Pub Date : 2023-10-19DOI: 10.1088/1751-8121/ad0533
Ilon Joseph
Abstract The phase space Koopman-van Hove (KvH) equation can be derived from the asymptotic semiclassical analysis of partial differential equations.
Semiclassical theory yields the Hamilton-Jacobi equation for the complex phase factor and the transport equation for the amplitude.
These two equations can be combined to form a nonlinear semiclassical version of the KvH equation in configuration space.
Every solution of the configuration space KvH equation satisfies both the semiclassical phase space KvH equation and the Hamilton-Jacobi constraint.
For configuration space solutions, this constraint resolves the paradox that there are two different conserved densities in phase space.
For integrable systems, the KvH spectrum is the Cartesian product of a classical and a semiclassical spectrum.
If the classical spectrum is eliminated, then, with the correct choice of Jeffreys-Wentzel-Kramers-Brillouin (JWKB)
matching conditions, the semiclassical spectrum satisfies the Einstein-Brillouin-Keller quantization conditions which include the correction due to the Maslov index.
However, semiclassical analysis uses different choices for boundary conditions, continuity requirements, and the domain of definition. 
For example, use of the complex JWKB method allows for the treatment of tunneling through the complexification of phase space.
Finally, although KvH wavefunctions include the possibility of interference effects, interference is not observable when all observables are 
approximated as local operators on phase space.
Observing interference effects requires consideration of nonlocal operations, e.g. through higher orders in the asymptotic theory.
{"title":"Semiclassical theory and the Koopman-van Hove equation","authors":"Ilon Joseph","doi":"10.1088/1751-8121/ad0533","DOIUrl":"https://doi.org/10.1088/1751-8121/ad0533","url":null,"abstract":"Abstract The phase space Koopman-van Hove (KvH) equation can be derived from the asymptotic semiclassical analysis of partial differential equations.
Semiclassical theory yields the Hamilton-Jacobi equation for the complex phase factor and the transport equation for the amplitude.
These two equations can be combined to form a nonlinear semiclassical version of the KvH equation in configuration space.
Every solution of the configuration space KvH equation satisfies both the semiclassical phase space KvH equation and the Hamilton-Jacobi constraint.
For configuration space solutions, this constraint resolves the paradox that there are two different conserved densities in phase space.
For integrable systems, the KvH spectrum is the Cartesian product of a classical and a semiclassical spectrum.
If the classical spectrum is eliminated, then, with the correct choice of Jeffreys-Wentzel-Kramers-Brillouin (JWKB)
matching conditions, the semiclassical spectrum satisfies the Einstein-Brillouin-Keller quantization conditions which include the correction due to the Maslov index.
However, semiclassical analysis uses different choices for boundary conditions, continuity requirements, and the domain of definition. 
For example, use of the complex JWKB method allows for the treatment of tunneling through the complexification of phase space.
Finally, although KvH wavefunctions include the possibility of interference effects, interference is not observable when all observables are 
approximated as local operators on phase space.
Observing interference effects requires consideration of nonlocal operations, e.g. through higher orders in the asymptotic theory.","PeriodicalId":16785,"journal":{"name":"Journal of Physics A","volume":"141 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135729481","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 : 2023-10-18DOI: 10.1088/1751-8121/ad0189
Daria Holdenried-Chernoff, David A King, Bruce Buffett
Abstract Variations in the geomagnetic field occur on a vast range of time scales, from milliseconds to millions of years. The advent of satellite measurements has allowed for detailed studies of short timescale geomagnetic field behaviour, but understanding its long timescale evolution remains challenging due to the sparsity of the paleomagnetic record. This paper introduces a field theory framework for studying magnetic field generation as a result of stochastic fluid motions. Starting from a stochastic kinematic dynamo model (the Kazantsev kinematic model), we derive statistical properties of the magnetic field that may be compared to observations from the paleomagnetic record. The fluid velocity is taken to be a Kraichnan field with general covariance, which acts as a random forcing obeying Gaussian statistics. Using the Martin-Siggia-Rose-Janssen-de Dominicis formalism, we compute the average magnetic field response function for fluid velocities with short correlation time. From this we obtain an estimate for the turbulent contribution to the magnetic diffusivity, and find that it is consistent with results from mean-field dynamo theory. This framework presents much promise for studying the geomagnetic field in a stochastic context.
{"title":"A field theory approach to the statistical kinematic dynamo","authors":"Daria Holdenried-Chernoff, David A King, Bruce Buffett","doi":"10.1088/1751-8121/ad0189","DOIUrl":"https://doi.org/10.1088/1751-8121/ad0189","url":null,"abstract":"Abstract Variations in the geomagnetic field occur on a vast range of time scales, from milliseconds to millions of years. The advent of satellite measurements has allowed for detailed studies of short timescale geomagnetic field behaviour, but understanding its long timescale evolution remains challenging due to the sparsity of the paleomagnetic record. This paper introduces a field theory framework for studying magnetic field generation as a result of stochastic fluid motions. Starting from a stochastic kinematic dynamo model (the Kazantsev kinematic model), we derive statistical properties of the magnetic field that may be compared to observations from the paleomagnetic record. The fluid velocity is taken to be a Kraichnan field with general covariance, which acts as a random forcing obeying Gaussian statistics. Using the Martin-Siggia-Rose-Janssen-de Dominicis formalism, we compute the average magnetic field response function for fluid velocities with short correlation time. From this we obtain an estimate for the turbulent contribution to the magnetic diffusivity, and find that it is consistent with results from mean-field dynamo theory. This framework presents much promise for studying the geomagnetic field in a stochastic context.","PeriodicalId":16785,"journal":{"name":"Journal of Physics A","volume":"75 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135823172","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 : 2023-10-18DOI: 10.1088/1751-8121/ad018b
Giandomenico Palumbo
Abstract It is well known that noncommutative geometry naturally emerges in the quantum Hall states due to the presence of strong and constant magnetic fields. Here, we discuss the underlying noncommutative geometry of quantum Hall fluids in which the magnetic fields are spatially inhomogenoeus. We analyze these cases by employing symplectic geometry and Fedosov’s deformation quantization, which rely on symplectic connections and Fedosov’s star-product. Through this formalism, we unveil some new features concerning the static limit of the Haldane’s unimodular metric and the Girvin–MacDonald–Platzman algebra of the density operators, which plays a central role in the fractional quantum Hall effect.
{"title":"Noncommutative geometry and deformation quantization in the quantum Hall fluids with inhomogeneous magnetic fields","authors":"Giandomenico Palumbo","doi":"10.1088/1751-8121/ad018b","DOIUrl":"https://doi.org/10.1088/1751-8121/ad018b","url":null,"abstract":"Abstract It is well known that noncommutative geometry naturally emerges in the quantum Hall states due to the presence of strong and constant magnetic fields. Here, we discuss the underlying noncommutative geometry of quantum Hall fluids in which the magnetic fields are spatially inhomogenoeus. We analyze these cases by employing symplectic geometry and Fedosov’s deformation quantization, which rely on symplectic connections and Fedosov’s star-product. Through this formalism, we unveil some new features concerning the static limit of the Haldane’s unimodular metric and the Girvin–MacDonald–Platzman algebra of the density operators, which plays a central role in the fractional quantum Hall effect.","PeriodicalId":16785,"journal":{"name":"Journal of Physics A","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135823375","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 : 2023-10-18DOI: 10.1088/1751-8121/ad00f0
Benjamin Claude Bazin Symons, David Galvin, Emre Sahin, Vassil Alexandrov, Stefano Mensa
Abstract Quantum computing is gaining popularity across a wide range of scientific disciplines due to its potential to solve long-standing computational problems that are considered intractable with classical computers. One promising area where quantum computing has potential is in the speed-up of NP -hard optimisation problems that are common in industrial areas such as logistics and finance. Newcomers to the field of quantum computing who are interested in using this technology to solve optimisation problems do not have an easily accessible source of information on the current capabilities of quantum computers and algorithms. This paper aims to provide a comprehensive overview of the theory of quantum optimisation techniques and their practical application, focusing on their near-term potential for noisy intermediate scale quantum devices. The paper starts by drawing parallels between classical and quantum optimisation problems, highlighting their conceptual similarities and differences. Two main paradigms for quantum hardware are then discussed: analogue and gate-based quantum computers. While analog devices such as quantum annealers are effective for some optimisation problems, they have limitations and cannot be used for universal quantum computation. In contrast, gate-based quantum computers offer the potential for universal quantum computation, but they face challenges with hardware limitations and accurate gate implementation. The paper provides a detailed mathematical discussion with references to key works in the field, as well as a more practical discussion with relevant examples. The most popular techniques for quantum optimisation on gate-based quantum computers, the quantum approximate optimisation algorithm and the quantum alternating operator ansatz framework, are discussed in detail. However, it is still unclear whether these techniques will yield quantum advantage, even with advancements in hardware and noise reduction. The paper concludes with a discussion of the challenges facing quantum optimisation techniques and the need for further research and development to identify new, effective methods for achieving quantum advantage.
{"title":"A practitioner’s guide to quantum algorithms for optimisation problems","authors":"Benjamin Claude Bazin Symons, David Galvin, Emre Sahin, Vassil Alexandrov, Stefano Mensa","doi":"10.1088/1751-8121/ad00f0","DOIUrl":"https://doi.org/10.1088/1751-8121/ad00f0","url":null,"abstract":"Abstract Quantum computing is gaining popularity across a wide range of scientific disciplines due to its potential to solve long-standing computational problems that are considered intractable with classical computers. One promising area where quantum computing has potential is in the speed-up of NP -hard optimisation problems that are common in industrial areas such as logistics and finance. Newcomers to the field of quantum computing who are interested in using this technology to solve optimisation problems do not have an easily accessible source of information on the current capabilities of quantum computers and algorithms. This paper aims to provide a comprehensive overview of the theory of quantum optimisation techniques and their practical application, focusing on their near-term potential for noisy intermediate scale quantum devices. The paper starts by drawing parallels between classical and quantum optimisation problems, highlighting their conceptual similarities and differences. Two main paradigms for quantum hardware are then discussed: analogue and gate-based quantum computers. While analog devices such as quantum annealers are effective for some optimisation problems, they have limitations and cannot be used for universal quantum computation. In contrast, gate-based quantum computers offer the potential for universal quantum computation, but they face challenges with hardware limitations and accurate gate implementation. The paper provides a detailed mathematical discussion with references to key works in the field, as well as a more practical discussion with relevant examples. The most popular techniques for quantum optimisation on gate-based quantum computers, the quantum approximate optimisation algorithm and the quantum alternating operator ansatz framework, are discussed in detail. However, it is still unclear whether these techniques will yield quantum advantage, even with advancements in hardware and noise reduction. The paper concludes with a discussion of the challenges facing quantum optimisation techniques and the need for further research and development to identify new, effective methods for achieving quantum advantage.","PeriodicalId":16785,"journal":{"name":"Journal of Physics A","volume":"476 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135824154","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 Quantum measurement is one of the most fascinating and discussed phenomena in quantum physics, due to the impact on the system of the measurement action and the resulting interpretation issues. Scholars proposed weak measurements to amplify measured signals by exploiting a quantity called a weak value, but also to overcome philosophical difficulties related to the system perturbation induced by the measurement process. The method finds many applications and raises many philosophical questions as well, especially about the proper interpretation of the observations. In this paper, we show that any weak value can be expressed as the expectation value of a suitable non-normal operator. We propose a preliminary explanation of their anomalous and amplification behavior based on the theory of non-normal matrices and their link with non-normality: the weak value is different from an eigenvalue when the operator involved in the expectation value is non-normal. Our study paves the way for a deeper understanding of the measurement phenomenon, helps the design of experiments, and it is a call for collaboration to researchers in both fields to unravel new quantum phenomena induced by non-normality.
{"title":"Revisiting weak values through non-normality","authors":"Lorena Ballesteros Ferraz, Riccardo Muolo, Yves Caudano, Timoteo Carletti","doi":"10.1088/1751-8121/ad04a4","DOIUrl":"https://doi.org/10.1088/1751-8121/ad04a4","url":null,"abstract":"Abstract Quantum measurement is one of the most fascinating and discussed phenomena in quantum physics, due to the impact on the system of the measurement action and the resulting interpretation issues. Scholars proposed weak measurements to amplify measured signals by exploiting a quantity called a weak value, but also to overcome philosophical difficulties related to the system perturbation induced by the measurement process. The method finds many applications and raises many philosophical questions as well, especially about the proper interpretation of the observations. In this paper, we show that any weak value can be expressed as the expectation value of a suitable non-normal operator. We propose a preliminary explanation of their anomalous and amplification behavior based on the theory of non-normal matrices and their link with non-normality: the weak value is different from an eigenvalue when the operator involved in the expectation value is non-normal. Our study paves the way for a deeper understanding of the measurement phenomenon, helps the design of experiments, and it is a call for collaboration to researchers in both fields to unravel new quantum phenomena induced by non-normality.
","PeriodicalId":16785,"journal":{"name":"Journal of Physics A","volume":"183 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135824753","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 : 2023-10-18DOI: 10.1088/1751-8121/acfbc9
Junjie Wang, Fude Li, Xuexi Yi
Abstract A notable feature of non-Hermitian systems with skin effects is the sensitivity of their spectra and eigenstates to the boundary conditions. In the literature, three types of boundary conditions–periodic boundary condition, open boundary condition (OBC) and a defect in the system as a boundary, are explored. In this work we introduce the other type of boundary condition provided by a giant atom. The giant atom couples to a nonreciprocal Su–Schrieffer–Heeger (SSH) chain at two points and plays the role of defects. We study the spectrum and localization of eigenstates of the system and find that the giant atom can induce asymmetric zero modes. A remarkable feature is that bulk states might localize at the left or the right chain-atom coupling sites in weak localization regimes. This bipolar localization leads to Bloch-like states, even though translational invariance is broken. Moreover, we find that the localization is obviously weaker than the case with two small atoms or OBCs even in strong coupling regimes. These intriguing results indicate that nonlocal coupling of the giant atom to a nonreciprocal SSH chain weakens the localization of the eigenstates. We also show that the Lyapunov exponent in the long-time dynamics in real space can act as a witness of the localized bulk states.
{"title":"Giant atom induced zero modes and localization in the nonreciprocal Su-Schrieffer-Heeger chain","authors":"Junjie Wang, Fude Li, Xuexi Yi","doi":"10.1088/1751-8121/acfbc9","DOIUrl":"https://doi.org/10.1088/1751-8121/acfbc9","url":null,"abstract":"Abstract A notable feature of non-Hermitian systems with skin effects is the sensitivity of their spectra and eigenstates to the boundary conditions. In the literature, three types of boundary conditions–periodic boundary condition, open boundary condition (OBC) and a defect in the system as a boundary, are explored. In this work we introduce the other type of boundary condition provided by a giant atom. The giant atom couples to a nonreciprocal Su–Schrieffer–Heeger (SSH) chain at two points and plays the role of defects. We study the spectrum and localization of eigenstates of the system and find that the giant atom can induce asymmetric zero modes. A remarkable feature is that bulk states might localize at the left or the right chain-atom coupling sites in weak localization regimes. This bipolar localization leads to Bloch-like states, even though translational invariance is broken. Moreover, we find that the localization is obviously weaker than the case with two small atoms or OBCs even in strong coupling regimes. These intriguing results indicate that nonlocal coupling of the giant atom to a nonreciprocal SSH chain weakens the localization of the eigenstates. We also show that the Lyapunov exponent in the long-time dynamics in real space can act as a witness of the localized bulk states.","PeriodicalId":16785,"journal":{"name":"Journal of Physics A","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135823142","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 : 2023-10-18DOI: 10.1088/1751-8121/ad04a5
Francesco Giglio, Giulio Landolfi, L Martina
Abstract We investigate real solutions of a C-integrable non-evolutionary partial differential equation in the form of a scalar conservation law where the flux density depends both on the density and on its first derivatives with respect to the local variables. By performing a similarity reduction dictated by one of its local symmetry generators, a nonlinear ordinary differential equation arises that is connected to the Painlevé III equation. Exact solutions are secured and described provided a constraint holds among the coefficients of the original equation. In the most general case, we pinpoint the generation of additional singularities by numerical integration. Then, we discuss the evolution of given initial profiles. Finally, we mention aspects concerning rational solutions with a finite number of poles.
{"title":"On solutions to a novel non-evolutionary integrable 1+1 PDE","authors":"Francesco Giglio, Giulio Landolfi, L Martina","doi":"10.1088/1751-8121/ad04a5","DOIUrl":"https://doi.org/10.1088/1751-8121/ad04a5","url":null,"abstract":"Abstract We investigate real solutions of a C-integrable non-evolutionary partial differential equation in the form of a scalar conservation law where the flux density depends both on the density and on its first derivatives with respect to the local variables. By performing a similarity reduction dictated by one of its local symmetry generators, a nonlinear ordinary differential equation arises that is connected to the Painlevé III equation. Exact solutions are secured and described provided a constraint holds among the coefficients of the original equation. In the most general case, we pinpoint the generation of additional singularities by numerical integration. Then, we discuss the evolution of given initial profiles. Finally, we mention aspects concerning rational solutions with a finite number of poles.","PeriodicalId":16785,"journal":{"name":"Journal of Physics A","volume":"74 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135824927","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 : 2023-10-17DOI: 10.1088/1751-8121/acfd6b
Jan Wasilewski, Tomasz Paterek, Karol Horodecki
Abstract The usual figure of merit characterizing the performance of neural networks applied to problems in the quantum domain is their accuracy, being the probability of a correct answer on a previously unseen input. Here we append this parameter with the uncertainty of the prediction, characterizing the degree of confidence in the answer. A powerful technique for estimating uncertainty is provided by Bayesian neural networks (BNNs). We first give simple illustrative examples of advantages brought forward by BNNs, out of which we wish to highlight their ability of reliable uncertainty estimation even after training with biased datasets. Then we apply BNNs to the problem of recognition of quantum contextuality, which shows that the uncertainty itself is an independent parameter identifying the chance of misclassification of contextuality.
{"title":"Uncertainty of feed forward neural networks recognizing quantum contextuality","authors":"Jan Wasilewski, Tomasz Paterek, Karol Horodecki","doi":"10.1088/1751-8121/acfd6b","DOIUrl":"https://doi.org/10.1088/1751-8121/acfd6b","url":null,"abstract":"Abstract The usual figure of merit characterizing the performance of neural networks applied to problems in the quantum domain is their accuracy, being the probability of a correct answer on a previously unseen input. Here we append this parameter with the uncertainty of the prediction, characterizing the degree of confidence in the answer. A powerful technique for estimating uncertainty is provided by Bayesian neural networks (BNNs). We first give simple illustrative examples of advantages brought forward by BNNs, out of which we wish to highlight their ability of reliable uncertainty estimation even after training with biased datasets. Then we apply BNNs to the problem of recognition of quantum contextuality, which shows that the uncertainty itself is an independent parameter identifying the chance of misclassification of contextuality.","PeriodicalId":16785,"journal":{"name":"Journal of Physics A","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135945116","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 : 2023-10-17DOI: 10.1088/1751-8121/ad043a
Jaco van Tonder, Jon Links
Abstract Several studies have exploited the integrable structure of central spin models to deepen understanding of these fundamental systems. In recent years, an underlying supersymmetry for systems with XX interactions has been uncovered. Here we report that a class of central spin models with XY interactions is also supersymmetric and integrable. The associated Bethe Ansatz solution is presented for the case where all particles are spin-1/2.
{"title":"Supersymmetry and integrability for a class of <i>XY</i> central spin models","authors":"Jaco van Tonder, Jon Links","doi":"10.1088/1751-8121/ad043a","DOIUrl":"https://doi.org/10.1088/1751-8121/ad043a","url":null,"abstract":"Abstract Several studies have exploited the integrable structure of central spin models to deepen understanding of these fundamental systems. In recent years, an underlying supersymmetry for systems with XX interactions has been uncovered. Here we report that a class of central spin models with XY interactions is also supersymmetric and integrable. The associated Bethe Ansatz solution is presented for the case where all particles are spin-1/2.&#xD;","PeriodicalId":16785,"journal":{"name":"Journal of Physics A","volume":"21 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135945192","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 : 2023-10-17DOI: 10.1088/1751-8121/ad0439
Aaron Villanueva, Peyman Najafi, Hilbert Kappen
Abstract We study quantum annealing for combinatorial optimization with Hamiltonian $H = H_0 + z H_f$ where $H_f$ is diagonal, $H_0=-ket{phi}bra{phi}$ is the equal superposition state projector and $z$ the annealing parameter.
We analytically compute the minimal spectral gap, which is $Omega(1/sqrt{N})$ with $N$ the total number of states, and its location $z_*$.
We show that quantum speed-up requires an annealing schedule which demands a precise knowledge of $z_*$, which can be computed only if the density of states of the optimization problem is known.
However, in general the density of states is intractable to compute, making quadratic speed-up unfeasible for any practical combinatorial optimization problems. 
We conjecture that it is likely that this negative result also applies for any other instance independent transverse Hamiltonians such as $H_0 = -sum_{i=1}^n sigma_i^x$.
{"title":"Why adiabatic quantum annealing is unlikely to yield speed-up","authors":"Aaron Villanueva, Peyman Najafi, Hilbert Kappen","doi":"10.1088/1751-8121/ad0439","DOIUrl":"https://doi.org/10.1088/1751-8121/ad0439","url":null,"abstract":"Abstract We study quantum annealing for combinatorial optimization with Hamiltonian $H = H_0 + z H_f$ where $H_f$ is diagonal, $H_0=-ket{phi}bra{phi}$ is the equal superposition state projector and $z$ the annealing parameter.&#xD;We analytically compute the minimal spectral gap, which is $Omega(1/sqrt{N})$ with $N$ the total number of states, and its location $z_*$.&#xD;We show that quantum speed-up requires an annealing schedule which demands a precise knowledge of $z_*$, which can be computed only if the density of states of the optimization problem is known.&#xD;However, in general the density of states is intractable to compute, making quadratic speed-up unfeasible for any practical combinatorial optimization problems. &#xD;We conjecture that it is likely that this negative result also applies for any other instance independent transverse Hamiltonians such as $H_0 = -sum_{i=1}^n sigma_i^x$.","PeriodicalId":16785,"journal":{"name":"Journal of Physics A","volume":"31 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135994746","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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