Pub Date : 2023-10-27DOI: 10.1088/1751-8121/ad07c9
Konstantin Klemm, Anita Mehta, Peter F Stadler
Abstract Difficult, in particular NP-complete, optimization problems are traditionally solved approximately using search heuristics. These are usually slowed down by the rugged landscapes encountered, because local minima arrest the search process. Cover-encoding maps were devised to circumvent this problem by transforming the original landscape to one that is free of local minima and enriched in near-optimal solutions. By definition, these involve the mapping of the original (larger) search space into smaller subspaces, by processes that typically amount to a form of coarse-graining. In this paper, we explore the details of this coarse-graining using formal arguments, as well as concrete examples of cover-encoding maps, that are investigated analytically as well as computationally. Our results strongly suggest that the coarse-graining involved in cover-encoding maps bears a strong resemblance to that encountered in renormalisation group schemes. Given the apparently disparate nature of these two formalisms, these strong similarities are rather startling, and suggest deep mathematical underpinnings that await further exploration.
{"title":"Optimisation via encodings: a renormalisation group perspective","authors":"Konstantin Klemm, Anita Mehta, Peter F Stadler","doi":"10.1088/1751-8121/ad07c9","DOIUrl":"https://doi.org/10.1088/1751-8121/ad07c9","url":null,"abstract":"Abstract Difficult, in particular NP-complete, optimization problems are traditionally solved approximately using search heuristics. These are usually slowed down by the rugged landscapes encountered, because local minima arrest the search process. Cover-encoding maps were devised to circumvent this problem by transforming the original landscape to one that is free of local minima and enriched in near-optimal solutions. By definition, these involve the mapping of the original (larger) search space into smaller subspaces, by processes that typically amount to a form of coarse-graining. In this paper, we explore the details of this coarse-graining using formal arguments, as well as concrete examples of cover-encoding maps, that are investigated analytically as well as computationally. Our results strongly suggest that the coarse-graining involved in cover-encoding maps bears a strong resemblance to that encountered in renormalisation group schemes. Given the apparently disparate nature of these two formalisms, these strong similarities are rather startling, and suggest deep mathematical underpinnings that await further exploration.","PeriodicalId":16785,"journal":{"name":"Journal of Physics A","volume":"120 3-4","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136318154","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-27DOI: 10.1088/1751-8121/ad07c8
Krzysztof Szczygielski
Abstract We propose and explore a notion of decomposably divisible (D-divisible) differentiable quantum evolution families on matrix algebras. This is achieved by replacing the complete positivity requirement, imposed on the propagator, by more general condition of decomposability. It is shown that such D-divisible dynamical maps satisfy a generalized version of Master Equation and are totally characterized by their time-local generators. Necessary and sufficient conditions for D-divisibility are found. Additionally, decomposable trace preserving semigroups are examined.
{"title":"D-divisible quantum evolution families","authors":"Krzysztof Szczygielski","doi":"10.1088/1751-8121/ad07c8","DOIUrl":"https://doi.org/10.1088/1751-8121/ad07c8","url":null,"abstract":"Abstract We propose and explore a notion of decomposably divisible (D-divisible) differentiable quantum evolution families on matrix algebras. This is achieved by replacing the complete positivity requirement, imposed on the propagator, by more general condition of decomposability. It is shown that such D-divisible dynamical maps satisfy a generalized version of Master Equation and are totally characterized by their time-local generators. Necessary and sufficient conditions for D-divisibility are found. Additionally, decomposable trace preserving semigroups are examined.","PeriodicalId":16785,"journal":{"name":"Journal of Physics A","volume":"34 2","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136234985","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-26DOI: 10.1088/1751-8121/ad075e
Zolotaryuk, A. V., Zolotaryuk, Y., Gusynin, V. P.
Abstract The spectrum of a one-dimensional pseudospin-one Hamiltonian with a three-component potential is studied for two configurations: (i) all the potential components are constants over the whole coordinate space and (ii) the profile of some components is of a rectangular form. In case (i), it is illustrated how the structure of three (lower, middle and upper) bands depends on the configuration of potential strengths including the appearance of flat bands at some special values of these strengths. In case (ii), the set of two equations for finding bound states is derived. The spectrum of bound-state energies is shown to depend crucially on the configuration of potential strengths. Each of these configurations is specified by a single strength parameter V . The bound-state energies are calculated as functions of the strength V and a one-point approach is developed realizing correspondent point interactions. For different potential configurations, the energy dependence on the strength V is described in detail, including its one-point approximation. From a whole variety of bound-state spectra, four characteristic types are singled out.
{"title":"Bound states and point interactions of the one-dimensional pseudospin-one Hamiltonian","authors":"Zolotaryuk, A. V., Zolotaryuk, Y., Gusynin, V. P.","doi":"10.1088/1751-8121/ad075e","DOIUrl":"https://doi.org/10.1088/1751-8121/ad075e","url":null,"abstract":"Abstract The spectrum of a one-dimensional pseudospin-one Hamiltonian with a three-component potential is studied for two configurations: (i) all the potential components are constants over the whole coordinate space and (ii) the profile of some components is of a rectangular form. In case (i), it is illustrated how the structure of three (lower, middle and upper) bands depends on the configuration of potential strengths including the appearance of flat bands at some special values of these strengths. In case (ii), the set of two equations for finding bound states is derived. The spectrum of bound-state energies is shown to depend crucially on the configuration of potential strengths. Each of these configurations is specified by a single strength parameter V . The bound-state energies are calculated as functions of the strength V and a one-point approach is developed realizing correspondent point interactions. For different potential configurations, the energy dependence on the strength V is described in detail, including its one-point approximation. From a whole variety of bound-state spectra, four characteristic types are singled out.
","PeriodicalId":16785,"journal":{"name":"Journal of Physics A","volume":"30 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134907746","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-26DOI: 10.1088/1751-8121/acfcf6
Benoit Estienne, Yacine Ikhlef, Andrei Rotaru
Abstract We identify the maximal chiral algebra of conformal cyclic orbifolds. In terms of this extended algebra, the orbifold is a rational and diagonal conformal field theory, provided the mother theory itself is also rational and diagonal. The operator content and operator product expansion of the cyclic orbifolds are revisited in terms of this algebra. The fusion rules and fusion numbers are computed via the Verlinde formula. This allows one to predict which conformal blocks appear in a given four-point function of twisted or untwisted operators, which is relevant for the computation of various entanglement measures in one-dimensional critical systems.
{"title":"The operator algebra of cyclic orbifolds","authors":"Benoit Estienne, Yacine Ikhlef, Andrei Rotaru","doi":"10.1088/1751-8121/acfcf6","DOIUrl":"https://doi.org/10.1088/1751-8121/acfcf6","url":null,"abstract":"Abstract We identify the maximal chiral algebra of conformal cyclic orbifolds. In terms of this extended algebra, the orbifold is a rational and diagonal conformal field theory, provided the mother theory itself is also rational and diagonal. The operator content and operator product expansion of the cyclic orbifolds are revisited in terms of this algebra. The fusion rules and fusion numbers are computed via the Verlinde formula. This allows one to predict which conformal blocks appear in a given four-point function of twisted or untwisted operators, which is relevant for the computation of various entanglement measures in one-dimensional critical systems.","PeriodicalId":16785,"journal":{"name":"Journal of Physics A","volume":"60 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136377039","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-26DOI: 10.1088/1751-8121/ad076e
Naruhiko Aizawa, Ren Ito
Abstract We investigate the possibilities of integration on the minimal Z 2 2 -superspace. Two definitions are taken from the works by Poncin and Schouten and we examine their generalizations. It is shown that these definitions impose some restrictions on the integrable functions. We then introduce a new definition of integral, which is inspired by our previous work, and show that the definition does not impose restrictions on the integrable functions. An interesting feature of this definition is the emergence of a spatial coordinate which means that the integral is defined on R 2 despite the fact that the (0,0) part of the minimal Z 2 2 -superspace is R
{"title":"Integration on minimal Z<sub>2</sub> <sup>2</sup>-superspace and emergence of space","authors":"Naruhiko Aizawa, Ren Ito","doi":"10.1088/1751-8121/ad076e","DOIUrl":"https://doi.org/10.1088/1751-8121/ad076e","url":null,"abstract":"Abstract We investigate the possibilities of integration on the minimal Z 2 2 -superspace. Two definitions are taken from the works by Poncin and Schouten and we examine their generalizations. It is shown that these definitions impose some restrictions on the integrable functions. We then introduce a new definition of integral, which is inspired by our previous work, and show that the definition does not impose restrictions on the integrable functions. An interesting feature of this definition is the emergence of a spatial coordinate which means that the integral is defined on R <sup>2 despite the fact that the (0,0) part of the minimal Z 2 2 -superspace is R","PeriodicalId":16785,"journal":{"name":"Journal of Physics A","volume":"42 5","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134907531","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-26DOI: 10.1088/1751-8121/ad075d
Paul E Lammert
Abstract Informed by an abstraction of Kohn-Sham computation called a KS machine, a functional analytic perspective is developed on mathematical aspects of density functional theory. A natural semantics for the machine is bivariate, consisting of a sequence of potentials paired with a ground density. Although the question of when the KS machine can converge to a solution (where the potential component matches a designated target) is not resolved here, a number of related ones are. For instance: Can the machine progress toward a solution? Barring presumably exceptional circumstances, yes in an energetic sense, but using a potential-mixing scheme rather than the usual density-mixing variety. Are energetic and function space distance notions of proximity-to-solution commensurate? Yes, to a significant degree. If the potential components of a sequence of ground pairs converges to a target density, do the density components cluster on ground densities thereof? Yes, barring particle number drifting to infinity.
{"title":"Kohn-Sham computation and the bivariate view of density functional theory","authors":"Paul E Lammert","doi":"10.1088/1751-8121/ad075d","DOIUrl":"https://doi.org/10.1088/1751-8121/ad075d","url":null,"abstract":"Abstract Informed by an abstraction of Kohn-Sham computation called a KS machine, a functional analytic perspective is developed on mathematical aspects of density functional theory. A natural semantics for the machine is bivariate, consisting of a sequence of potentials paired with a ground density. Although the question of when the KS machine can converge to a solution (where the potential component matches a designated target) is not resolved here, a number of related ones are. For instance: Can the machine progress toward a solution? Barring presumably exceptional circumstances, yes in an energetic sense, but using a potential-mixing scheme rather than the usual density-mixing variety. Are energetic and function space distance notions of proximity-to-solution commensurate? Yes, to a significant degree. If the potential components of a sequence of ground pairs converges to a target density, do the density components cluster on ground densities thereof? Yes, barring particle number drifting to infinity.&#xD;","PeriodicalId":16785,"journal":{"name":"Journal of Physics A","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134907678","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We extend the notion of quantum reading to the case where the information to be retrieved, which is encoded into a set of quantum channels, is of quantum nature. We use two-qubit unitaries describing the system environment interaction, with the initial environment state determining the system’s input-output channel and hence the encoded information. The performance of the most relevant two-qubit unitaries is determined with two different approaches: i) one-shot quantum capacity of the channel arising between environment and system’s output; ii) estimation of parameters characterizing the initial quantum state of the environment. The obtained results are mostly in (qualitative) agreement, with some distinguishing features that include the CNOT unitary.
{"title":"Quantum reading of quantum information","authors":"Samad Khabbazi Oskouei, Stefano Mancini, Milajiguli Rexiti","doi":"10.1088/1751-8121/ad075f","DOIUrl":"https://doi.org/10.1088/1751-8121/ad075f","url":null,"abstract":"Abstract We extend the notion of quantum reading to the case where the information to be retrieved, which is encoded into a set of quantum channels, is of quantum nature. We use two-qubit unitaries describing the system environment interaction, with the initial environment state determining the system’s input-output channel and hence the encoded information. The performance of the most relevant two-qubit unitaries is determined with two different approaches: i) one-shot quantum capacity of the channel arising between environment and system’s output; ii) estimation of parameters characterizing the initial quantum state of the environment. The obtained results are mostly in (qualitative) agreement, with some distinguishing features that include the CNOT unitary.","PeriodicalId":16785,"journal":{"name":"Journal of Physics A","volume":"21 2","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134907015","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-25DOI: 10.1088/1751-8121/ad034a
Hiroshi Miki, Satoshi Tsujimoto, Da Zhao
Abstract In this paper, we study quantum walks on the extension of association schemes. Various state transfers can be achieved on these graphs, such as multiple state transfer among extreme points of a simplex, fractional revival on subsimplexes. Since only few examples of multiple (perfect) state transfer are known, we aim to make some additions in this collection.
{"title":"Quantum walks on simplexes and multiple perfect state transfer","authors":"Hiroshi Miki, Satoshi Tsujimoto, Da Zhao","doi":"10.1088/1751-8121/ad034a","DOIUrl":"https://doi.org/10.1088/1751-8121/ad034a","url":null,"abstract":"Abstract In this paper, we study quantum walks on the extension of association schemes. Various state transfers can be achieved on these graphs, such as multiple state transfer among extreme points of a simplex, fractional revival on subsimplexes. Since only few examples of multiple (perfect) state transfer are known, we aim to make some additions in this collection.","PeriodicalId":16785,"journal":{"name":"Journal of Physics A","volume":"15 2","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134972746","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-25DOI: 10.1088/1751-8121/ad0348
Ying-Xiang Wang, Sheng-Chen Liu, Lin Cheng, Liang-You Peng
Abstract Ions confined in a Paul trap serve as crucial platforms in various research fields, including quantum computing and precision spectroscopy. However, the ion dynamics is inevitably influenced by different types of noise, which require accurate computations and general analytical analysis to facilitate diverse applications based on trapped ions with white or colored noise. In the present work, we investigate the motion of ions in a Paul trap via the Langevin equation using both analytical and numerical methods, systematically studying three different types of noise: the white noise, the colored noise via the Ornstein–Uhlenbeck process and the Wiener process. For the white noise of the case, we provide a recursion method to calculate ion motion for a wide range of parameters. Furthermore, we present an analytical solution to the more realistic stochastic process associated with the colored noise, verified by the Monte Carlo simulation. By comparing the results of the colored noise with those of the white noise, and additionally considering another limit of noise parameters corresponding to the Wiener process, we summarize the effects of different noise types on the ion dynamics.
{"title":"Systematic investigations on ion dynamics with noises in Paul trap","authors":"Ying-Xiang Wang, Sheng-Chen Liu, Lin Cheng, Liang-You Peng","doi":"10.1088/1751-8121/ad0348","DOIUrl":"https://doi.org/10.1088/1751-8121/ad0348","url":null,"abstract":"Abstract Ions confined in a Paul trap serve as crucial platforms in various research fields, including quantum computing and precision spectroscopy. However, the ion dynamics is inevitably influenced by different types of noise, which require accurate computations and general analytical analysis to facilitate diverse applications based on trapped ions with white or colored noise. In the present work, we investigate the motion of ions in a Paul trap via the Langevin equation using both analytical and numerical methods, systematically studying three different types of noise: the white noise, the colored noise via the Ornstein–Uhlenbeck process and the Wiener process. For the white noise of the case, we provide a recursion method to calculate ion motion for a wide range of parameters. Furthermore, we present an analytical solution to the more realistic stochastic process associated with the colored noise, verified by the Monte Carlo simulation. By comparing the results of the colored noise with those of the white noise, and additionally considering another limit of noise parameters corresponding to the Wiener process, we summarize the effects of different noise types on the ion dynamics.","PeriodicalId":16785,"journal":{"name":"Journal of Physics A","volume":"44 11","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135218532","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-25DOI: 10.1088/1751-8121/ad06fc
Etera R. Livine
Abstract In quantum mechanics, a classical particle is raised to a wave-function, thereby acquiring many more degrees of freedom. For instance, in the semi-classical regime, while the position and momentum expectation values follow the classical trajectory, the uncertainty of a wave-packet can evolve and beat independently. We use this insight to revisit the dynamics of a 1d particle in a time-dependent harmonic well. One can solve it by considering time reparameterizations and the Virasoro group action to map the system to the harmonic oscillator with constant frequency. We prove that identifying such a simplifying time variable is naturally solved by quantizing the system and looking at the evolution of the width of a Gaussian wave-packet. We further show that the Ermakov-Lewis invariant for the classical evolution in a time-dependent harmonic potential is actually the quantum uncertainty of a Gaussian wave-packet. This naturally extends the classical Ermakov-Lewis invariant to a constant of motion for quantum systems following Schrodinger equation. We conclude with a discussion of potential applications to quantum gravity and quantum cosmology.
{"title":"Quantum uncertainty as an intrinsic clock","authors":"Etera R. Livine","doi":"10.1088/1751-8121/ad06fc","DOIUrl":"https://doi.org/10.1088/1751-8121/ad06fc","url":null,"abstract":"Abstract In quantum mechanics, a classical particle is raised to a wave-function, thereby acquiring many more degrees of freedom. For instance, in the semi-classical regime, while the position and momentum expectation values follow the classical trajectory, the uncertainty of a wave-packet can evolve and beat independently. We use this insight to revisit the dynamics of a 1d particle in a time-dependent harmonic well. One can solve it by considering time reparameterizations and the Virasoro group action to map the system to the harmonic oscillator with constant frequency. We prove that identifying such a simplifying time variable is naturally solved by quantizing the system and looking at the evolution of the width of a Gaussian wave-packet. We further show that the Ermakov-Lewis invariant for the classical evolution in a time-dependent harmonic potential is actually the quantum uncertainty of a Gaussian wave-packet. This naturally extends the classical Ermakov-Lewis invariant to a constant of motion for quantum systems following Schrodinger equation. We conclude with a discussion of potential applications to quantum gravity and quantum cosmology.","PeriodicalId":16785,"journal":{"name":"Journal of Physics A","volume":"26 16","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134972241","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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