level of the complexity serving as effective indicators. Our findings underscore the role of non-local interactions in accelerating the onset of chaos and modifying dynamical complexity in quantum spin chains.
Time New Roman 10 We investigate signatures of quantum chaos within Ising spin chains subjected to transverse and longitudinal fields, incorporating both local (nearest-neighbor) and non-local (long-range) couplings. While local Ising models may exhibit integrable or chaotic dynamics contingent on interaction strengths and field parameters, systems with non-local interactions generally display a stronger propensity toward chaos, even when the non-local couplings are weak. By examining the distribution of energy level spacings through the level spacing ratio, we delineate the transition from integrable to chaotic regimes and characterize the emergence of quantum chaos in these systems. Our analysis demonstrates that non-local couplings facilitate faster operator spreading and more intricate dynamical behavior, enabling these systems to approach maximal chaos more readily than their local counterparts. Additionally, we analyze Krylov complexity as a dynamical probe of chaos, observing a characteristic peak followed by a plateau at late times in chaotic regimes. This behavior provides a quantitative means to distinguish between integrable and chaotic phases, with the growth rate and saturation level of the complexity serving as effective indicators. Our findings underscore the role of non-local interactions in accelerating the onset of chaos and modifying dynamical complexity in quantum spin chains.
We study a pair of exactly solvable, isospectral fermion chains, one strongly interacting and one quadratic, that nevertheless display remarkably different phase structures and operator dynamics. A nonlocal nonlinear unitary transformation maps one onto the other while preserving the entire many-body spectrum and converting local fermion operators into extended many-body strings. Thus, operators that evolve within a closed linear subspace in the quadratic model become interacting operators that generate increasingly higher-body terms and exhibit asymptotic Lanczos growth bn∝nb_n\propto\sqrt nbn​∝n​. Despite their identical spectra, the two models realize distinct phases and sharply different notions of operator complexity. Our results demonstrate that free many-body spectra and interacting operator dynamics are fundamentally compatible.
Abstract The concepts of operator size and computational complexity play important roles in the study of quantum chaos and holographic duality because they help characterize the structure of time-evolving Heisenberg operators. It is particularly important to understand how these microscopically defined measures of complexity are related to notions of complexity defined in terms of a dual holographic geometry, such as complexity-volume (CV) duality. Here we study partially entangled thermal states in the Sachdev-Ye-Kitaev (SYK) model and their dual description in terms of operators inserted in the interior of a black hole in Jackiw-Teitelboim (JT) gravity. We compare a microscopic definition of complexity in the SYK model known as K-complexity to calculations using CV duality in JT gravity and find that both quantities show an exponential-to-linear growth behavior. We also calculate the growth of operator size under time evolution and find connections between size and complexity. While the notion of operator size saturates at the scrambling time, our study suggests that complexity, which is well defined in both quantum systems and gravity theories, can serve as a useful measure of operator evolution at both early and late times.
Krylov complexity is a measure of operator complexity that exhibits universal behavior and bounds a large class of other measures. In this paper, we generalize Krylov complexity from a closed system to an open system coupled to a Markovian bath, where Lindbladian evolution replaces Hamiltonian evolution. We show that Krylov complexity in open systems can be mapped to a non-Hermitian tight-binding model in a half-infinite chain. We discuss the properties of the non-Hermitian terms and show that the strengths of the non-Hermitian terms increase linearly with the increase of the Krylov basis index nnn. Such a non-Hermitian tight-binding model can exhibit localized edge modes that determine the long-time behavior of Krylov complexity. Hence, the growth of Krylov complexity is suppressed by dissipation, and at long times, Krylov complexity saturates at a finite value much smaller than that of a closed system with the same Hamiltonian. Our conclusions are supported by numerical results on several models, such as the Sachdev-Ye-Kitaev model and the interacting fermion model. Our work provides insights for discussing complexity, chaos, and holography for open quantum systems.