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

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Interdisciplinary Mathematics

Infinite-dimensional Hamiltonian systems in quantum computing simulations

Hemant Kumar Balasaheb Jadhav, Dinesh Goyal, Tanya Singh, Yatri Davda, Suman Kumar Swarnkar, Dharmesh Dhabliya

Published Jan 1, 2025• 0 citations
Publisher

Abstract

There is a powerful mathematical method for simulating quantum dynamics that goes beyond finite qubit representations called infinite-dimensional Hamiltonian systems. These systems make it possible to describe quantum fields, open quantum systems, and hybrid designs more accurately by simulating unbounded operators, spectral decompositions, and continuous-variable interactions. Quantum algorithms for partial differential equations, many-body physics, and error-tolerant models of infinite Hilbert spaces are all areas where they are useful. Adding Hamiltonians with infinite dimensions to computer systems makes it possible to use both classical approximation and quantum hardware. This method makes it easier to scale up, improves accuracy when dynamics aren’t simple, and sets the stage for more advanced robust quantum computing models in chemistry, information sciences, and physics.

DOI
10.47974/jim-2378

Related papers

Infinite-dimensional Extension of the Linear Combination of Hamiltonian Simulation: Theorems and Applications

Lu, Rundi, Hao-En Li, Zhengwei Liu +1 more

We generalize the Linear Combination of Hamiltonian Simulation (LCHS) formula [An, Liu, Lin, Phys. Rev. Lett. 2023] to simulate time-evolution operators in infinite-dimensional spaces, including scenarios involving unbounded operators. This extension, named Inf-LCHS for short, bridges the gap between finite-dimensional quantum simulations and the broader class of infinite-dimensional quantum dynamics governed by partial differential equations (PDEs). Furthermore, we propose two sampling methods by integrating the infinite-dimensional LCHS with Gaussian quadrature schemes (Inf-LCHS-Gaussian) or Monte Carlo integration schemes (Inf-LCHS-MC). We demonstrate the applicability of the Inf-LCHS theorem to a wide range of non-Hermitian dynamics, including linear parabolic PDEs, queueing models (birth-or-death processes), Schrödinger equations with complex potentials, Lindblad equations, and black hole thermal field equations. Our analysis provides insights into simulating general linear dynamics using a finite number of quantum dynamics and includes cost estimates for the corresponding quantum algorithms.

arXiv:2502.19688•Feb 27, 2025•0 citations

Quantum Simulations

Antti Regelin

Quantum simulation involves using a quantum system to study another quantum system. The proposal to simulate quantum physics using quantum mechanical devices, as outlined by Richard Feynman in 1982, stems from classical computers' challenges in simulating quantum systems. The pursuit of simulating complex quantum systems has led to the intersection of quantum mechanics and quantum computing. This offers new opportunities to understand and manipulate the quantum world. Central to describing quantum dynamics is the Hamiltonian operator $\hat{H}$, which governs the time evolution of physical systems and serves as the mathematical foundation for quantum simulations. This thesis explores the methodologies and advancements in simulating Hamiltonian operators on quantum computers, focusing on decomposing these operators into quantum gates and the efficiency gains over classical computational approaches. Initially, the thesis explains the fundamental theory and basics of quantum computing. Further it dives into the foundations of the Hamiltonian operator within quantum mechanics, highlighting its role in describing the dynamics of quantum systems. The complexity and high-dimensional nature of Hamiltonian matrices necessitate using quantum computers to simulate quantum phenomena accurately and efficiently. The process of simulating the Hamiltonian operator on a quantum computer involves a detailed examination of the strategies for decomposing the operator into a sequence of quantum gates. This decomposition is crucial for translating the abstract mathematical description of quantum systems into the physical operations of a quantum computer. By utilizing the principles of quantum mechanics, such as superposition and entanglement, quantum gates manipulate qubits to mimic the evolution of quantum states as the Hamiltonian operator dictates. Latter portions of the thesis are dedicated to comparing the efficiency of quantum simulations against traditional classical simulations and inspecting efficient quantum simulations. Quantum computers offer a polynomial or an exponential speedup for certain classes of problems, such as those involving highly entangled quantum states or requiring the simulation of complex quantum dynamics. This advantage is due to the quantum computer's ability to naturally represent and process quantum information - a task for which classical computers require exponential resources as the system size increases. The efficiency of simulating the Hamiltonian operator on a quantum computer is further enhanced through optimization techniques and algorithmic improvements. These include error mitigation strategies and the development of quantum algorithms tailored to specific types of Hamiltonian operators. The thesis presents a comprehensive analysis of these methods, demonstrating how they make quantum simulations more practical and accessible for scientific research and technological development. In conclusion this thesis is a literature review that examines the transformative potential of quantum simulations. By efficiently simulating the Hamiltonian operator on quantum computers researchers can explore quantum phenomena with precision, paving the way for breakthroughs in quantum physics, materials science, and beyond. The advancements in quantum computing technology and algorithmic approaches discussed here highlight the current capabilities and future explorations in the quantum realm.

•Sep 9, 2026•0 citations

Operator Algebras and Quantum Dynamics: Semigroup Methods in Infinite Dimensions

Anam Ghafoor, Ume Roman, Sawaiba Yousaf

Background: Operator algebras and quantum dynamical semigroups are the mathematical tools of choice for understanding open quantum systems, especially in infinite-dimensional spaces. Such tools are extremely important for the time development of quantum systems out of equilibium under the influence of environment, where conventional finite dimensional methods do not include all possibilities for interaction. Objective: The objects of the present research are the r-properties and the decompositions of generators of quantum dynamical semigroups on infinite-dimensional spaces including their Hamiltonian and dissipative part, semigroup continuity, stationary states, and positivity improvement. The aim is to set a fundament in order to think about long-time dynamics in open quantum systems. Methods: The theorem Hille–Yosida is used to get conditions for the semigroup strongly continuous generated by unbounded operators. The time-evolution operator of the system is approximated with both Gaussian quadrature and Monte Carlo integration during infinitesimal time slices, by using Inf-LCHS in conjunction. Generators were studied under decomposition of their Hamiltonian and dissipative parts, while commutant algebras were investigated for irreducibility and positivity enhancement. Results: Most systems studied for these properties presented a singular decomposition, strong and weak semigroup continuity, existence of stationary states and positivity improvement. The counterexamples were the systems with in a lack of verification for dissipative parts, and they lost smoothing property, uniqueness of decomposition, partial semigroup continuity or large errors in approximation when we perform numerically. Conclusions: The combination of the methods of operator algebra with state-of-the-art simulation techniques yields a powerful approach to study infinite-dimensional quantum dynamics. Although dissipative components are not yet completely characterized, the results improve our understanding of generator structures and long-term system behavior which will help future work in quantum statistical mechanics and quantum information theory.

Annual Methodological Archive Research Review•Dec 16, 2025•0 citations

Dilation theorem via Schrödingerisation, with applications to the quantum simulation of differential equations

Junpeng Hu, Jin, Shi, Nana Liu +1 more

Nagy's unitary dilation theorem in operator theory asserts the possibility of dilating a contraction into a unitary operator. When used in quantum computing, its practical implementation primarily relies on block-encoding techniques, based on finite-dimensional scenarios. In this study, we delve into the recently devised Schrödingerisation approach and demonstrate its viability as an alternative dilation technique. This approach is applicable to operators in the form of $V(t)=\exp(-At)$, which arises in wide-ranging applications, particularly in solving linear ordinary and partial differential equations. Importantly, the Schrödingerisation approach is adaptable to both finite and infinite-dimensional cases, in both countable and uncountable domains. For quantum systems lying in infinite dimensional Hilbert space, the dilation involves adding a single infinite dimensional mode, and this is the continuous-variable version of the Schrödingerisation procedure which makes it suitable for analog quantum computing. Furthermore, by discretising continuous variables, the Schrödingerisation method can also be effectively employed in finite-dimensional scenarios suitable for qubit-based quantum computing.

arXiv:2309.16262•Sep 28, 2023•1 citation

Quantum Computing and Simulation of Many-Body Systems

Dr.Prakash Dubey

Quantum computing introduces a revolutionary approach to solving problems that are computationally beyond the reach of classical systems. Among its most promising applications is the simulation of many-body systems, where strong particle interactions lead to complex quantum behaviors. Traditional computational techniques often fail due to exponential resource requirements, but quantum simulators can efficiently model these interactions through entanglement and superposition. Recent advances in superconducting qubits, trapped ions, and photonic quantum devices have enabled small- scale experimental demonstrations of spin models, Hubbard systems, and correlated electron dynamics. These developments offer new insights into condensed matter physics and quantum phase transitions. As quantum technologies continue to evolve, they hold the potential to transform the study of materials, superconductivity, and other many-body phenomena with remarkable precision and scalability.

International Journal on Science and Technology•Feb 10, 2026•0 citations

Computational Complexity in the Analysis of Quantum Operations

Miosz Michalski

Mathematical theory of infinite dimensional Hilbert spaces and the theory of operator algebras acting in such spaces (or C algebras in a more abstract approach) provide a standard setting for the formulation of modern quantum mechanics. On the other hand, experimental and theoretical progress achieved in the field of quantum information theory in the last two decades has indicated the practical and technological importance of low-dimensional quantum systems, where only a few basic modes play a significant role. Such modes can often be effectively decoupled from the rest of the system and controlled separately, providing physical realizations of qubits, qutrits and other basic information carriers. Regardless of concrete physical realization, be it photon polarization, electron or nuclear spin, charge in Josephson junctions to name just a few, the mathematical description of such systems requires only finite-dimensional Hilbert space language and finite-dimensional matrix algebras. Such structures are in principle computationally manageable in sharp contrast to the infinite dimensional ones.

InTech eBooks•Jul 24, 2013•3 citations
There is a powerful mathematical method for simulating quantum dynamics that goes beyond finite qubit representations called infinite-dimensional Hamiltonian systems. These systems make it possible to describe quantum fields, open quantum systems, and hybrid designs more accurately by simulating unbounded operators, spectral decompositions, and continuous-variable interactions. Quantum algorithms for partial differential equations, many-body physics, and error-tolerant models of infinite Hilbert spaces are all areas where they are useful. Adding Hamiltonians with infinite dimensions to computer systems makes it possible to use both classical approximation and quantum hardware. This method makes it easier to scale up, improves accuracy when dynamics aren’t simple, and sets the stage for more advanced robust quantum computing models in chemistry, information sciences, and physics.
We generalize the Linear Combination of Hamiltonian Simulation (LCHS) formula [An, Liu, Lin, Phys. Rev. Lett. 2023] to simulate time-evolution operators in infinite-dimensional spaces, including scenarios involving unbounded operators. This extension, named Inf-LCHS for short, bridges the gap between finite-dimensional quantum simulations and the broader class of infinite-dimensional quantum dynamics governed by partial differential equations (PDEs). Furthermore, we propose two sampling methods by integrating the infinite-dimensional LCHS with Gaussian quadrature schemes (Inf-LCHS-Gaussian) or Monte Carlo integration schemes (Inf-LCHS-MC). We demonstrate the applicability of the Inf-LCHS theorem to a wide range of non-Hermitian dynamics, including linear parabolic PDEs, queueing models (birth-or-death processes), Schrödinger equations with complex potentials, Lindblad equations, and black hole thermal field equations. Our analysis provides insights into simulating general linear dynamics using a finite number of quantum dynamics and includes cost estimates for the corresponding quantum algorithms.
Quantum simulation involves using a quantum system to study another quantum system. The proposal to simulate quantum physics using quantum mechanical devices, as outlined by Richard Feynman in 1982, stems from classical computers' challenges in simulating quantum systems. The pursuit of simulating complex quantum systems has led to the intersection of quantum mechanics and quantum computing. This offers new opportunities to understand and manipulate the quantum world. Central to describing quantum dynamics is the Hamiltonian operator H^\hat{H}H^, which governs the time evolution of physical systems and serves as the mathematical foundation for quantum simulations. This thesis explores the methodologies and advancements in simulating Hamiltonian operators on quantum computers, focusing on decomposing these operators into quantum gates and the efficiency gains over classical computational approaches. Initially, the thesis explains the fundamental theory and basics of quantum computing. Further it dives into the foundations of the Hamiltonian operator within quantum mechanics, highlighting its role in describing the dynamics of quantum systems. The complexity and high-dimensional nature of Hamiltonian matrices necessitate using quantum computers to simulate quantum phenomena accurately and efficiently. The process of simulating the Hamiltonian operator on a quantum computer involves a detailed examination of the strategies for decomposing the operator into a sequence of quantum gates. This decomposition is crucial for translating the abstract mathematical description of quantum systems into the physical operations of a quantum computer. By utilizing the principles of quantum mechanics, such as superposition and entanglement, quantum gates manipulate qubits to mimic the evolution of quantum states as the Hamiltonian operator dictates. Latter portions of the thesis are dedicated to comparing the efficiency of quantum simulations against traditional classical simulations and inspecting efficient quantum simulations. Quantum computers offer a polynomial or an exponential speedup for certain classes of problems, such as those involving highly entangled quantum states or requiring the simulation of complex quantum dynamics. This advantage is due to the quantum computer's ability to naturally represent and process quantum information - a task for which classical computers require exponential resources as the system size increases. The efficiency of simulating the Hamiltonian operator on a quantum computer is further enhanced through optimization techniques and algorithmic improvements. These include error mitigation strategies and the development of quantum algorithms tailored to specific types of Hamiltonian operators. The thesis presents a comprehensive analysis of these methods, demonstrating how they make quantum simulations more practical and accessible for scientific research and technological development. In conclusion this thesis is a literature review that examines the transformative potential of quantum simulations. By efficiently simulating the Hamiltonian operator on quantum computers researchers can explore quantum phenomena with precision, paving the way for breakthroughs in quantum physics, materials science, and beyond. The advancements in quantum computing technology and algorithmic approaches discussed here highlight the current capabilities and future explorations in the quantum realm.
Background: Operator algebras and quantum dynamical semigroups are the mathematical tools of choice for understanding open quantum systems, especially in infinite-dimensional spaces. Such tools are extremely important for the time development of quantum systems out of equilibium under the influence of environment, where conventional finite dimensional methods do not include all possibilities for interaction. Objective: The objects of the present research are the r-properties and the decompositions of generators of quantum dynamical semigroups on infinite-dimensional spaces including their Hamiltonian and dissipative part, semigroup continuity, stationary states, and positivity improvement. The aim is to set a fundament in order to think about long-time dynamics in open quantum systems. Methods: The theorem Hille–Yosida is used to get conditions for the semigroup strongly continuous generated by unbounded operators. The time-evolution operator of the system is approximated with both Gaussian quadrature and Monte Carlo integration during infinitesimal time slices, by using Inf-LCHS in conjunction. Generators were studied under decomposition of their Hamiltonian and dissipative parts, while commutant algebras were investigated for irreducibility and positivity enhancement. Results: Most systems studied for these properties presented a singular decomposition, strong and weak semigroup continuity, existence of stationary states and positivity improvement. The counterexamples were the systems with in a lack of verification for dissipative parts, and they lost smoothing property, uniqueness of decomposition, partial semigroup continuity or large errors in approximation when we perform numerically. Conclusions: The combination of the methods of operator algebra with state-of-the-art simulation techniques yields a powerful approach to study infinite-dimensional quantum dynamics. Although dissipative components are not yet completely characterized, the results improve our understanding of generator structures and long-term system behavior which will help future work in quantum statistical mechanics and quantum information theory.
Nagy's unitary dilation theorem in operator theory asserts the possibility of dilating a contraction into a unitary operator. When used in quantum computing, its practical implementation primarily relies on block-encoding techniques, based on finite-dimensional scenarios. In this study, we delve into the recently devised Schrödingerisation approach and demonstrate its viability as an alternative dilation technique. This approach is applicable to operators in the form of V(t)=exp⁡(−At)V(t)=\exp(-At)V(t)=exp(−At), which arises in wide-ranging applications, particularly in solving linear ordinary and partial differential equations. Importantly, the Schrödingerisation approach is adaptable to both finite and infinite-dimensional cases, in both countable and uncountable domains. For quantum systems lying in infinite dimensional Hilbert space, the dilation involves adding a single infinite dimensional mode, and this is the continuous-variable version of the Schrödingerisation procedure which makes it suitable for analog quantum computing. Furthermore, by discretising continuous variables, the Schrödingerisation method can also be effectively employed in finite-dimensional scenarios suitable for qubit-based quantum computing.
Quantum computing introduces a revolutionary approach to solving problems that are computationally beyond the reach of classical systems. Among its most promising applications is the simulation of many-body systems, where strong particle interactions lead to complex quantum behaviors. Traditional computational techniques often fail due to exponential resource requirements, but quantum simulators can efficiently model these interactions through entanglement and superposition. Recent advances in superconducting qubits, trapped ions, and photonic quantum devices have enabled small- scale experimental demonstrations of spin models, Hubbard systems, and correlated electron dynamics. These developments offer new insights into condensed matter physics and quantum phase transitions. As quantum technologies continue to evolve, they hold the potential to transform the study of materials, superconductivity, and other many-body phenomena with remarkable precision and scalability.
Mathematical theory of infinite dimensional Hilbert spaces and the theory of operator algebras acting in such spaces (or C algebras in a more abstract approach) provide a standard setting for the formulation of modern quantum mechanics. On the other hand, experimental and theoretical progress achieved in the field of quantum information theory in the last two decades has indicated the practical and technological importance of low-dimensional quantum systems, where only a few basic modes play a significant role. Such modes can often be effectively decoupled from the rest of the system and controlled separately, providing physical realizations of qubits, qutrits and other basic information carriers. Regardless of concrete physical realization, be it photon polarization, electron or nuclear spin, charge in Josephson junctions to name just a few, the mathematical description of such systems requires only finite-dimensional Hilbert space language and finite-dimensional matrix algebras. Such structures are in principle computationally manageable in sharp contrast to the infinite dimensional ones.