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Search Results (232)

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23 pages, 5646 KiB  
Article
Enhancing Security and Authenticity in Immersive Environments
by Rebecca Acheampong, Dorin-Mircea Popovici, Titus Balan, Alexandre Rekeraho and Manuel Soto Ramos
Information 2025, 16(3), 191; https://doi.org/10.3390/info16030191 (registering DOI) - 1 Mar 2025
Abstract
Immersive environments have brought a great transformation in human–computer interaction by enabling realistic and interactive experiences within simulated or augmented spaces. In these immersive environments, virtual assets such as custom avatars, digital artwork, and virtual real estate play an important role, often holding [...] Read more.
Immersive environments have brought a great transformation in human–computer interaction by enabling realistic and interactive experiences within simulated or augmented spaces. In these immersive environments, virtual assets such as custom avatars, digital artwork, and virtual real estate play an important role, often holding a substantial value in both virtual and real worlds. However, this value also makes them attractive to fraudulent activities. As a result, ensuring the authenticity and integrity of virtual assets is of concern. This study proposes a cryptographic solution that leverages digital signatures and hash algorithms to secure virtual assets in immersive environments. The system employs RSA-2048 for signing and SHA-256 hashing for binding the digital signature to the asset’s data to prevent tampering and forgery. Our experimental evaluation demonstrates that the signing process operates with remarkable efficiency; over ten trials, the signing time averaged 17.3 ms, with a narrow range of 16–19 ms and a standard deviation of 1.1 ms. Verification times were near-instantaneous (0–1 ms), ensuring real-time responsiveness. Moreover, the signing process incurred a minimal memory footprint of approximately 4 KB, highlighting the system’s suitability for resource-constrained VR applications. Simulations of tampering and forgery attacks further validated the system’s capability to detect unauthorized modifications, with a 100% detection rate observed across multiple trials. While the system currently employs RSA, which may be vulnerable to quantum computing in the future, its modular design ensures crypto-agility, allowing for the integration of quantum-resistant algorithms as needed. This work not only addresses immediate security challenges in immersive environments but also lays the groundwork for broader applications, including regulatory compliance for financial virtual assets. Full article
(This article belongs to the Collection Augmented Reality Technologies, Systems and Applications)
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Figure 1

Figure 1
<p>Cryptographic signature process.</p>
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<p>Layered architecture diagram of the proposed system.</p>
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<p>Player input binding SignItem and VerifyItem to controller’s primary and secondary buttons, respectively.</p>
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<p>The main architecture of the proposed system.</p>
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<p>Logs of the user’s actions during the signing process.</p>
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<p>Logs of user’s actions during the verification process.</p>
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<p>In the virtual room, when the asset is verified by a user, feedback is displayed on the board with the name of the signer and the validity message. The signature is just displayed for research purposes.</p>
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<p>Asset configuration settings in the inspector referencing the feedback interface and the controller input actions.</p>
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<p>Feedback to the user interface when tampering is detected.</p>
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<p>Console logs displaying the comparison in the hash of the original data and the modified data to provide traceability to tampering detection.</p>
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<p>Forgery detection logged in the console.</p>
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<p>Time it took to verify an asset’s validity.</p>
Full article ">
30 pages, 1684 KiB  
Article
Efficient GPU Implementation of the McMurchie–Davidson Method for Shell-Based ERI Computations
by Haruto Fujii, Yasuaki Ito, Nobuya Yokogawa, Kanta Suzuki, Satoki Tsuji, Koji Nakano, Victor Parque and Akihiko Kasagi
Appl. Sci. 2025, 15(5), 2572; https://doi.org/10.3390/app15052572 - 27 Feb 2025
Viewed by 68
Abstract
Quantum chemistry offers the formal machinery to derive molecular and physical properties arising from (sub)atomic interactions. However, as molecules of practical interest are largely polyatomic, contemporary approximation schemes such as the Hartree–Fock scheme are computationally expensive due to the large number of electron [...] Read more.
Quantum chemistry offers the formal machinery to derive molecular and physical properties arising from (sub)atomic interactions. However, as molecules of practical interest are largely polyatomic, contemporary approximation schemes such as the Hartree–Fock scheme are computationally expensive due to the large number of electron repulsion integrals (ERIs). Central to the Hartree–Fock method is the efficient computation of ERIs over Gaussian functions (GTO-ERIs). Here, the well-known McMurchie–Davidson method (MD) offers an elegant formalism by incrementally extending Hermite Gaussian functions and auxiliary tabulated functions. Although the MD method offers a high degree of versatility to acceleration schemes through Graphics Processing Units (GPUs), the current GPU implementations limit the practical use of supported values of the azimuthal quantum number. In this paper, we propose a generalized framework capable of computing GTO-ERIs for arbitrary azimuthal quantum numbers, provided that the intermediate terms of the MD method can be stored. Our approach benefits from extending the MD recurrence relations through shells, batches, and triple-buffering of the shared memory, and ordering similar ERIs, thus enabling the effective parallelization and use of GPU resources. Furthermore, our approach proposes four GPU implementation schemes considering the suitable mappings between Gaussian basis and CUDA blocks and threads. Our computational experiments involving the GTO-ERI computations of molecules of interest on an NVIDIA A100 Tensor Core GPU (NVIDIA, Santa Clara, CA, USA) have revealed the merits of the proposed acceleration schemes in terms of computation time, including up to a 72× improvement over our previous GPU implementation and up to a 4500× speedup compared to a naive CPU implementation, highlighting the effectiveness of our method in accelerating ERI computations for both monatomic and polyatomic molecules. Our work has the potential to explore new parallelization schemes of distinct and complex computation paths involved in ERI computation. Full article
(This article belongs to the Special Issue Data Structures for Graphics Processing Units (GPUs))
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Figure 1

Figure 1
<p>Example of the configuration of two basis functions <math display="inline"><semantics> <msub> <mi>χ</mi> <mn>1</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>χ</mi> <mn>2</mn> </msub> </semantics></math> (<math display="inline"><semantics> <mrow> <mi>M</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>). (<b>a</b>) The quartet combinations, and (<b>b</b>) the symmetry-based combinations for Basis-ERIs when <math display="inline"><semantics> <mrow> <mi>M</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>. By considering symmetrical relations, it becomes possible to reduce the number of Basis-ERIs, as shown by the upper triangular matrix in (<b>b</b>).</p>
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<p>Example of the relation between the Basis-ERIs and the GTO-ERIs. The row (column) directions represent the <span class="html-italic">bra</span> (<span class="html-italic">ket</span>) Basis-ERIs. Each cell in the upper triangular matrix corresponds to a single Basis-ERI.</p>
Full article ">Figure 3
<p>Basic idea behind the definition of shell-based ERIs. The term <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>s</mi> <mi>s</mi> <mo>|</mo> </mrow> </semantics></math> implies that a <span class="html-italic">bra</span> consists of two s-shells, and the term <math display="inline"><semantics> <mrow> <mo>|</mo> <mi>s</mi> <mi>p</mi> <mo>)</mo> </mrow> </semantics></math> implies that a <span class="html-italic">ket</span> consists of one s-shell and one p-shell. The integral <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>s</mi> <mi>s</mi> <mo>|</mo> <mi>s</mi> <mi>p</mi> <mo>)</mo> </mrow> </semantics></math> consists of three GTO-ERIs: <math display="inline"><semantics> <mrow> <mo>[</mo> <mi>s</mi> <mi>s</mi> <mo>|</mo> <mi>s</mi> <msub> <mi>p</mi> <mi>x</mi> </msub> <mo>]</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>[</mo> <mi>s</mi> <mi>s</mi> <mo>|</mo> <mi>s</mi> <msub> <mi>p</mi> <mi>y</mi> </msub> <mo>]</mo> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mo>[</mo> <mi>s</mi> <mi>s</mi> <mo>|</mo> <mi>s</mi> <msub> <mi>p</mi> <mi>z</mi> </msub> <mo>]</mo> </mrow> </semantics></math>.</p>
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<p>Basic idea of the dependencies, denoted by arrows, behind computing the values of the corresponding recurrences <span class="html-italic">R</span> by using <span class="html-italic">batch</span> concepts when <math display="inline"><semantics> <mrow> <mi>K</mi> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math>. The values required by the MD method are highlighted in <span style="color: #FF0000">red</span>.</p>
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<p>Basic idea of the computation of <span class="html-italic">R</span> values for each batch using triple-buffering of the shared memory.</p>
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<p>Comparison of the required size of <span class="html-italic">shared memory</span> to store <span class="html-italic">R</span> values.</p>
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<p>Basic idea of the parallel thread assignment of CUDA blocks and CUDA threads to the Basis-ERI computation in BBM.</p>
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<p>Basic idea of the parallel thread assignment of CUDA blocks and CUDA threads to the Basis-ERI computation in BTM.</p>
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<p>Parallel thread assignment of CUDA blocks and CUDA threads to the shell-based ERI computation in SBM.</p>
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<p>Basic idea behind the parallel thread assignment of CUDA blocks and CUDA threads to the shell-based ERI computation in STM.</p>
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<p>Schematic of the 64-bit key for sorting the Basis-ERIs.</p>
Full article ">
18 pages, 1334 KiB  
Article
Transient Dynamics and Homogenization in Incoherent Collision Models
by Göktuğ Karpat and Barış Çakmak
Entropy 2025, 27(2), 206; https://doi.org/10.3390/e27020206 - 15 Feb 2025
Viewed by 252
Abstract
Collision models have attracted significant attention in recent years due to their versatility to simulate open quantum systems in different dynamical regimes. They have been used to study various interesting phenomena such as the dynamical emergence of non-Markovian memory effects and the spontaneous [...] Read more.
Collision models have attracted significant attention in recent years due to their versatility to simulate open quantum systems in different dynamical regimes. They have been used to study various interesting phenomena such as the dynamical emergence of non-Markovian memory effects and the spontaneous establishment of synchronization in open quantum systems. In such models, the repeated pairwise interactions between the system and the environment and also the possible coupling between different environmental units are typically modeled using the coherent partial SWAP (PSWAP) operation as it is known to be a universal homogenizer. In this study, we investigate the dynamical behavior of incoherent collision models, where the interactions between different units are modeled by the incoherent controlled SWAP (CSWAP) operation, which is also a universal homogenizer. Even though the asymptotic dynamics of the open system in cases of both coherent and incoherent swap interactions appear to be identical, its transient dynamics turns out to be significantly different. Here, we present a comparative analysis of the consequences of having coherent or incoherent couplings in collision models, namely, PSWAP or CSWAP interactions, respectively, for the emergence of memory effects for a single-qubit system and for the onset synchronization between a pair of qubits, both of which are strictly determined by the transient dynamics of the open system. Full article
(This article belongs to the Special Issue Simulation of Open Quantum Systems)
Show Figures

Figure 1

Figure 1
<p>For the coherent PSWAP interaction between the system qubit s and the environment qubits <math display="inline"><semantics> <msub> <mi>e</mi> <mi>i</mi> </msub> </semantics></math> without intra–environment couplings, <math display="inline"><semantics> <mrow> <msub> <mi>γ</mi> <mrow> <mi>e</mi> <mi>e</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>γ</mi> <mrow> <mi>s</mi> <mi>e</mi> </mrow> </msub> <mo>=</mo> <mn>0.05</mn> <mrow> <mo>(</mo> <mi>π</mi> <mo>/</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow> </semantics></math>, (<b>a</b>) shows the evolution of the coherence <math display="inline"><semantics> <mrow> <mi>C</mi> <mo>(</mo> <msub> <mi>ρ</mi> <mi>s</mi> </msub> <mo>)</mo> </mrow> </semantics></math> in the open system, the entropy <math display="inline"><semantics> <mrow> <mi>S</mi> <mo>(</mo> <msub> <mi>ρ</mi> <mi>s</mi> </msub> <mo>)</mo> </mrow> </semantics></math> of the open system, and the fidelity <math display="inline"><semantics> <mrow> <mi>F</mi> <mo>(</mo> <msub> <mi>ρ</mi> <mi>s</mi> </msub> <mo>,</mo> <msub> <mi>ρ</mi> <mi>e</mi> </msub> <mo>)</mo> </mrow> </semantics></math> between the open system and the initial state of the environment qubits <math display="inline"><semantics> <mrow> <msub> <mi>ρ</mi> <mi>e</mi> </msub> <mo>=</mo> <mrow> <mo>|</mo> <mn>0</mn> <mo>〉</mo> <mo>〈</mo> <mn>0</mn> <mo>|</mo> </mrow> </mrow> </semantics></math> for <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>1100</mn> </mrow> </semantics></math> collisions. (<b>b</b>) displays the path of the open system state s through the Bloch ball starting from the state <math display="inline"><semantics> <mrow> <msub> <mi>ρ</mi> <mi>s</mi> </msub> <mo>=</mo> <mrow> <mo>|</mo> <mo>+</mo> <mo>〉</mo> <mo>〈</mo> <mo>+</mo> <mo>|</mo> </mrow> </mrow> </semantics></math>. On the other hand, (<b>c</b>) and (<b>d</b>) display the same set of plots as in (<b>a</b>) and (<b>b</b>) when the interaction between the system and the environment is described by the incoherent CSWAP coupling with the same interaction parameters, that is, <math display="inline"><semantics> <mrow> <msub> <mi>γ</mi> <mrow> <mi>e</mi> <mi>e</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>γ</mi> <mrow> <mi>s</mi> <mi>e</mi> </mrow> </msub> <mo>=</mo> <mn>0.05</mn> <mrow> <mo>(</mo> <mi>π</mi> <mo>/</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow> </semantics></math>.</p>
Full article ">Figure 2
<p>While (<b>a</b>) shows the dynamics of the trace distance for the PSWAP–PSWAP collision model, together with the evolution of the non-Markovianity measure <math display="inline"><semantics> <msub> <mi mathvariant="script">N</mi> <mi>D</mi> </msub> </semantics></math> shown in the inset for <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>1200</mn> </mrow> </semantics></math> collisions, (<b>c</b>) displays the same set of plots in the case of the PSWAP–CSWAP collision model for same number of collisions. We take the initial system state pair as <math display="inline"><semantics> <mrow> <mrow> <mo>|</mo> <mo>±</mo> <mo>〉</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>/</mo> <msqrt> <mn>2</mn> </msqrt> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mo>|</mo> <mn>0</mn> <mo>〉</mo> </mrow> <mo>±</mo> <mrow> <mo>|</mo> <mn>1</mn> <mo>〉</mo> <mo>)</mo> </mrow> </mrow> </semantics></math>, and set the system–environment and the intra-environment coupling strengths identically in both models as <math display="inline"><semantics> <mrow> <msub> <mi>γ</mi> <mrow> <mi>s</mi> <mi>e</mi> </mrow> </msub> <mo>=</mo> <mn>0.05</mn> <mrow> <mo>(</mo> <mi>π</mi> <mo>/</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>γ</mi> <mrow> <mi>e</mi> <mi>e</mi> </mrow> </msub> <mo>=</mo> <mn>0.93</mn> <mrow> <mo>(</mo> <mi>π</mi> <mo>/</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow> </semantics></math>. In (<b>b</b>) and (<b>d</b>), we demonstrate the paths of evolution of the Bloch vectors through the Bloch ball for the PSWAP–PSWAP and PSWAP–CSWAP models, respectively, starting from the initial system state <math display="inline"><semantics> <mrow> <msub> <mi>ρ</mi> <mi>s</mi> </msub> <mo>=</mo> <mrow> <mo>|</mo> <mo>+</mo> <mo>〉</mo> <mo>〈</mo> <mo>+</mo> <mo>|</mo> </mrow> </mrow> </semantics></math>, with the same interaction parameters.</p>
Full article ">Figure 3
<p>Non-Markovianity diagrams for (<b>a</b>) the PSWAP-PSWAP and (<b>b</b>) the PSWAP-CSWAP collision models in terms of the system–environment and the intra-environment interaction parameters, <math display="inline"><semantics> <msub> <mi>γ</mi> <mrow> <mi>s</mi> <mi>e</mi> </mrow> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>γ</mi> <mrow> <mi>e</mi> <mi>e</mi> </mrow> </msub> </semantics></math>. For both models, we simulate the dynamics for <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>12,000</mn> </mrow> </semantics></math> iterations, and the state pair used in the calculation of the non-Markovianity measure <math display="inline"><semantics> <msub> <mi mathvariant="script">N</mi> <mi>D</mi> </msub> </semantics></math> is fixed as <math display="inline"><semantics> <mrow> <mrow> <mo>|</mo> <mo>±</mo> <mo>〉</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>/</mo> <msqrt> <mn>2</mn> </msqrt> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mo>|</mo> <mn>0</mn> <mo>〉</mo> </mrow> <mo>±</mo> <mrow> <mo>|</mo> <mn>1</mn> <mo>〉</mo> <mo>)</mo> </mrow> </mrow> </semantics></math>.</p>
Full article ">Figure 4
<p>While (<b>a</b>) shows the dynamics of the trace distance for the CSWAP–CSWAP collision model, together with the evolution of the non–Markovianity measure <math display="inline"><semantics> <msub> <mi mathvariant="script">N</mi> <mi>D</mi> </msub> </semantics></math> shown in the inset for <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>1200</mn> </mrow> </semantics></math> collisions, (<b>c</b>) displays the same set of plots in the case of the CSWAP–PSWAP collision model for same number of collisions. We take the initial system state pair as <math display="inline"><semantics> <mrow> <mrow> <mo>|</mo> <mo>±</mo> <mo>〉</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>/</mo> <msqrt> <mn>2</mn> </msqrt> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mo>|</mo> <mn>0</mn> <mo>〉</mo> </mrow> <mo>±</mo> <mrow> <mo>|</mo> <mn>1</mn> <mo>〉</mo> <mo>)</mo> </mrow> </mrow> </semantics></math>, and set the system–environment and the intra-environment coupling strengths identically in both models as <math display="inline"><semantics> <mrow> <msub> <mi>γ</mi> <mrow> <mi>s</mi> <mi>e</mi> </mrow> </msub> <mo>=</mo> <mn>0.05</mn> <mrow> <mo>(</mo> <mi>π</mi> <mo>/</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>γ</mi> <mrow> <mi>e</mi> <mi>e</mi> </mrow> </msub> <mo>=</mo> <mn>0.93</mn> <mrow> <mo>(</mo> <mi>π</mi> <mo>/</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow> </semantics></math>. In (<b>b</b>) and (<b>d</b>), we demonstrate the paths of evolution of the Bloch vectors through the Bloch ball for the CSWAP–CSWAP and CSWAP–PSWAP models, respectively, starting from the initial system state <math display="inline"><semantics> <mrow> <msub> <mi>ρ</mi> <mi>s</mi> </msub> <mo>=</mo> <mrow> <mo>|</mo> <mo>+</mo> <mo>〉</mo> <mo>〈</mo> <mo>+</mo> <mo>|</mo> </mrow> </mrow> </semantics></math>, with the same interaction parameters.</p>
Full article ">Figure 5
<p>System particles are initialized as <math display="inline"><semantics> <mrow> <msub> <mi>ρ</mi> <mrow> <msub> <mi>s</mi> <mn>1</mn> </msub> <msub> <mi>s</mi> <mn>2</mn> </msub> </mrow> </msub> <mo>=</mo> <mrow> <mo>|</mo> <mo>+</mo> <mo>〉</mo> <mo>|</mo> <mi>L</mi> <mo>〉</mo> <mo>〈</mo> <mi>L</mi> <mo>|</mo> <mo>〈</mo> <mo>+</mo> <mo>|</mo> </mrow> </mrow> </semantics></math> and resonant such that <math display="inline"><semantics> <mrow> <msub> <mi>ω</mi> <mn>1</mn> </msub> <mspace width="-0.166667em"/> <mo>=</mo> <mspace width="-0.166667em"/> <msub> <mi>ω</mi> <mn>2</mn> </msub> <mspace width="-0.166667em"/> <mo>=</mo> <mspace width="-0.166667em"/> <mn>1</mn> </mrow> </semantics></math>, and both interact with a common environmental unit through a coherent PSWAP having strength <math display="inline"><semantics> <mrow> <msub> <mi>γ</mi> <mrow> <mi>s</mi> <mi>e</mi> </mrow> </msub> <mo>=</mo> <mn>0.03</mn> <mrow> <mo>(</mo> <mi>π</mi> <mo>/</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow> </semantics></math>. While (<b>a</b>) displays the dynamics of <math display="inline"><semantics> <mrow> <mo>〈</mo> <msubsup> <mi>σ</mi> <msub> <mi>s</mi> <mn>1</mn> </msub> <mi>x</mi> </msubsup> <mo>〉</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mo>〈</mo> <msubsup> <mi>σ</mi> <msub> <mi>s</mi> <mn>2</mn> </msub> <mi>x</mi> </msubsup> <mo>〉</mo> </mrow> </semantics></math>, (<b>b</b>) shows the corresponding Pearson coefficient <math display="inline"><semantics> <msub> <mi>C</mi> <mrow> <mi>x</mi> <mi>y</mi> </mrow> </msub> </semantics></math> between these two data sets settling to <math display="inline"><semantics> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </semantics></math> signaling anti-synchronization, which is plotted considering data windows of 100 collisions with partial overlaps of 50 collisions for N = 2500. In (<b>c</b>), we show the fidelity F between the state of the environmental units <math display="inline"><semantics> <msub> <mi>ρ</mi> <mi>e</mi> </msub> </semantics></math> and both the local states of the system particles <math display="inline"><semantics> <msub> <mi>ρ</mi> <msub> <mi>s</mi> <mn>1</mn> </msub> </msub> </semantics></math>, <math display="inline"><semantics> <msub> <mi>ρ</mi> <msub> <mi>s</mi> <mn>2</mn> </msub> </msub> </semantics></math> and their global state <math display="inline"><semantics> <msub> <mi>ρ</mi> <mrow> <msub> <mi>s</mi> <mn>1</mn> </msub> <msub> <mi>s</mi> <mn>2</mn> </msub> </mrow> </msub> </semantics></math>. We observe a clear convergence towards <math display="inline"><semantics> <mrow> <mi>F</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, which indicates homogenization of system particles with the environment.</p>
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<p>System particles are initialized as <math display="inline"><semantics> <mrow> <msub> <mi>ρ</mi> <mrow> <msub> <mi>s</mi> <mn>1</mn> </msub> <msub> <mi>s</mi> <mn>2</mn> </msub> </mrow> </msub> <mo>=</mo> <mrow> <mo>|</mo> <mo>+</mo> <mo>〉</mo> <mo>|</mo> <mi>L</mi> <mo>〉</mo> <mo>〈</mo> <mi>L</mi> <mo>|</mo> <mo>〈</mo> <mo>+</mo> <mo>|</mo> </mrow> </mrow> </semantics></math> and resonant such that <math display="inline"><semantics> <mrow> <msub> <mi>ω</mi> <mn>1</mn> </msub> <mspace width="-0.166667em"/> <mo>=</mo> <mspace width="-0.166667em"/> <msub> <mi>ω</mi> <mn>2</mn> </msub> <mspace width="-0.166667em"/> <mo>=</mo> <mspace width="-0.166667em"/> <mn>1</mn> </mrow> </semantics></math>, and both interact with a common environmental unit through an incoherent CSWAP having strength <math display="inline"><semantics> <mrow> <msub> <mi>γ</mi> <mrow> <mi>s</mi> <mi>e</mi> </mrow> </msub> <mo>=</mo> <mn>0.03</mn> <mrow> <mo>(</mo> <mi>π</mi> <mo>/</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow> </semantics></math>. While (<b>a</b>) displays the dynamics of <math display="inline"><semantics> <mrow> <mo>〈</mo> <msubsup> <mi>σ</mi> <msub> <mi>s</mi> <mn>1</mn> </msub> <mi>x</mi> </msubsup> <mo>〉</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mo>〈</mo> <msubsup> <mi>σ</mi> <msub> <mi>s</mi> <mn>2</mn> </msub> <mi>x</mi> </msubsup> <mo>〉</mo> </mrow> </semantics></math>, (<b>b</b>) displays the corresponding Pearson coefficient <math display="inline"><semantics> <msub> <mi>C</mi> <mrow> <mi>x</mi> <mi>y</mi> </mrow> </msub> </semantics></math> between these two data sets, which is plotted considering data windows of 100 collisions with partial overlaps of 50 collisions for N = 2500, showing no sign of synchronization. In (<b>c</b>), we show the fidelity F between the state of the environmental units <math display="inline"><semantics> <msub> <mi>ρ</mi> <mi>e</mi> </msub> </semantics></math> and both the local states of the system particles <math display="inline"><semantics> <msub> <mi>ρ</mi> <msub> <mi>s</mi> <mn>1</mn> </msub> </msub> </semantics></math>, <math display="inline"><semantics> <msub> <mi>ρ</mi> <msub> <mi>s</mi> <mn>2</mn> </msub> </msub> </semantics></math> and their global state <math display="inline"><semantics> <msub> <mi>ρ</mi> <mrow> <msub> <mi>s</mi> <mn>1</mn> </msub> <msub> <mi>s</mi> <mn>2</mn> </msub> </mrow> </msub> </semantics></math>. We observe a clear convergence towards <math display="inline"><semantics> <mrow> <mi>F</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, which indicates the homogenization of system particles with the environment.</p>
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<p>System particles are initialized as <math display="inline"><semantics> <mrow> <msub> <mi>ρ</mi> <mrow> <msub> <mi>s</mi> <mn>1</mn> </msub> <msub> <mi>s</mi> <mn>2</mn> </msub> </mrow> </msub> <mo>=</mo> <mrow> <mo>|</mo> <mo>+</mo> <mo>〉</mo> <mo>|</mo> <mi>L</mi> <mo>〉</mo> <mo>〈</mo> <mi>L</mi> <mo>|</mo> <mo>〈</mo> <mo>+</mo> <mo>|</mo> </mrow> </mrow> </semantics></math> and resonant such that <math display="inline"><semantics> <mrow> <msub> <mi>ω</mi> <mn>1</mn> </msub> <mspace width="-0.166667em"/> <mo>=</mo> <mspace width="-0.166667em"/> <msub> <mi>ω</mi> <mn>2</mn> </msub> <mspace width="-0.166667em"/> <mo>=</mo> <mspace width="-0.166667em"/> <mn>1</mn> </mrow> </semantics></math>. As the first particle interacts with environmental units through incoherent CSWAP, the second one interacts with the same environment units via a coherent PSWAP interaction, with identical strength <math display="inline"><semantics> <mrow> <msub> <mi>γ</mi> <mrow> <mi>s</mi> <mi>e</mi> </mrow> </msub> <mo>=</mo> <mn>0.03</mn> <mrow> <mo>(</mo> <mi>π</mi> <mo>/</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow> </semantics></math>. While (<b>a</b>) displays the dynamics of <math display="inline"><semantics> <mrow> <mo>〈</mo> <msubsup> <mi>σ</mi> <msub> <mi>s</mi> <mn>1</mn> </msub> <mi>x</mi> </msubsup> <mo>〉</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mo>〈</mo> <msubsup> <mi>σ</mi> <msub> <mi>s</mi> <mn>2</mn> </msub> <mi>x</mi> </msubsup> <mo>〉</mo> </mrow> </semantics></math>, (<b>b</b>) displays the corresponding Pearson coefficient <math display="inline"><semantics> <msub> <mi>C</mi> <mrow> <mi>x</mi> <mi>y</mi> </mrow> </msub> </semantics></math> between these two data sets, which is plotted considering data windows of 100 collisions with partial overlaps of 50 collisions for N = 2500, showing no sign of synchronization. In (<b>c</b>), we show the fidelity F between the state of the environmental units <math display="inline"><semantics> <msub> <mi>ρ</mi> <mi>e</mi> </msub> </semantics></math> and both the local states of the system particles <math display="inline"><semantics> <msub> <mi>ρ</mi> <msub> <mi>s</mi> <mn>1</mn> </msub> </msub> </semantics></math>, <math display="inline"><semantics> <msub> <mi>ρ</mi> <msub> <mi>s</mi> <mn>2</mn> </msub> </msub> </semantics></math> and their global state <math display="inline"><semantics> <msub> <mi>ρ</mi> <mrow> <msub> <mi>s</mi> <mn>1</mn> </msub> <msub> <mi>s</mi> <mn>2</mn> </msub> </mrow> </msub> </semantics></math>. We observe a clear convergence towards <math display="inline"><semantics> <mrow> <mi>F</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, confirming the homogenization of system particles with the environment.</p>
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29 pages, 5846 KiB  
Article
Explainable AI-Driven Quantum Deep Neural Network for Fault Location in DC Microgrids
by Amir Hossein Poursaeed and Farhad Namdari
Energies 2025, 18(4), 908; https://doi.org/10.3390/en18040908 - 13 Feb 2025
Viewed by 539
Abstract
Fault location in DC microgrids (DCMGs) is a critical challenge due to the system’s inherent complexities and the demand for high reliability in modern power systems. This study proposes an explainable artificial intelligence (XAI)-based quantum deep neural network (QDNN) framework to address fault [...] Read more.
Fault location in DC microgrids (DCMGs) is a critical challenge due to the system’s inherent complexities and the demand for high reliability in modern power systems. This study proposes an explainable artificial intelligence (XAI)-based quantum deep neural network (QDNN) framework to address fault localization challenges in DCMGs. First, voltage signals from the DCMG are collected and analyzed using high-order synchrosqueezing transform to detect traveling waves (TWs) and extract critical fault parameters such as time of arrival, magnitude, and polarity of the first and second TWs. These features are fed into the proposed QDNN model that integrates advanced learning techniques for accurate fault localization. The cumulative distance from the fault point to the bus connecting the DCMG to the power network is considered the output vector. The model uses a combination of deep learning and quantum computing techniques to extract features and improve accuracy. To ensure transparency, an XAI technique called Shapley additive explanations (SHAP) is applied, enabling system operators to identify critical fault features. The SHAP-based explainability framework plays a critical role in translating the model’s predictions into actionable insights, ensuring that the proposed solution is not only accurate but also practically implementable in real-world scenarios. The results demonstrate the QDNN framework’s superior accuracy in fault localization even in noisy environments and with high-resistance faults, independent of voltage levels and DCMG configurations, making it a robust solution for modern power systems. Full article
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<p>Schematic representation of HOSST.</p>
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<p>Proposed QDNN Architecture for fault detection and location in DCMG. The architecture consists of two primary processing paths: (1) a CNN-based feature extraction module, responsible for capturing spatial patterns in fault signal characteristics, and (2) a QNN-enhanced quantum feature extraction module, leveraging quantum principles to improve fault location accuracy. The extracted features are then combined and processed by a BD-LSTM layer, which learns sequential dependencies from past and future data points. The attention mechanism further refines the learned features by emphasizing the most relevant temporal characteristics, ensuring robust decision-making under varying fault conditions.</p>
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<p>The flowchart of the proposed method.</p>
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<p>Overview of the proposed method.</p>
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<p>Schematic representation of the case study system, a medium-voltage DCMG.</p>
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<p>Visualization of QNN-extracted features in a 3D PCA-projected space.</p>
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<p>SHAP analysis of the DNN-based fault location model, demonstrating the influence of different features on model predictions. (<b>a</b>) The SHAP summary plot illustrates that the TOA of the first TW has the highest impact on the model’s output, with its wide SHAP value range confirming its critical role in fault localization. Other important features include the magnitude of the first and second TWs, while TW polarity has a relatively smaller impact. (<b>b</b>) The feature importance bar chart confirms that the DNN primarily relies on temporal features, which may limit its ability to capture more complex relationships compared to the QDNN.</p>
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<p>SHAP analysis for the QDNN-based fault location model, highlighting the advantages of quantum-enhanced features. (<b>a</b>) The SHAP summary plot shows that QNN Feature 2 contributes the most to fault location predictions, reflecting the power of quantum feature extraction in capturing complex fault patterns. Other QNN-derived features, including QNN Features 1, 3, 4, 5, and 6, also play a role but with a slightly lower impact. The model exhibits more concentrated SHAP values compared to DNN, indicating its improved fault representation capability. (<b>b</b>) The feature importance bar chart confirms that QDNN prioritizes quantum-derived features over conventional temporal indicators, showcasing its robustness in fault identification across varying conditions.</p>
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<p>Correlation plots comparing predicted versus actual values for the test dataset across six models: (<b>a</b>) QDNN; (<b>b</b>) XGB; (<b>c</b>) KNN; (<b>d</b>) RNN; (<b>e</b>) SVR; (<b>f</b>) MLP.</p>
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<p>Correlation plots comparing predicted versus actual values for the test dataset across six models: (<b>a</b>) QDNN; (<b>b</b>) XGB; (<b>c</b>) KNN; (<b>d</b>) RNN; (<b>e</b>) SVR; (<b>f</b>) MLP.</p>
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<p>RROC curves for various datasets: (<b>a</b>) Training; (<b>b</b>) Validation; (<b>c</b>) Test.</p>
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<p>SHAP summary plots for the QDNN model under varying fault resistances: (<b>a</b>) 10 Ω; (<b>b</b>) <span style="lang:ar">50</span> Ω; (<b>c</b>) 1<span style="lang:ar">0</span>0 Ω.</p>
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<p>SHAP summary plots for the QDNN model under different noise levels: (<b>a</b>) 5 dB; (<b>b</b>) 10 dB; (<b>c</b>) 20 dB.</p>
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20 pages, 355 KiB  
Article
On the Yang-Mills Propagator at Strong Coupling
by Yves Gabellini, Thierry Grandou and Ralf Hofmann
Universe 2025, 11(2), 56; https://doi.org/10.3390/universe11020056 - 10 Feb 2025
Viewed by 335
Abstract
About twelve years ago, the use of standard functional manipulations was demonstrated to imply an unexpected property satisfied by the fermionic Green’s functions of QCD. This non-perturbative phenomenon has been dubbed an effective locality. In a much simpler way [...] Read more.
About twelve years ago, the use of standard functional manipulations was demonstrated to imply an unexpected property satisfied by the fermionic Green’s functions of QCD. This non-perturbative phenomenon has been dubbed an effective locality. In a much simpler way than in QCD, the most remarkable and intriguing aspects of effective locality have been presented in a recent publication on the Yang-Mills theory on Minkowski spacetime. While quickly recalled in the current paper, these results are used to calculate the problematic gluonic propagator in the Yang-Mills non-perturbative regime. This paper is dedicated to the memory of Professor Herbert M. Fried (1929–2023), whose inspiring manner, impressive command of functional methods in quantum field theories, enthusiasm for a broad range of topics in Theoretical Physics, and warm friendship are missed greatly by the authors. Full article
(This article belongs to the Special Issue Quantum Field Theory, 2nd Edition)
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<p>The integration contour displaying the analyticity domain of the integrals of (<a href="#FD54-universe-11-00056" class="html-disp-formula">54</a>) and (<a href="#FD55-universe-11-00056" class="html-disp-formula">55</a>).</p>
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22 pages, 2728 KiB  
Article
Hybrid Dynamic Galois Field with Quantum Resilience for Secure IoT Data Management and Transmission in Smart Cities Using Reed–Solomon (RS) Code
by Abdullah Aljuhni, Amer Aljaedi, Adel R. Alharbi, Ahmed Mubaraki and Moahd K. Alghuson
Symmetry 2025, 17(2), 259; https://doi.org/10.3390/sym17020259 - 8 Feb 2025
Viewed by 540
Abstract
The Internet of Things (IoT), which is characteristic of the current industrial revolutions, is the connection of physical devices through different protocols and sensors to share information. Even though the IoT provides revolutionary opportunities, its connection to the current Internet for smart cities [...] Read more.
The Internet of Things (IoT), which is characteristic of the current industrial revolutions, is the connection of physical devices through different protocols and sensors to share information. Even though the IoT provides revolutionary opportunities, its connection to the current Internet for smart cities brings new opportunities for security threats, especially with the appearance of new threats like quantum computing. Current approaches to protect IoT data are not immune to quantum attacks and are not designed to offer the best data management for smart city applications. Thus, post-quantum cryptography (PQC), which is still in its research stage, aims to solve these problems. To this end, this research introduces the Dynamic Galois Reed–Solomon with Quantum Resilience (DGRS-QR) system to improve the secure management and communication of data in IoT smart cities. The data preprocessing includes K-Nearest Neighbors (KNN) and min–max normalization and then applying the Galois Field Adaptive Expansion (GFAE). Optimization of the quantum-resistant keys is accomplished by applying Artificial Bee Colony (ABC) and Moth Flame Optimization (MFO) algorithms. Also, role-based access control provides strong cloud data security, and quantum resistance is maintained by refreshing keys every five minutes of the active session. For error correction, Reed–Solomon (RS) codes are used which provide data reliability. Data management is performed using an attention-based Bidirectional Long Short-Term Memory (Att-Bi-LSTM) model with skip connections to provide optimized city management. The proposed approach was evaluated using key performance metrics: a key generation time of 2.34 s, encryption time of 4.56 s, decryption time of 3.56 s, PSNR of 33 dB, and SSIM of 0.99. The results show that the proposed system is capable of protecting IoT data from quantum threats while also ensuring optimal data management and processing. Full article
(This article belongs to the Special Issue New Advances in Symmetric Cryptography)
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<p>Overview of IoT-enabled smart city ecosystem.</p>
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<p>Architecture of proposed DGRS-QR technique.</p>
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<p>Att-Bi-LSTM network architecture.</p>
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<p>Comparison of encryption time.</p>
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<p>Comparison of decryption time.</p>
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<p>Comparison of key generation time.</p>
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<p>Comparison of PSNR.</p>
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<p>Comparison of SSIM.</p>
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23 pages, 909 KiB  
Article
Extending the QMM Framework to the Strong and Weak Interactions
by Florian Neukart, Eike Marx and Valerii Vinokur
Entropy 2025, 27(2), 153; https://doi.org/10.3390/e27020153 - 2 Feb 2025
Viewed by 387
Abstract
We extend the Quantum Memory Matrix (QMM) framework, originally developed to reconcile quantum mechanics and general relativity by treating space–time as a dynamic information reservoir, to incorporate the full suite of Standard Model gauge interactions. In this discretized, Planck-scale formulation, each space–time cell [...] Read more.
We extend the Quantum Memory Matrix (QMM) framework, originally developed to reconcile quantum mechanics and general relativity by treating space–time as a dynamic information reservoir, to incorporate the full suite of Standard Model gauge interactions. In this discretized, Planck-scale formulation, each space–time cell possesses a finite-dimensional Hilbert space that acts as a local memory, or quantum imprint, for matter and gauge field configurations. We focus on embedding non-Abelian SU(3)c (quantum chromodynamics) and SU(2)L × U(1)Y (electroweak interactions) into QMM by constructing gauge-invariant imprint operators for quarks, gluons, electroweak bosons, and the Higgs mechanism. This unified approach naturally enforces unitarity by allowing black hole horizons, or any high-curvature region, to store and later retrieve quantum information about color and electroweak charges, thereby preserving subtle non-thermal correlations in evaporation processes. Moreover, the discretized nature of QMM imposes a Planck-scale cutoff, potentially taming UV divergences and modifying running couplings at trans-Planckian energies. We outline major challenges, such as the precise formulation of non-Abelian imprint operators and the integration of QMM with loop quantum gravity, as well as possible observational strategies—ranging from rare decay channels to primordial black hole evaporation spectra—that could provide indirect probes of this discrete, memory-based view of quantum gravity and the Standard Model. Full article
(This article belongs to the Section Astrophysics, Cosmology, and Black Holes)
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<p>A 2D representation of discretized space–time at the Planck scale, where each cell is associated with a finite-dimensional Hilbert space <math display="inline"><semantics> <msub> <mi mathvariant="script">H</mi> <mi>x</mi> </msub> </semantics></math>. The label <math display="inline"><semantics> <msub> <mi mathvariant="script">H</mi> <mi>x</mi> </msub> </semantics></math> indicates the local degrees of freedom stored in that cell, while small icons (or “<math display="inline"><semantics> <msub> <mi>I</mi> <mi>x</mi> </msub> </semantics></math>” marks) represent possible imprint operators recording matter or gauge interactions. Time and space directions are only schematic here.</p>
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<p>A conceptual diagram of SU(3) color flux confinement in QMM. Two quarks (red and blue) are connected by a color flux tube. Discrete QMM cells along the line (represented by small squares) store local color imprints <math display="inline"><semantics> <mrow> <mo>{</mo> <msub> <mi>I</mi> <msub> <mi>x</mi> <mn>1</mn> </msub> </msub> <mo>,</mo> <msub> <mi>I</mi> <msub> <mi>x</mi> <mn>2</mn> </msub> </msub> <mo>,</mo> <mo>…</mo> <mo>}</mo> </mrow> </semantics></math> that enforce confinement.</p>
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<p>Flowchart illustrating the construction of the unified QMM Hamiltonian. Each sector (<span class="html-italic">Gravity</span>, <span class="html-italic">QCD</span>, <span class="html-italic">Electroweak</span>, and <span class="html-italic">QMM Memory</span>) feeds into the total Hamiltonian <math display="inline"><semantics> <mover accent="true"> <mi>H</mi> <mo stretchy="false">^</mo> </mover> </semantics></math>. The interaction Hamiltonian <math display="inline"><semantics> <msub> <mover accent="true"> <mi>H</mi> <mo stretchy="false">^</mo> </mover> <mi>int</mi> </msub> </semantics></math> couples the imprint degrees of freedom to the gauge and gravitational fields, ensuring local unitarity.</p>
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<p>Schematic view of a black hole formed from colored and electroweak-charged matter. As it evaporates, outgoing quanta (gluons, photons, leptons) interact with QMM horizon cells that store color/electroweak imprints, introducing subtle non-thermal correlations in the Hawking radiation.</p>
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10 pages, 349 KiB  
Article
Temperature Dependence of the Dynamical and DC Conductivity in 2D Dirac Systems: Self-Consistent Random-Phase-Approximation Approach
by Ivan Kupčić and Patrik Papac
Condens. Matter 2025, 10(1), 9; https://doi.org/10.3390/condmat10010009 - 1 Feb 2025
Viewed by 322
Abstract
We studied relaxation processes in heavily doped two-dimensional Dirac systems associated with electron scattering by acoustic and optical phonons and by static disorder. The frequency dependence of the real and imaginary parts of the relaxation function is calculated for different temperatures. The two-component [...] Read more.
We studied relaxation processes in heavily doped two-dimensional Dirac systems associated with electron scattering by acoustic and optical phonons and by static disorder. The frequency dependence of the real and imaginary parts of the relaxation function is calculated for different temperatures. The two-component low-frequency dynamical conductivity is found to be strongly dependent on temperature. At low temperatures, the imaginary part of the zero-frequency relaxation function and the DC resistivity are characterized by the scaling law aTx with the exponent x between 2.5 and 3. Full article
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<p>The expansion of the current–dipole correlation function <math display="inline"><semantics> <mrow> <msub> <mi>σ</mi> <mrow> <mi>α</mi> <mi>α</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">q</mi> <mo>,</mo> <mi>ω</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> in powers of <math display="inline"><semantics> <mrow> <msub> <mi>G</mi> <mi>ν</mi> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">k</mi> <mo>,</mo> <msup> <mrow> <mi mathvariant="bold">k</mi> </mrow> <mo>′</mo> </msup> <mo>)</mo> </mrow> </mrow> </semantics></math> [<a href="#B3-condensedmatter-10-00009" class="html-bibr">3</a>,<a href="#B13-condensedmatter-10-00009" class="html-bibr">13</a>]. The solid/dashed lines illustrate the bare electron–phonon propagators. (<b>a</b>) The first two <math display="inline"><semantics> <msup> <mi>λ</mi> <mn>2</mn> </msup> </semantics></math> contributions (<math display="inline"><semantics> <mrow> <mn>2</mn> <msub> <mi>A</mi> <mn>1</mn> </msub> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mn>2</mn> <msub> <mi>A</mi> <mn>2</mn> </msub> </mrow> </semantics></math>) represent two self-energy contributions and the last one (<math display="inline"><semantics> <mrow> <mn>2</mn> <mi>B</mi> </mrow> </semantics></math>) the corresponding vertex correction. (<b>b</b>) The first quartet of <math display="inline"><semantics> <msup> <mi>λ</mi> <mn>4</mn> </msup> </semantics></math> contributions consists of one self-energy term and three terms with vertex corrections.</p>
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<p>The real and imaginary parts of the <math display="inline"><semantics> <msup> <mi>λ</mi> <mn>2</mn> </msup> </semantics></math> memory function <math display="inline"><semantics> <mrow> <mi>M</mi> <mo>(</mo> <mi>ω</mi> <mo>)</mo> </mrow> </semantics></math> as a function of temperature. (<b>a</b>) The 2D Holstein model with the parabolic electron dispersion (<math display="inline"><semantics> <msub> <mi>λ</mi> <mn>0</mn> </msub> </semantics></math> is a convenient energy scale from Ref. [<a href="#B18-condensedmatter-10-00009" class="html-bibr">18</a>]). (<b>b</b>) Heavily electron-doped 2D massless Dirac system, the electron scattering by optical phonons, for <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>2.52</mn> </mrow> </semantics></math> eV, <math display="inline"><semantics> <mrow> <msub> <mi>E</mi> <mi mathvariant="normal">F</mi> </msub> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math> eV, <math display="inline"><semantics> <mrow> <mrow> <mo>|</mo> </mrow> <msub> <mi>G</mi> <mi>op</mi> </msub> <msup> <mrow> <mo>|</mo> </mrow> <mn>2</mn> </msup> <mo>≡</mo> <msub> <mi>α</mi> <mi>op</mi> </msub> <mo>=</mo> <mn>0.25</mn> </mrow> </semantics></math> eV<sup>2</sup>, <math display="inline"><semantics> <mrow> <mo>ℏ</mo> <msub> <mi>ω</mi> <mi mathvariant="normal">E</mi> </msub> <mo>=</mo> <mn>0.2</mn> </mrow> </semantics></math> eV, and <math display="inline"><semantics> <mrow> <mo>ℏ</mo> <msubsup> <mo>Σ</mo> <mrow> <mi>im</mi> </mrow> <mi>dis</mi> </msubsup> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math> meV.</p>
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<p>(<b>a</b>) The real and imaginary parts of the <math display="inline"><semantics> <msup> <mi>λ</mi> <mn>2</mn> </msup> </semantics></math> memory function <math display="inline"><semantics> <mrow> <mi>M</mi> <mo>(</mo> <mi>ω</mi> <mo>,</mo> <mi>T</mi> <mo>)</mo> </mrow> </semantics></math> as a function of temperature in heavily electron-doped 2D Dirac systems for the electron scattering by acoustic phonons, for <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>2.52</mn> </mrow> </semantics></math> eV, <math display="inline"><semantics> <mrow> <msub> <mi>E</mi> <mi mathvariant="normal">F</mi> </msub> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math> eV, <math display="inline"><semantics> <mrow> <mrow> <mo>|</mo> </mrow> <msub> <mi>G</mi> <mi>ac</mi> </msub> <msup> <mrow> <mrow> <mo>(</mo> <mi mathvariant="bold">q</mi> <mo>)</mo> </mrow> <mo>|</mo> </mrow> <mn>2</mn> </msup> <mo>/</mo> <mrow> <mo>(</mo> <mi>q</mi> <mi>a</mi> <mo>)</mo> </mrow> <mo>≡</mo> <msub> <mi>α</mi> <mi>ac</mi> </msub> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math> eV<sup>2</sup>, <math display="inline"><semantics> <mrow> <mo>ℏ</mo> <msub> <mi>ω</mi> <mi mathvariant="normal">D</mi> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">q</mi> <mo>)</mo> </mrow> <mo>/</mo> <mrow> <mo>(</mo> <mi>q</mi> <mi>a</mi> <mo>)</mo> </mrow> <mo>≡</mo> <msub> <mover accent="true"> <mi>c</mi> <mo>˜</mo> </mover> <mi mathvariant="normal">s</mi> </msub> <mo>=</mo> <mn>30</mn> </mrow> </semantics></math> meV, and <math display="inline"><semantics> <mrow> <mo>ℏ</mo> <msubsup> <mo>Σ</mo> <mrow> <mi>im</mi> </mrow> <mi>dis</mi> </msubsup> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math> meV. (<b>b</b>) The temperature dependence of <math display="inline"><semantics> <mrow> <mi>M</mi> <mo>(</mo> <mn>0</mn> <mo>,</mo> <mi>T</mi> <mo>)</mo> </mrow> </semantics></math> for three different values of <math display="inline"><semantics> <msub> <mover accent="true"> <mi>c</mi> <mo>˜</mo> </mover> <mi mathvariant="normal">s</mi> </msub> </semantics></math><math display="inline"><semantics> <mrow> <mo>(</mo> <mn>30</mn> <mo>,</mo> <mn>25</mn> </mrow> </semantics></math>, and 20 meV), and <math display="inline"><semantics> <mrow> <mo>ℏ</mo> <msubsup> <mo>Σ</mo> <mrow> <mi>im</mi> </mrow> <mi>dis</mi> </msubsup> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math> meV. (<b>c</b>) The temperature dependence of <math display="inline"><semantics> <mrow> <mi>M</mi> <mo>(</mo> <mn>0</mn> <mo>,</mo> <mi>T</mi> <mo>)</mo> </mrow> </semantics></math> for four different values of <math display="inline"><semantics> <mrow> <mo>ℏ</mo> <msubsup> <mo>Σ</mo> <mrow> <mi>im</mi> </mrow> <mi>dis</mi> </msubsup> </mrow> </semantics></math> (<math display="inline"><semantics> <mrow> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> </mrow> </semantics></math>, and 5 meV), and <math display="inline"><semantics> <mrow> <msub> <mover accent="true"> <mi>c</mi> <mo>˜</mo> </mover> <mi mathvariant="normal">s</mi> </msub> <mo>=</mo> <mn>30</mn> </mrow> </semantics></math> meV. The other parameters in (<b>b</b>,<b>c</b>) are the same as in (<b>a</b>). The temperature <math display="inline"><semantics> <msub> <mi>T</mi> <mn>2</mn> </msub> </semantics></math> is illustrated in (<b>b</b>) by arrows. At low temperatures, <math display="inline"><semantics> <mrow> <msub> <mi>M</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mn>0</mn> <mo>,</mo> <mi>T</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> follows the scaling law <math display="inline"><semantics> <mrow> <mo>ℏ</mo> <msub> <mi>M</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mn>0</mn> <mo>,</mo> <mi>T</mi> <mo>)</mo> </mrow> <mo>−</mo> <mo>ℏ</mo> <msub> <mi>M</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>)</mo> </mrow> <mo>≈</mo> <mi>a</mi> <msup> <mi>T</mi> <mi>x</mi> </msup> </mrow> </semantics></math>. In (<b>b</b>), the exponent <span class="html-italic">x</span> is approximately equal to 2.7 for all three curves. The characteristic temperatures <math display="inline"><semantics> <msub> <mi>T</mi> <mn>1</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>T</mi> <mn>2</mn> </msub> </semantics></math> are introduced in the text.</p>
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<p>The real part of the intraband dynamical conductivity <math display="inline"><semantics> <mrow> <msub> <mi>σ</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>ω</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> as a function of temperature calculated by using <math display="inline"><semantics> <mrow> <mi>M</mi> <mo>(</mo> <mi>ω</mi> <mo>)</mo> </mrow> </semantics></math> from <a href="#condensedmatter-10-00009-f003" class="html-fig">Figure 3</a>a. The dotted line is the ordinary Drude conductivity for <math display="inline"><semantics> <mrow> <mo>ℏ</mo> <mo>Γ</mo> <mo>=</mo> <mn>17.94</mn> </mrow> </semantics></math> meV.</p>
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17 pages, 4690 KiB  
Article
Advantages of Density in Tensor Network Geometries for Gradient-Based Training
by Sergi Masot-Llima and Artur Garcia-Saez
Algorithms 2025, 18(2), 70; https://doi.org/10.3390/a18020070 - 31 Jan 2025
Viewed by 480
Abstract
Tensor networks are a very powerful data structure tool originating from simulations of quantum systems. In recent years, they have seen increased use in machine learning, mostly in trainings with gradient-based techniques, due to their flexibility and performance achieved by exploiting hardware acceleration. [...] Read more.
Tensor networks are a very powerful data structure tool originating from simulations of quantum systems. In recent years, they have seen increased use in machine learning, mostly in trainings with gradient-based techniques, due to their flexibility and performance achieved by exploiting hardware acceleration. As ansatzes, tensor networks can be used with flexible geometries, and it is known that for highly regular ones, their dimensionality has a large impact on performance and representation power. For heterogeneous structures, however, these effects are not completely characterized. In this article, we train tensor networks with different geometries to encode a random quantum state, and see that densely connected structures achieve better infidelities than more sparse structures, with higher success rates and less time. Additionally, we give some general insight on how to improve the memory requirements of these sparse structures and the impact of such improvement on the trainings. Finally, as we use HPC resources for the calculations, we discuss the requirements for this approach and showcase performance improvements with GPU acceleration on a last-generation supercomputer. Full article
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<p>The TN geometries used in this work, with a few tree-like TNs (including an MPS) and the PEPS ansatz. The highlight indicates the longest path between tensors, which we use as a measure of density. For the PEPS, instead, it shows the longest <span class="html-italic">minimal</span> path between nodes when considering all node combinations. At the top is a single tensor equivalent to the contraction of any of the other geometries.</p>
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<p>The usual training process of a TN model using samples of a target unknown physical system, to the left, compared to our method with a surrogate TN, and to the right, where we control the number of correlations in a TN with its bond dimension, and then we contract it into a single dense tensor. Both approaches output a model of the physical system in the form of another TN (in the example, an MPS).</p>
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<p>Example using an MPS of the “compact” TN approach, where bonds smaller than <math display="inline"><semantics> <mi>χ</mi> </semantics></math> are contracted. This reduces the memory needed to store the tensor network, but increases the cost of the contraction for the compacted sites.</p>
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<p>Training performance measured with infidelity as a function of the largest tensor in the TN, for <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>12</mn> </mrow> </semantics></math> sites. The size of the tensors was controlled with bond dimension for each of the geometries in <a href="#algorithms-18-00070-f001" class="html-fig">Figure 1</a>. Trainings in the blue shaded area use a bond dimension <math display="inline"><semantics> <mrow> <mi>χ</mi> <mo>&gt;</mo> <msub> <mi>χ</mi> <mi>S</mi> </msub> </mrow> </semantics></math>. In the inset, the best infidelity reached against the maximum node distance of the TN structure is given. The ordering and colors follow that of the legend.</p>
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<p>Training performance measured with infidelity as a function of bond dimension, for <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>16</mn> </mrow> </semantics></math> sites, in (<b>a</b>). For some bond dimensions in the MPS, none of the trainings reach a good infidelity. The number of trainings that cross a <math display="inline"><semantics> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> </semantics></math> threshold for infidelity is plotted in (<b>b</b>), showing a decrease with the amount of information in the structure, as measured using the bond dimension <math display="inline"><semantics> <mi>χ</mi> </semantics></math>. The blue shaded area indicates <math display="inline"><semantics> <mrow> <mi>χ</mi> <mo>&gt;</mo> <msub> <mi>χ</mi> <mi>S</mi> </msub> </mrow> </semantics></math>, and colors in both plots follow the same legend.</p>
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<p>Training performance measured with infidelity as a function of the full size of the tensor network for the examples in <a href="#algorithms-18-00070-f001" class="html-fig">Figure 1</a>, and the compact version of each geometry that follows from <a href="#algorithms-18-00070-f003" class="html-fig">Figure 3</a>. The last data points for the compact TNs decrease in size because the compact threshold is high enough to contract them into a single tensor.</p>
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<p>Example of training with <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>16</mn> </mrow> </semantics></math> sites for each geometry and using low bond dimension to compare them with the compact tensor network structures. Solid (dashed) lines represent the best regular (compact) training, and dotted (dot-dashed) represent the regular (compact) median training, ordered according to final infidelity.</p>
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<p>The time cost (left axis, points) and energy cost (right axis, bars) of the training in this work for different sizes of the system, adding together all reviewed geometries. We use 20 CPU cores and 20 CPU cores + GPU, both with high and reduced precision, labeled as “CPU”, “GPU”, and “GPU low”, respectively. In the inset, the average power needed for each training is presented.</p>
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29 pages, 32678 KiB  
Article
An Active Control Method for a Lower Limb Rehabilitation Robot with Human Motion Intention Recognition
by Zhuangqun Song, Peng Zhao, Xueji Wu, Rong Yang and Xueshan Gao
Sensors 2025, 25(3), 713; https://doi.org/10.3390/s25030713 - 24 Jan 2025
Viewed by 383
Abstract
This study presents a method for the active control of a follow-up lower extremity exoskeleton rehabilitation robot (LEERR) based on human motion intention recognition. Initially, to effectively support body weight and compensate for the vertical movement of the human center of mass, a [...] Read more.
This study presents a method for the active control of a follow-up lower extremity exoskeleton rehabilitation robot (LEERR) based on human motion intention recognition. Initially, to effectively support body weight and compensate for the vertical movement of the human center of mass, a vision-driven follow-and-track control strategy is proposed. Subsequently, an algorithm for recognizing human motion intentions based on machine learning is proposed for human-robot collaboration tasks. A muscle–machine interface is constructed using a bi-directional long short-term memory (BiLSTM) network, which decodes multichannel surface electromyography (sEMG) signals into flexion and extension angles of the hip and knee joints in the sagittal plane. The hyperparameters of the BiLSTM network are optimized using the quantum-behaved particle swarm optimization (QPSO) algorithm, resulting in a QPSO-BiLSTM hybrid model that enables continuous real-time estimation of human motion intentions. Further, to address the uncertain nonlinear dynamics of the wearer-exoskeleton robot system, a dual radial basis function neural network adaptive sliding mode Controller (DRBFNNASMC) is designed to generate control torques, thereby enabling the precise tracking of motion trajectories generated by the muscle–machine interface. Experimental results indicate that the follow-up-assisted frame can accurately track human motion trajectories. The QPSO-BiLSTM network outperforms traditional BiLSTM and PSO-BiLSTM networks in predicting continuous lower limb motion, while the DRBFNNASMC controller demonstrates superior gait tracking performance compared to the fuzzy compensated adaptive sliding mode control (FCASMC) algorithm and the traditional proportional–integral–derivative (PID) control algorithm. Full article
(This article belongs to the Section Wearables)
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<p>Active control framework of LEERR.</p>
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<p>OpenPose network structure.</p>
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<p>Actual results of human joint and limb recognition based on Python-OpenPose. (The green lines represent the main skeleton of the human body, and the red dots represent the main key points of the human body).</p>
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<p>Principle of binocular ranging.</p>
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<p>Human follow-up-assisted frame kinematic model.</p>
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<p>LSTM unit structure.</p>
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<p>BiLSTM model structure.</p>
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<p>QPSO-BiLSTM gait trajectory prediction framework.</p>
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<p>DRBFNNASMC controller structure.</p>
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<p>Attachment positions of the six-channel sEMG sensors on the human body.</p>
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<p>Follow-up assistive frame and human follow-up tracking experiment results.</p>
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<p>Comparison of joint angle predictions by three network models and IMU measured values. (IMU is an abbreviation for inertial measurement unit, BiLSTM stands for bi-directional long short-term memory network, PSO-BiLSTM denotes particle swarm optimization bi-directional long short-term memory network, and QPSO-BiLSTM refers to quantum-behaved particle swarm optimization bi-directional long short-term memory network).</p>
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<p>Diagram of the walking experiment. (The green lines represent the main skeleton of the human body, and the red dots represent the main key points of the human body).</p>
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<p>Trajectory tracking control results based on continuous motion intention estimation. RF stands for rectus femoris, TA refers to tibialis anterior, GM denotes gluteus medius, Gmax represents gluteus maximus, HS indicates hamstrings, and GA stands for gastrocnemius.</p>
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<p>Comparison of the control effectiveness of three different controllers. (DRBFNNASMC stands for dual radial basis function neural network adaptive sliding mode controller, FCASMC refers to fuzzy compensated adaptive sliding mode controller, and PID denotes proportional–integral–derivative controller).</p>
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23 pages, 5469 KiB  
Article
Shutter-Synchronized Molecular Beam Epitaxy for Wafer-Scale Homogeneous GaAs and Telecom Wavelength Quantum Emitter Growth
by Elias Kersting, Hans-Georg Babin, Nikolai Spitzer, Jun-Yong Yan, Feng Liu, Andreas D. Wieck and Arne Ludwig
Nanomaterials 2025, 15(3), 157; https://doi.org/10.3390/nano15030157 - 21 Jan 2025
Viewed by 446
Abstract
Quantum dot (QD)-based single-photon emitter devices today are based on self-assembled random position nucleated QDs emitting at random wavelengths. Deterministic QD growth in position and emitter wavelength would be highly appreciated for industry-scale high-yield device manufacturing from wafers. Local droplet etching during molecular [...] Read more.
Quantum dot (QD)-based single-photon emitter devices today are based on self-assembled random position nucleated QDs emitting at random wavelengths. Deterministic QD growth in position and emitter wavelength would be highly appreciated for industry-scale high-yield device manufacturing from wafers. Local droplet etching during molecular beam epitaxy is an all in situ method that allows excellent density control and predetermines the nucleation site of quantum dots. This method can produce strain-free GaAs QDs with excellent photonic and spin properties. Here, we focus on the emitter wavelength homogeneity. By wafer rotation-synchronized shutter opening time and adapted growth parameters, we grow QDs with a narrow peak emission wavelength homogeneity with no more than 1.2 nm shifts on a 45 mm diameter area and a narrow inhomogeneous ensemble broadening of only 2 nm at 4 K. The emission wavelength of these strain-free GaAs QDs is <800 nm, attractive for quantum optics experiments and quantum memory applications. We can use a similar random local droplet nucleation, nanohole drilling, and now, InAs infilling to produce QDs emitting in the telecommunication optical fiber transparency window around 1.3 µm, the so-called O-band. For this approach, we demonstrate good wavelength homogeneity and excellent density homogeneity beyond the possibilities of standard Stranski–Krastanov self-assembly. We discuss our methodology, structural and optical properties, and limitations set by our current setup capabilities. Full article
(This article belongs to the Section Nanophotonics Materials and Devices)
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Figure 1
<p>Schematic representation of material deposition: (<b>a</b>) Top view of the resulting material amount profile when the wafer is aligned in a fixed position to the effusion cell. (<b>b</b>) Side view of the material amount profile during gradient deposition and alignment of the cell to the substrate. (<b>c</b>) Side view of the material amount profile of a homogeneous layer grown under rotation. <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>t</mi> </mrow> <mrow> <mi>rot</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow> <mi>t</mi> </mrow> <mrow> <mi>shutter</mi> </mrow> </msub> </mrow> </semantics></math> applies to the shutter-synchronous deposition.</p>
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<p>Growth and gradients: (<b>a</b>) Layer structure of GaAs QDs samples (not to scale). (<b>b</b>) Alignment of the material gradients.</p>
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<p>Wafer maps showing the s-peak photoluminescence emission wavelength distribution at ~100 K. In all samples, the filling material was deposited as a gradient with the following amounts in the center of the samples: (<b>a</b>) 1 nm GaAs; (<b>b</b>) 0.60 nm GaAs; (<b>c</b>) 0.44 nm GaAs.</p>
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<p>AFM and PL study of sample <math display="inline"><semantics> <mrow> <mo>#</mo> <msubsup> <mrow> <mi mathvariant="normal">E</mi> </mrow> <mrow> <mn>0.60</mn> <mo>/</mo> <mi mathvariant="normal">G</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">s</mi> <mo>/</mo> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">t</mi> </mrow> <mrow> <mn>0.4</mn> <mo>/</mo> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">l</mi> <mn>2</mn> <mo>/</mo> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">t</mi> </mrow> </msubsup> </mrow> </semantics></math> to determine the QD density: (<b>a</b>) Surface hole density and integrated QD PL intensity in horizontal (h) and vertical (v) directions. (<b>b</b>) Representative 5 × 5 µm<sup>2</sup> image of the surface morphology at center position (38|38). The color scale ranges from 0 nm (dark) to 8 nm (bright).</p>
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<p>Photoluminescence results at ~90 K of the shutter-synchronized grown GaAs QDs: (<b>a</b>) Integrated intensity in a wavelength range of (750–820) nm of sample #F. (<b>b</b>) Wafer map of the s-peak emission wavelength of sample #F. The areas under investigation are marked by circles with diameters of 30, 45, and 60 mm. (<b>c</b>) Representative spectrum and Gaussian fitting from the center of the wafer (sample #F). The red shaded area corresponds to the investigated energy range (here, for example, a total tuning range of ±5 meV). (<b>d</b>) Emission wavelength profile in vertical (dashed) and horizontal (solid line) directions through the center of the wafer for samples #D, #E, and #F. The shaded areas correspond to the FWHM area of the ensemble PL.</p>
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<p>The proportion of QDs emitting in the range accessible by Stark-shift tuning for wafers #D, #E, and #F: (<b>a</b>) Proportion of QDs that are tunable to the wafer center peak wavelength, calculated from the PL measured at 90 K. A maximum tuning range of 25 meV has been reported. We assume a statistical distribution of the tuning range and calculate for a ±12.5 meV and smaller tuning ranges. (<b>b</b>) The PL ground state transition energy of a total of 794 individual QDs at position (38/42) near the center of wafer #F at a sample temperature of 4 K. The excitation was performed with a 637 nm laser operated at a power of 7 µW. (<b>c</b>) Proportion of QDs that can be tuned to the wafer center peak emission wavelength for a ±1 meV tuning range.</p>
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<p>Growth and gradients: (<b>a</b>) Layer structure of the LDE InAs QDs (not to scale). (<b>b</b>) Alignment of the PDL and the InAs filling gradient.</p>
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<p>Photoluminescence results at ~80 K of the gradient-grown InAs QDs: (<b>a</b>) Integrated intensity in a wavelength range of (990–1360) nm. A, B, and C mark the 3 different areas. (<b>b</b>) Wafer map of the s-peak emission wavelength. (<b>c</b>–<b>e</b>) One representative spectrum each from areas A, B, and C. (<b>f</b>) Spectral heatmap in the vertical direction for x = 38 mm. (<b>g</b>) Wavelength at the maximum intensity over the InAs material amount.</p>
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<p>Photoluminescence results at ~80 K of the shutter-synchronized grown InAs QDs: (<b>a</b>) Integrated intensity in a wavelength range of (1100–1350) nm. For the offset plots in (<b>c</b>), orientation points are marked by I and XIII. (<b>b</b>) Wafer map of the s-peak emission wavelength. The areas under investigation are marked by circles with diameters of 30, 45, and 60 mm. (<b>c</b>) Offset spectra plots along the horizontal line from I to XIII with an increment of 5 mm. (<b>d</b>) Horizontal and vertical line profiles of the emission wavelength and FWHM (shaded area) through the center of the wafer. (<b>e</b>) Density of buried QDs and surface LDE QDs in horizontal (h) and vertical (v) directions through the center. (<b>f</b>) Proportion of QDs that are tunable to the wafer center peak wavelength for three different tuning ranges.</p>
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<p>Normalized material amount profiles along the center of the wafer for gradient deposition (<b>left</b>) and shutter-synchronous deposition (<b>right</b>) for the different effusion cells. The profiles are calculated from quantum well PL measurements. The dip in the Ga cell flux distribution results from a too-hot center wafer temperature for the evaluated QW.</p>
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<p>Surface morphology of sample <math display="inline"><semantics> <mrow> <mo>#</mo> <msubsup> <mrow> <mi mathvariant="normal">G</mi> </mrow> <mrow> <mn>12.25</mn> <mo>/</mo> <mi mathvariant="normal">I</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">s</mi> <mo>,</mo> <mi mathvariant="normal">g</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mn>0.84</mn> <mo>/</mo> <mi mathvariant="normal">G</mi> <mi mathvariant="normal">a</mi> <mo>/</mo> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">t</mi> </mrow> </msubsup> </mrow> </semantics></math> (#15736) from area B. The buried QDs (elongated mounds) have a height of ~4 nm and the LDE QDs of ~15 nm. The color scale ranges from 0.5 nm (dark) to 5 nm (bright).</p>
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<p>Surface morphology of sample <math display="inline"><semantics> <mrow> <mo>#</mo> <msubsup> <mrow> <mi mathvariant="normal">F</mi> </mrow> <mrow> <mn>0.60</mn> <mo>/</mo> <mi mathvariant="normal">G</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">s</mi> <mo>/</mo> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">t</mi> </mrow> <mrow> <mn>0.4</mn> <mo>/</mo> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">l</mi> <mn>1</mn> <mo>/</mo> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">t</mi> </mrow> </msubsup> </mrow> </semantics></math> (#15744) at the center (38|38) and at position (38|8): (<b>a</b>) The color scale ranges from 0.4 nm (dark) to 6 nm (bright). The buried QDs and the LDE QDs have the same height as described in <a href="#nanomaterials-15-00157-f0A2" class="html-fig">Figure A2</a>. The SK QDs have a height of ~13 nm. (<b>b</b>) The color ranges from 0.3 nm (dark) to 10 nm (bright). The small bright dots have a height of ~12 nm and the large dots up to 35 nm.</p>
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27 pages, 3978 KiB  
Article
Dissipation Alters Modes of Information Encoding in Small Quantum Reservoirs near Criticality
by Krai Cheamsawat and Thiparat Chotibut
Entropy 2025, 27(1), 88; https://doi.org/10.3390/e27010088 - 18 Jan 2025
Viewed by 531
Abstract
Quantum reservoir computing (QRC) has emerged as a promising paradigm for harnessing near-term quantum devices to tackle temporal machine learning tasks. Yet, identifying the mechanisms that underlie enhanced performance remains challenging, particularly in many-body open systems where nonlinear interactions and dissipation intertwine in [...] Read more.
Quantum reservoir computing (QRC) has emerged as a promising paradigm for harnessing near-term quantum devices to tackle temporal machine learning tasks. Yet, identifying the mechanisms that underlie enhanced performance remains challenging, particularly in many-body open systems where nonlinear interactions and dissipation intertwine in complex ways. Here, we investigate a minimal model of a driven-dissipative quantum reservoir described by two coupled Kerr-nonlinear oscillators, an experimentally realizable platform that features controllable coupling, intrinsic nonlinearity, and tunable photon loss. Using Partial Information Decomposition (PID), we examine how different dynamical regimes encode input drive signals in terms of redundancy (information shared by each oscillator) and synergy (information accessible only through their joint observation). Our key results show that, near a critical point marking a dynamical bifurcation, the system transitions from predominantly redundant to synergistic encoding. We further demonstrate that synergy amplifies short-term responsiveness, thereby enhancing immediate memory retention, whereas strong dissipation leads to more redundant encoding that supports long-term memory retention. These findings elucidate how the interplay of instability and dissipation shapes information processing in small quantum systems, providing a fine-grained, information-theoretic perspective for analyzing and designing QRC platforms. Full article
(This article belongs to the Special Issue Quantum Computing in the NISQ Era)
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Figure 1
<p>Schematic of two coupled Kerr-nonlinear oscillators. Each cavity <span class="html-italic">i</span> features a Kerr nonlinearity <math display="inline"><semantics> <msub> <mi>U</mi> <mi>i</mi> </msub> </semantics></math> and a photon loss rate <math display="inline"><semantics> <msub> <mi>γ</mi> <mi>i</mi> </msub> </semantics></math>. A time-dependent drive <math display="inline"><semantics> <mrow> <mi>F</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math> (green arrows) injects identical signals into both cavities, while the coherent tunneling of strength <span class="html-italic">J</span> (violet arrow) couples the two modes. We measure the mean fields <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mi>i</mi> </msub> <mo>=</mo> <mrow> <mo>〈</mo> <msub> <mover accent="true"> <mi>a</mi> <mo stretchy="false">^</mo> </mover> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>〉</mo> </mrow> </mrow> </semantics></math> to probe the system’s response. The Hamiltonian is specified by Equation (<a href="#FD1-entropy-27-00088" class="html-disp-formula">1</a>), and the Lindblad Equation (<a href="#FD2-entropy-27-00088" class="html-disp-formula">2</a>) governs this driven-dissipative dynamics. We assume both cavities have the same detuning <math display="inline"><semantics> <mo>Δ</mo> </semantics></math> from the drive frequency. This work investigates how the readouts <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> encode the time-dependent drive <math display="inline"><semantics> <mrow> <mi>F</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math> across different dynamical regimes of the coupled Kerr oscillators.</p>
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<p>Classical mutual information <math display="inline"><semantics> <mrow> <mi>I</mi> <mfenced separators="" open="(" close=")"> <mi>s</mi> <mo>:</mo> <mo>(</mo> <msub> <mi>X</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>X</mi> <mn>2</mn> </msub> <mo>)</mo> </mfenced> </mrow> </semantics></math>, compared to <math display="inline"><semantics> <mrow> <mi>I</mi> <mo>(</mo> <mi>s</mi> <mo>:</mo> <msub> <mi>X</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>I</mi> <mo>(</mo> <mi>s</mi> <mo>:</mo> <msub> <mi>X</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> </semantics></math> in (<b>Left</b>) a mean-field regime and (<b>Right</b>) a quantum regime. In the mean-field regime, <math display="inline"><semantics> <mrow> <mi>I</mi> <mfenced separators="" open="(" close=")"> <mi>s</mi> <mo>:</mo> <mo>(</mo> <msub> <mi>X</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>X</mi> <mn>2</mn> </msub> <mo>)</mo> </mfenced> </mrow> </semantics></math> exceeds <math display="inline"><semantics> <mrow> <mi>I</mi> <mo>(</mo> <mi>s</mi> <mo>:</mo> <msub> <mi>X</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> </semantics></math> or <math display="inline"><semantics> <mrow> <mi>I</mi> <mo>(</mo> <mi>s</mi> <mo>:</mo> <msub> <mi>X</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> </semantics></math> alone near <math display="inline"><semantics> <mrow> <mi>J</mi> <mo>=</mo> <mo>|</mo> <mo>Δ</mo> <mo>|</mo> </mrow> </semantics></math>, hinting at synergy. On the other hand, in the quantum regime, <math display="inline"><semantics> <mrow> <mi>I</mi> <mfenced separators="" open="(" close=")"> <mi>s</mi> <mo>:</mo> <mo>(</mo> <msub> <mi>X</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>X</mi> <mn>2</mn> </msub> <mo>)</mo> </mfenced> </mrow> </semantics></math> is comparable to, but not always exceeding, the sum of <math display="inline"><semantics> <mrow> <mi>I</mi> <mo>(</mo> <mi>s</mi> <mo>:</mo> <msub> <mi>X</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>I</mi> <mo>(</mo> <mi>s</mi> <mo>:</mo> <msub> <mi>X</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> </semantics></math>.</p>
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<p>(<b>Left</b>) Normalized synergy vs. the coupling <span class="html-italic">J</span>. (<b>Right</b>) Normalized redundancy vs. the coupling <span class="html-italic">J</span>. A pronounced peak near <math display="inline"><semantics> <mrow> <mi>J</mi> <mo>≈</mo> <mo>|</mo> <mo>Δ</mo> <mo>|</mo> </mrow> </semantics></math> marks the crossover from predominantly redundant to more synergistic encoding. In the fully quantum description, enhanced quantum correlations can favor redundancy even at the transition, whereas second-order cumulants interpolate between mean-field and quantum descriptions.</p>
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<p>Effective potential around the steady state <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mrow> <mn>1</mn> <mo>,</mo> <mi>c</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>α</mi> <mrow> <mn>2</mn> <mo>,</mo> <mi>c</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> (red dot) in the <math display="inline"><semantics> <mrow> <mo>Δ</mo> <mo>&lt;</mo> <mn>0</mn> <mo>,</mo> <mi>J</mi> <mo>&gt;</mo> <mn>0</mn> </mrow> </semantics></math> regime, projected onto <math display="inline"><semantics> <mrow> <mi>Im</mi> <mrow> <mo>(</mo> <msub> <mi>α</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mi>Im</mi> <mrow> <mo>(</mo> <msub> <mi>α</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>. (<b>Left</b>) When <math display="inline"><semantics> <mrow> <mi>J</mi> <mo>&lt;</mo> <mo>|</mo> <mo>Δ</mo> <mo>|</mo> </mrow> </semantics></math>, the steady state is weakly unstable in all directions, with no soft modes present, and the system predominantly encodes inputs redundantly. (<b>Center</b>) At the critical point <math display="inline"><semantics> <mrow> <mi>J</mi> <mo>=</mo> <mo>|</mo> <mo>Δ</mo> <mo>|</mo> </mrow> </semantics></math>, flat directions appear, marking the onset of soft modes. In this near-critical regime, collective oscillations enhance synergistic encoding. (<b>Right</b>) For <math display="inline"><semantics> <mrow> <mi>J</mi> <mo>&gt;</mo> <mo>|</mo> <mo>Δ</mo> <mo>|</mo> </mrow> </semantics></math>, the potential deforms into a saddle, with two stable and two unstable directions. Here, the soft modes again disappear, and the system encodes inputs redundantly.</p>
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<p>Retarded Green’s function poles (A16) in the complex-frequency plane as <span class="html-italic">J</span> increases, illustrating the evolution of slow modes <math display="inline"><semantics> <msub> <mi>ω</mi> <mi>s</mi> </msub> </semantics></math> (orange dots) and fast modes <math display="inline"><semantics> <msub> <mi>ω</mi> <mi>f</mi> </msub> </semantics></math> (blue dots). For <math display="inline"><semantics> <mrow> <mi>J</mi> <mo>&lt;</mo> <mo>|</mo> <mo>Δ</mo> <mo>|</mo> </mrow> </semantics></math>, both slow and fast modes coexist, and the system tends to encode inputs more redundantly. Near the critical point <math display="inline"><semantics> <mrow> <mi>J</mi> <mo>=</mo> <mo>|</mo> <mo>Δ</mo> <mo>|</mo> </mrow> </semantics></math>, the real part of <math display="inline"><semantics> <msub> <mi>ω</mi> <mi>s</mi> </msub> </semantics></math> approaches zero, indicating a disappearance of slow collective oscillations and an overdamped decay. In this near-critical regime, collective oscillations due to underdamped fast modes dominate and enhance synergistic encoding. For <math display="inline"><semantics> <mrow> <mi>J</mi> <mo>&gt;</mo> <mo>|</mo> <mo>Δ</mo> <mo>|</mo> </mrow> </semantics></math>, the slow modes shift away from zero frequency, reducing synergy and transitioning the system back toward redundant encoding. This interplay between slow and fast modes governs how the system transitions from predominantly redundant to synergistic processing and back again as <span class="html-italic">J</span> crosses the critical point.</p>
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<p>Impact of uniform, uncorrelated noise input on information encoding at the quantum regime (<math display="inline"><semantics> <mrow> <mo>Δ</mo> <mo>=</mo> <mo>−</mo> <mn>2</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>F</mi> <mo>=</mo> <mn>0.2</mn> </mrow> </semantics></math>). (<b>Left</b>) Normalized PID with uniform noise input. (<b>Right</b>) Comparison of total MI and partial MI contributions. The left panel shows a clear transition to synergistic encoding near <math display="inline"><semantics> <mrow> <mi>J</mi> <mo>=</mo> <mo>|</mo> <mo>Δ</mo> <mo>|</mo> </mrow> </semantics></math>. The right panel compares <math display="inline"><semantics> <mrow> <mi>I</mi> <mfenced separators="" open="(" close=")"> <mi>S</mi> <mo>:</mo> <mo>(</mo> <msub> <mi>X</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>X</mi> <mn>2</mn> </msub> <mo>)</mo> </mfenced> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>I</mi> <mrow> <mo>(</mo> <mi>S</mi> <mo>:</mo> <msub> <mi>X</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mi>I</mi> <mrow> <mo>(</mo> <mi>S</mi> <mo>:</mo> <msub> <mi>X</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> </mrow> </semantics></math>, highlighting the emergence of synergy near the critical coupling.</p>
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<p>Impact of increasing <math display="inline"><semantics> <mi>γ</mi> </semantics></math> on information metrics in the quantum regime (<math display="inline"><semantics> <mrow> <mo>Δ</mo> <mo>=</mo> <mo>−</mo> <mn>2</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>F</mi> <mo>=</mo> <mn>0.2</mn> </mrow> </semantics></math>). (<b>a</b>) Time-averaged quantum mutual information (QMI) between the two oscillators as a function of <span class="html-italic">J</span> for different <math display="inline"><semantics> <mi>γ</mi> </semantics></math>. (<b>b</b>) Classical mutual information (MI) between the input signal and the outputs vs. <span class="html-italic">J</span>. (<b>c</b>) Absolute synergy vs. <span class="html-italic">J</span>. (<b>d</b>) Absolute redundancy vs. <span class="html-italic">J</span>. These plots illustrate the transition from low-synergistic to high-redundant encoding with increasing <math display="inline"><semantics> <mi>γ</mi> </semantics></math>. At large dissipation, the two oscillators approach a product state, indicated by low QMI. Interestingly, despite the redundancy dominating at higher <math display="inline"><semantics> <mi>γ</mi> </semantics></math>, the total mutual information near <math display="inline"><semantics> <mrow> <mi>J</mi> <mo>=</mo> <mo>|</mo> <mo>Δ</mo> <mo>|</mo> </mrow> </semantics></math> remains approximately constant.</p>
Full article ">Figure 8
<p>(<b>Left</b>) Memory capacity <math display="inline"><semantics> <mrow> <mi>MC</mi> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> </semantics></math> for <span class="html-italic">n</span> = 1–10 and <math display="inline"><semantics> <mrow> <mi>J</mi> <mo>∈</mo> <mo>[</mo> <mn>0</mn> <mo>,</mo> <mn>2</mn> <mo>]</mo> </mrow> </semantics></math>, showing an increase in short-term memory capacity as <math display="inline"><semantics> <mrow> <mi>J</mi> <mo>→</mo> <mo>|</mo> <mo>Δ</mo> <mo>|</mo> </mrow> </semantics></math>. (<b>Right</b>) Memory capacity for <span class="html-italic">n</span> = 1–20 and <math display="inline"><semantics> <mrow> <mi>J</mi> <mo>∈</mo> <mo>[</mo> <mn>1.96</mn> <mo>,</mo> <mn>2</mn> <mo>]</mo> </mrow> </semantics></math>, showing that the long-term capacity drops as <math display="inline"><semantics> <mrow> <mi>J</mi> <mo>→</mo> <mo>|</mo> <mo>Δ</mo> <mo>|</mo> </mrow> </semantics></math>. Parameters: <math display="inline"><semantics> <mrow> <mo>Δ</mo> <mo>=</mo> <mo>−</mo> <mn>2</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>F</mi> <mo>=</mo> <mn>0.2</mn> </mrow> </semantics></math>. These results are averaged over 50 input realizations.</p>
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<p>Memory capacity at the critical coupling <math display="inline"><semantics> <mrow> <mi>J</mi> <mo>=</mo> <mo>|</mo> <mo>Δ</mo> <mo>|</mo> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math> as <math display="inline"><semantics> <mi>γ</mi> </semantics></math> increases. (<b>Left</b>) Linear scale plot: <math display="inline"><semantics> <mrow> <mi>MC</mi> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> </semantics></math> vs. <span class="html-italic">n</span>. (<b>Right</b>) Log scale plot of <math display="inline"><semantics> <mrow> <mi>MC</mi> <mo>(</mo> <mi>n</mi> <mo>)</mo> <mo>∼</mo> <mo form="prefix">exp</mo> <mo>(</mo> <mo>−</mo> <mo>Γ</mo> <mi>n</mi> <mo>)</mo> </mrow> </semantics></math>. The left panel shows the average memory capacity (100 realizations). The right panel illustrates an exponential decay in <math display="inline"><semantics> <mrow> <mi>MC</mi> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> </semantics></math>. The exponential decay rate of the two-time correlation in the memory capacity, <math display="inline"><semantics> <mo>Γ</mo> </semantics></math>, remains approximately constant as <math display="inline"><semantics> <mi>γ</mi> </semantics></math> increases, indicating that dissipation uniformly governs the loss of correlations across different dissipation rates. Differences in memory capacity primarily arise from the proportionality factor, with larger <math display="inline"><semantics> <mi>γ</mi> </semantics></math> leading to a smaller variance in the output prediction in the denominator of Equation (<a href="#FD18-entropy-27-00088" class="html-disp-formula">18</a>), as the output rapidly stabilizes to the steady state. This rapid stabilization corresponds to highly redundant encoding, and in turn enhances the total memory capacity for memorizing uniformly random input time series. Notably, in this redundant coding regime, the highly dissipative dynamics improve the quantum reservoir’s memory capacity.</p>
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<p>Co-information <math display="inline"><semantics> <mrow> <msub> <mi>CoI</mi> <msub> <mi>Q</mi> <mi>α</mi> </msub> </msub> <mrow> <mo>(</mo> <mi>X</mi> <mo>;</mo> <mi>Y</mi> <mo>;</mo> <mi>Z</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> and mutual information <math display="inline"><semantics> <mrow> <msub> <mi>I</mi> <msub> <mi>Q</mi> <mi>α</mi> </msub> </msub> <mrow> <mo>(</mo> <mi>X</mi> <mo>:</mo> <mrow> <mo>(</mo> <mi>Y</mi> <mo>,</mo> <mi>Z</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math> for the AND-gate example, plotted as functions of <math display="inline"><semantics> <mi>α</mi> </semantics></math>. The optimum for redundancy (respectively, synergy) occurs at <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="false"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> </semantics></math>.</p>
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<p>(<b>Left</b>) <math display="inline"><semantics> <mrow> <msub> <mi>CoI</mi> <msub> <mi>Q</mi> <mrow> <mi>α</mi> <mo>,</mo> <mi>β</mi> </mrow> </msub> </msub> <mrow> <mo>(</mo> <mi>X</mi> <mo>;</mo> <mi>Y</mi> <mo>;</mo> <mi>Z</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>; (<b>Right</b>) <math display="inline"><semantics> <mrow> <msub> <mi>I</mi> <msub> <mi>Q</mi> <mrow> <mi>α</mi> <mo>,</mo> <mi>β</mi> </mrow> </msub> </msub> <mrow> <mo>(</mo> <mi>X</mi> <mo>:</mo> <mrow> <mo>(</mo> <mi>Y</mi> <mo>,</mo> <mi>Z</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math>; for the XOR-gate example. The optimum occurs at <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mi>β</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>.</p>
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<p>Classical mutual information for the 2nd-order cumulant description. This is to be contrasted with <a href="#entropy-27-00088-f002" class="html-fig">Figure 2</a> (left) to reveal how second-order description captures partial but nontrivial correlation effects correcting mean-field approximation. We set <math display="inline"><semantics> <mrow> <mo>Δ</mo> <mo>=</mo> <mo>−</mo> <mn>2</mn> <mo>,</mo> <mi>F</mi> <mo>=</mo> <mn>0.5</mn> <mo>,</mo> <mi>γ</mi> <mo>=</mo> <mn>0.5</mn> <mo>,</mo> <msub> <mi>U</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.2</mn> <mo>,</mo> <msub> <mi>U</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>2</mn> <msub> <mi>U</mi> <mn>1</mn> </msub> </mrow> </semantics></math> to represent a dynamical regime with non-negligible correlations, motivating the use of second-order cumulants description.</p>
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30 pages, 8556 KiB  
Article
Optimization of Microgrid Dispatching by Integrating Photovoltaic Power Generation Forecast
by Tianrui Zhang, Weibo Zhao, Quanfeng He and Jianan Xu
Sustainability 2025, 17(2), 648; https://doi.org/10.3390/su17020648 - 15 Jan 2025
Viewed by 683
Abstract
In order to address the impact of the uncertainty and intermittency of a photovoltaic power generation system on the smooth operation of the power system, a microgrid scheduling model incorporating photovoltaic power generation forecast is proposed in this paper. Firstly, the factors affecting [...] Read more.
In order to address the impact of the uncertainty and intermittency of a photovoltaic power generation system on the smooth operation of the power system, a microgrid scheduling model incorporating photovoltaic power generation forecast is proposed in this paper. Firstly, the factors affecting the accuracy of photovoltaic power generation prediction are analyzed by classifying the photovoltaic power generation data using cluster analysis, analyzing its important features using Pearson correlation coefficients, and downscaling the high-dimensional data using PCA. And based on the theories of the sparrow search algorithm, convolutional neural network, and bidirectional long- and short-term memory network, a combined SSA-CNN-BiLSTM prediction model is established, and the attention mechanism is used to improve the prediction accuracy. Secondly, a multi-temporal dispatch optimization model of the microgrid power system, which aims at the economic optimization of the system operation cost and the minimization of the environmental cost, is constructed based on the prediction results. Further, differential evolution is introduced into the QPSO algorithm and the model is solved using this improved quantum particle swarm optimization algorithm. Finally, the feasibility of the photovoltaic power generation forecasting model and the microgrid power system dispatch optimization model, as well as the validity of the solution algorithms, are verified through real case simulation experiments. The results show that the model in this paper has high prediction accuracy. In terms of scheduling strategy, the generation method with the lowest cost is selected to obtain an effective way to interact with the main grid and realize the stable and economically optimized scheduling of the microgrid system. Full article
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<p>Heat map of correlation coefficient of each variable.</p>
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<p>Data clustering result graph.</p>
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<p>Data preprocessing flow chart.</p>
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<p>Flow chart of optimization parameters of SSA algorithm.</p>
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<p>Structure diagram of attention mechanism.</p>
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<p>Flow chart of photovoltaic power generation prediction model based on SSA-CNN-BiLSTM-ATT.</p>
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<p>Improved QPSO flowchart.</p>
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<p>Microgrid scheduling strategy.</p>
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<p>Microturbine power cost curve.</p>
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<p>Fuel cell generation cost curve.</p>
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<p>The fitness curve of CNN-BiLSTM-ATT prediction model was optimized by different algorithms.</p>
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<p>Convergence diagram of different algorithms.</p>
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<p>Bar chart of error for different weather types under different forecast models.</p>
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<p>Microgrid power system.</p>
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<p>Daily heat load and electricity load in winter.</p>
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<p>Winter daily wind speed and temperature.</p>
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<p>Winter light intensity.</p>
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<p>Curve of microgrid generation.</p>
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<p>Microgrid interaction curve.</p>
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<p>Performance comparison of algorithms.</p>
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<p>Comparison of algorithms in the later stage.</p>
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20 pages, 12787 KiB  
Article
Exploring the Properties of Quantum Scars in a Toy Model
by Sudip Sinha and Subhasis Sinha
Condens. Matter 2025, 10(1), 5; https://doi.org/10.3390/condmat10010005 - 12 Jan 2025
Viewed by 301
Abstract
We introduce the concept of ergodicity and explore its deviation caused by quantum scars in an isolated quantum system, employing a pedagogical approach based on a toy model. Quantum scars, originally identified as traces of classically unstable orbits in certain wavefunctions of chaotic [...] Read more.
We introduce the concept of ergodicity and explore its deviation caused by quantum scars in an isolated quantum system, employing a pedagogical approach based on a toy model. Quantum scars, originally identified as traces of classically unstable orbits in certain wavefunctions of chaotic systems, have recently regained interest for their role in non-ergodic dynamics, as they retain memory of their initial states. We elucidate these features of quantum scars within the same framework of this toy model. The integrable part of the model consists of two large spins, with a classical counterpart, which we combine with a random matrix to induce ergodic behavior. Scarred states can be selectively generated from the integrable spin Hamiltonian by protecting them from the ergodic states using a projector method. Deformed projectors mimic the ‘quantum leakage’ of scarred states, enabling tunable mixing with ergodic states and thereby controlling the degree of scarring. In this simple model, we investigate various properties of quantum scarring and shed light on different aspects of many-body quantum scars observed in more complex quantum systems. Notably, the underlying classicality can be revealed through the entanglement spectrum and the dynamics of ‘out-of-time-ordered correlators’. Full article
(This article belongs to the Special Issue Non-equilibrium Dynamics in Ultra-Cold Quantum Gases)
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Figure 1

Figure 1
<p>Ergodic properties for <math display="inline"><semantics> <mrow> <mi>ϵ</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> are as follows: (<b>a</b>) Variation in Entanglement entropy (EE), <math display="inline"><semantics> <msub> <mi>S</mi> <mrow> <mi>e</mi> <mi>n</mi> </mrow> </msub> </semantics></math>, of the energy eigenstates with energy density, <math display="inline"><semantics> <mrow> <mi>E</mi> <mo>≡</mo> <msub> <mi mathvariant="script">E</mi> <mi>n</mi> </msub> <mo>/</mo> <mi>S</mi> </mrow> </semantics></math>. The pink dashed line corresponds to the page value of EE [see Equation (<a href="#FD14-condensedmatter-10-00005" class="html-disp-formula">14</a>)]. The insets (<b>a1</b>,<b>a2</b>) display the ergodic and projected states zoomed in a small window around <span class="html-italic">E</span> ∼ 0, respectively. (<b>b</b>) Distribution of the consecutive level spacing ratio <math display="inline"><semantics> <mrow> <mi>P</mi> <mo>(</mo> <mi>r</mi> <mo>)</mo> </mrow> </semantics></math> for 100 different realizations (ensembles) of random matrix <math display="inline"><semantics> <msub> <mover accent="true"> <mi mathvariant="script">H</mi> <mo stretchy="false">^</mo> </mover> <mi>GOE</mi> </msub> </semantics></math>, which agrees well with the GOE class as <math display="inline"><semantics> <mrow> <mo stretchy="false">〈</mo> <mi>r</mi> <mo stretchy="false">〉</mo> </mrow> </semantics></math> ∼ 0.523. We set <math display="inline"><semantics> <mrow> <mi>J</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>S</mi> <mo>=</mo> <mn>20</mn> </mrow> </semantics></math> in this and the rest of the figures unless otherwise specified.</p>
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<p>The properties of the scarred states for a single realization of random matrix: (<b>a1</b>,<b>b1</b>) EE of the energy eigenstates for different values of <math display="inline"><semantics> <mi>ϵ</mi> </semantics></math>. The black squares denote EE of the eigenstates for <math display="inline"><semantics> <mrow> <mi>ϵ</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>. Entanglement spectrum (ES) and Husimi distribution corresponding to the first spin sector of the energy eigenstates (blue circles) marked in (<b>a1</b>,<b>b1</b>) for (<b>a2</b>,<b>a3</b>) <math display="inline"><semantics> <mi>ϵ</mi> </semantics></math> ∼ 0.01 and (<b>b2</b>,<b>b3</b>) <math display="inline"><semantics> <mi>ϵ</mi> </semantics></math> ∼ 0.05, respectively. The double headed arrows mark the gap, <math display="inline"><semantics> <mrow> <mo>Δ</mo> <mi>λ</mi> </mrow> </semantics></math>, separating the largest eigenvalue and the extended tail in the ES. The red diamonds correspond to the ES of a random eigenstate of a GOE matrix of the same dimension, reflecting the ergodic behavior.</p>
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<p>Degree of scarring: (<b>a</b>) Variation in the average deviation of the EE <math display="inline"><semantics> <msub> <mover> <mrow> <mo>Δ</mo> <mi>S</mi> </mrow> <mo>¯</mo> </mover> <mrow> <mi>e</mi> <mi>n</mi> </mrow> </msub> </semantics></math> scaled by the maximum limit (page value), and the average gap <math display="inline"><semantics> <mover> <mrow> <mo>Δ</mo> <mi>λ</mi> </mrow> <mo>¯</mo> </mover> </semantics></math> in the ES (inset) at energy density <span class="html-italic">E</span> ∼ 0 with increasing deformation strength <math display="inline"><semantics> <mi>ϵ</mi> </semantics></math>, for <math display="inline"><semantics> <mrow> <mi>S</mi> <mo>=</mo> <mn>20</mn> </mrow> </semantics></math>. (<b>b</b>) Variation in <math display="inline"><semantics> <mrow> <mover> <mrow> <mo>Δ</mo> <msub> <mi>S</mi> <mrow> <mi>e</mi> <mi>n</mi> </mrow> </msub> </mrow> <mo>¯</mo> </mover> <mo>/</mo> <msub> <mi>S</mi> <mrow> <mi>p</mi> <mi>a</mi> <mi>g</mi> <mi>e</mi> </mrow> </msub> </mrow> </semantics></math> with increasing large spin magnitude <span class="html-italic">S</span> for different values of <math display="inline"><semantics> <mi>ϵ</mi> </semantics></math>. The averaging (indicated by <math display="inline"><semantics> <mover accent="true"> <mo>·</mo> <mo>¯</mo> </mover> </semantics></math> ) is performed in two steps: first, over the states with minimum EE in a small energy density window <math display="inline"><semantics> <mrow> <mo>Δ</mo> <mi>E</mi> <mo>=</mo> <mn>0.09</mn> <mo>/</mo> <mi>S</mi> <mo>&lt;</mo> <mn>1</mn> <mo>/</mo> <mi>S</mi> </mrow> </semantics></math> around <span class="html-italic">E</span> ∼ 0, and then over an ensemble of (<b>a</b>) 100 and (<b>b</b>) 1000 random matrices <math display="inline"><semantics> <msub> <mover accent="true"> <mi mathvariant="script">H</mi> <mo stretchy="false">^</mo> </mover> <mi>GOE</mi> </msub> </semantics></math>.</p>
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<p>Quantum dynamics starting from the initial state <math display="inline"><semantics> <mrow> <mrow> <mo>|</mo> <mi>ϕ</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mi>θ</mi> <mo>=</mo> <mi>π</mi> <mo>/</mo> <mn>2</mn> <mo>⟩</mo> </mrow> <mo>⊗</mo> <mrow> <mo>|</mo> <msub> <mi>m</mi> <mrow> <mn>2</mn> <mi>z</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>⟩</mo> </mrow> </mrow> </semantics></math> at <span class="html-italic">E</span> ∼ 0: Time evolution of survival probability <math display="inline"><semantics> <mrow> <mi>F</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mrow> <mo>|</mo> <mrow> <mo stretchy="false">〈</mo> <mi>ψ</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>|</mo> <mi>ψ</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> <mo stretchy="false">〉</mo> </mrow> <mo>|</mo> </mrow> <mn>2</mn> </msup> </mrow> </semantics></math> for (<b>a1</b>) <math display="inline"><semantics> <mrow> <mi>ϵ</mi> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math> and (<b>b1</b>) <math display="inline"><semantics> <mrow> <mi>ϵ</mi> <mo>=</mo> <mn>0.05</mn> </mrow> </semantics></math> (red lines), compared with the dynamics for <math display="inline"><semantics> <mrow> <mi>ϵ</mi> <mo>=</mo> <mn>0.0</mn> </mrow> </semantics></math> (blue line). Snapshots of Husimi distribution corresponding to the first spin sector at different times for (<b>a2</b>,<b>a3</b>) <math display="inline"><semantics> <mrow> <mi>ϵ</mi> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math> and (<b>b2</b>,<b>b3</b>) <math display="inline"><semantics> <mrow> <mi>ϵ</mi> <mo>=</mo> <mn>0.05</mn> </mrow> </semantics></math>. The green line in (<b>a1</b>) exhibits a rapid relaxation of <math display="inline"><semantics> <mrow> <mi>F</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math> starting from the initial state <math display="inline"><semantics> <mrow> <mrow> <mo>|</mo> <mi>ϕ</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mi>θ</mi> <mo>=</mo> <mi>π</mi> <mo>/</mo> <mn>2</mn> <mo>⟩</mo> </mrow> <mo>⊗</mo> <mrow> <mo>|</mo> <msub> <mi>m</mi> <mrow> <mn>2</mn> <mi>z</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>⟩</mo> </mrow> </mrow> </semantics></math> for <math display="inline"><semantics> <mrow> <mi>ϵ</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, indicating a complete loss of memory of the initial state (ergodic dynamics), as it is not a linear combination of the perfectly projected states.</p>
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<p>Quantum dynamics starting from the initial state <math display="inline"><semantics> <mrow> <mrow> <mi>ϕ</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mi>θ</mi> <mo>=</mo> <mi>π</mi> <mo>/</mo> <mn>2</mn> </mrow> <mo>⊗</mo> <mrow> <msub> <mi>m</mi> <mrow> <mn>2</mn> <mi>z</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mrow> </mrow> </semantics></math> at <span class="html-italic">E</span> ∼ 0: (<b>a</b>–<b>d</b>) Time evolution of <math display="inline"><semantics> <mrow> <mover> <mrow> <mo stretchy="false">〈</mo> <msub> <mover accent="true"> <mi>S</mi> <mo stretchy="false">^</mo> </mover> <mrow> <mn>1</mn> <mi>x</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>/</mo> <mi>S</mi> <mo stretchy="false">〉</mo> </mrow> <mo>¯</mo> </mover> </mrow> </semantics></math> for different values of <math display="inline"><semantics> <mi>ϵ</mi> </semantics></math>. Fluctuation dynamics of the spin operators (<b>e</b>) <math display="inline"><semantics> <mrow> <msubsup> <mover accent="true"> <mi>S</mi> <mo>¯</mo> </mover> <mrow> <mn>1</mn> <mi>z</mi> </mrow> <mrow> <mi>f</mi> <mi>l</mi> <mi>u</mi> <mi>c</mi> </mrow> </msubsup> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> and (<b>f</b>) <math display="inline"><semantics> <mrow> <msubsup> <mover accent="true"> <mi>S</mi> <mo>¯</mo> </mover> <mrow> <mn>1</mn> <mi>x</mi> </mrow> <mrow> <mi>f</mi> <mi>l</mi> <mi>u</mi> <mi>c</mi> </mrow> </msubsup> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>. The dotted line denotes the microcanonical value given by Equation (<a href="#FD28-condensedmatter-10-00005" class="html-disp-formula">28</a>). Note that, <math display="inline"><semantics> <mover accent="true"> <mrow> <mo stretchy="false">〈</mo> <mo>·</mo> <mo stretchy="false">〉</mo> </mrow> <mo>¯</mo> </mover> </semantics></math> and <math display="inline"><semantics> <mover accent="true"> <mo>·</mo> <mo>¯</mo> </mover> </semantics></math> represent averaging over an ensemble of 100 random matrices.</p>
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<p>OTOC dynamics: Time evolution of <math display="inline"><semantics> <mrow> <msub> <mover accent="true"> <mi>C</mi> <mo>¯</mo> </mover> <mrow> <mi>t</mi> <mi>o</mi> <mi>t</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> for eigenstate with minimum entanglement in a small energy density window <math display="inline"><semantics> <mrow> <mo>Δ</mo> <mi>E</mi> <mo>=</mo> <mn>0.045</mn> <mo>&lt;</mo> <mn>1</mn> <mo>/</mo> <mi>S</mi> </mrow> </semantics></math> around <span class="html-italic">E</span> ∼ 0. The inset shows the OTOC dynamics zoomed at long time and the dotted line denotes the microcanonical value given by Equation (<a href="#FD34-condensedmatter-10-00005" class="html-disp-formula">34</a>). Note that, the OTOC dynamics is averaged over 100 ensembles of random matrices.</p>
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17 pages, 1232 KiB  
Article
Simulation of Fidelity in Entanglement-Based Networks with Repeater Chains
by David Pérez Castro, Ana Fernández Vilas, Manuel Fernández Veiga, Mateo Blanco Rodríguez and Rebeca P. Díaz Redondo
Appl. Sci. 2024, 14(23), 11270; https://doi.org/10.3390/app142311270 - 3 Dec 2024
Viewed by 634
Abstract
We implement a set of simulation experiments in NetSquid specifically designed to estimate the end-to-end fidelity across a path of quantum repeaters or quantum switches. The switch model includes several generalizations that are not currently available in other tools and are useful for [...] Read more.
We implement a set of simulation experiments in NetSquid specifically designed to estimate the end-to-end fidelity across a path of quantum repeaters or quantum switches. The switch model includes several generalizations that are not currently available in other tools and are useful for gaining insight into practical and realistic quantum network engineering problems: an arbitrary number of memory registers at the switches, simplicity in including entanglement distillation mechanisms, arbitrary switching topologies, and routing protocols. An illustrative case study is presented, namely a comparison in terms of performance between a repeater chain where repeaters can only swap sequentially and a single switch equipped with multiple memory registers that is able to handle multiple swapping requests. Full article
(This article belongs to the Section Quantum Science and Technology)
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Figure 1

Figure 1
<p>Illustration of a quantum switch. In this case, there is one switch connected to multiple nodes, with one quantum memory position dedicated to each end node. Alice and Bob (highlighted) request communication with the network controller, which has knowledge of the state of the full network. By sharing an Einstein–Podolsky–Rose (EPR) pair with the switch and the action of entanglement swapping, they can establish a virtual link.</p>
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<p>Overview of software architecture in NetSquid (extracted from [<a href="#B13-applsci-14-11270" class="html-bibr">13</a>]).</p>
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<p>Scheme for the quantum communication scenario, including the main quantum mechanism, swapping and correct protocols and purification.</p>
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<p>Implementation architecture of the relay network in NetSquid.</p>
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<p>Architecture discussed in <a href="#sec6dot2-applsci-14-11270" class="html-sec">Section 6.2</a>. A two-hop network is established, and the program arranges and identifies the fidelity of the <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>18</mn> </mrow> </semantics></math> entangled pairs for each of the connections. The route with the highest fidelity is selected to drive the communication.</p>
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<p>(Color online) Fidelity average for the two different protocols discussed, PS and SP. These simulations have been performed in a network of 3 nodes. Error bars correspond to 1 standard deviation.</p>
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<p>Comparison of the impact on fidelity by varying the number of intermediate QR (<b>a</b>) and by varying the number of memory positions in a QR (<b>b</b>). (<b>a</b>) Fidelity by varying the number of QRs (4 memory positions) equally distributed across <math display="inline"><semantics> <mrow> <mi>L</mi> <mo>=</mo> <mn>1000</mn> </mrow> </semantics></math> km. (<b>b</b>) Fidelity by varying the number of memory positions in 2-hops with one QR and an internode distance of 500 km. Error bars represent the standard deviation of multiple simulations.</p>
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<p>Simulation for a three-node network applying one round of the DEJMPS distillation protocol.</p>
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<p>Representation of the theoretical increment in the fidelity after applying several rounds of purification. The total depolarizing probability of the channel is represented in the <span class="html-italic">x</span>-axis vs. the fidelity output of the process.</p>
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