quantum-computing

GPU-accelerated Effective Hamiltonian Calculator

Abhishek Chakraborty, Taylor L. Patti, Brucek Khailany, Andrew N. Jordan, and Anima Anandkumar

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⚡ Quantum Brief

Researchers developed a GPU-accelerated Python package for calculating effective Hamiltonians in large quantum systems, achieving up to 15x speedup for NPAD and 42x for Magnus expansion compared to CPU/QuTiP. The open-source tool, qCH_eff, uses CuPy for GPU optimization, enabling analysis of circuit-QED models like the Jaynes-Cummings-Hubbard system with minimal API complexity. NPAD implements iterative Schrieffer-Wolff transformations via Givens rotations, diagonalizing two-level subspaces early to reduce computational cost for many-body quantum systems. The Magnus expansion parallelizes time-evolution simulations, guaranteeing accuracy at characteristic intervals—ideal for rapidly driven quantum systems like 10-spin chains. Published in December 2025, the work demonstrates 100x faster diagonalization for modest systems, with GPU gains reaching 300x for small-to-medium sizes.

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AbstractEffective Hamiltonian calculations for large quantum systems can be both analytically intractable and numerically expensive using standard techniques. In this manuscript, we present numerical techniques inspired by Nonperturbative Analytical Diagonalization (NPAD) and the Magnus expansion for the efficient calculation of effective Hamiltonians. While these tools are appropriate for a wide array of applications, we here demonstrate their utility for models that can be realized in circuit-QED settings. Our numerical techniques are available as an open-source Python package, $\text{qCH}_\text{eff}$, which is available on GitHub and PyPI. We use the CuPy library for GPU-acceleration and report up to 15x speedup on GPU over CPU for NPAD, and up to 42x speedup for the Magnus expansion (compared to QuTiP), for large system sizes.Featured image: (a) Overview of the $\text{qCH}_\text{eff}$ package API. The package implements two different techniques, NPAD and Magnus, with a minimal API for users. (b) The NPAD algorithm implements an iterative Schrieffer-Wolff transformation (SWT). For a large many-body quantum system, NPAD can efficiently compute solutions leveraging symmetries thereof by iteratively targeting two-level subspaces of the full system. Operations called Givens rotations exactly diagonalize these two-level subspaces and iteration can be stopped early once the desired effective Hamiltonian is obtained. (c) The Magnus expansion can efficiently and accurately simulate time-evolution for quantum systems with rapid time dependence and guarantees accuracy at multiples of a characteristic time interval, called the Magnus interval. In contrast to sequential integration schemes, the Magnus expansion can be calculated in parallel for all intervals.Our numerical techniques are available as an open-source Python package, $\text{qCH}_\text{eff}$, which is available on GitHub and PyPI. Popular summaryAll but the simplest quantum systems require large amounts of computer memory and processing power for simulation and analysis. Many techniques have been developed to analyze such systems efficiently. Effective Hamiltonians capture the behavior of interest for quantum systems while reducing computational cost. Our work presents numerical implementations of two effective Hamiltonians calculation techniques, distributed as a Python package called $\text{qCH}_\text{eff}$. $\text{qCH}_\text{eff}$ uses NumPy and SciPy on CPU and CuPy on GPU efficient calculation. We leverage optimized matrix operations on GPUs to enable analysis of larger quantum systems than possible before. Many-body quantum systems grow exponentially in computer memory with their size. A complete description of such a system can be obtained using numerical diagonalization, but this becomes intractable for large systems. The Schrieffer-Wolff transformation (SWT) is a common but analytically tedious technique to compute a minimal effective description for such large systems. The first technique in our work, called Nonperturbative Analytical Diagonalization (NPAD) is an iterative SWT that leverages known symmetries to compute the solution efficiently. We show that NPAD can efficiently compute properties of the Jaynes-Cummings-Hubbard model, which describes the interaction between atom-cavity systems arranged in a ring. NPAD can be up to 100x faster than diagonalization even for modestly large quantum systems and runs up to 15x faster on GPU compared to CPU. The second technique, the Magnus expansion, applies to systems which vary rapidly in time. Typically, sequential integration schemes with a small time-step, which can be computationally expensive, are used to capture the rapid dynamics for such systems. The Magnus expansion guarantees an accurate solution at multiples of a characteristic time period, called the Magnus interval. Additionally, all Magnus intervals can be computed in parallel. The Magnus expansion is more accurate compared to some common analytical approximations and also sequential integration with large time steps. We optimize a control pulse for a linear chain of 10 spins using the Magnus expansion and observe up to a 42x speedup over conventional sequential integration.The GPU implementation is up to 300x faster compared to CPU for modest system sizes.► BibTeX data@article{Chakraborty2025gpuaccelerated, doi = {10.22331/q-2025-12-15-1946}, url = {https://doi.org/10.22331/q-2025-12-15-1946}, title = {{GPU}-accelerated {E}ffective {H}amiltonian {C}alculator}, author = {Chakraborty, Abhishek and Patti, Taylor L. and Khailany, Brucek and Jordan, Andrew N. and Anandkumar, Anima}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {9}, pages = {1946}, month = dec, year = {2025} }► References [1] Flemming Jørgensen. ``Effective hamiltonians''. 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Clerk. ``Designing high-fidelity two-qubit gates between fluxonium qubits''. PRX Quantum 5, 040317 (2024). arXiv:2403.07242. https://doi.org/10.1103/prxquantum.5.040317 arXiv:2403.07242Cited byOn Crossref's cited-by service no data on citing works was found (last attempt 2025-12-28 10:46:47). Could not fetch ADS cited-by data during last attempt 2025-12-28 10:46:48: No response from ADS or unable to decode the received json data when getting the list of citing works.This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions. AbstractEffective Hamiltonian calculations for large quantum systems can be both analytically intractable and numerically expensive using standard techniques. In this manuscript, we present numerical techniques inspired by Nonperturbative Analytical Diagonalization (NPAD) and the Magnus expansion for the efficient calculation of effective Hamiltonians. While these tools are appropriate for a wide array of applications, we here demonstrate their utility for models that can be realized in circuit-QED settings. Our numerical techniques are available as an open-source Python package, $\text{qCH}_\text{eff}$, which is available on GitHub and PyPI. We use the CuPy library for GPU-acceleration and report up to 15x speedup on GPU over CPU for NPAD, and up to 42x speedup for the Magnus expansion (compared to QuTiP), for large system sizes.Featured image: (a) Overview of the $\text{qCH}_\text{eff}$ package API. The package implements two different techniques, NPAD and Magnus, with a minimal API for users. (b) The NPAD algorithm implements an iterative Schrieffer-Wolff transformation (SWT). For a large many-body quantum system, NPAD can efficiently compute solutions leveraging symmetries thereof by iteratively targeting two-level subspaces of the full system. Operations called Givens rotations exactly diagonalize these two-level subspaces and iteration can be stopped early once the desired effective Hamiltonian is obtained. (c) The Magnus expansion can efficiently and accurately simulate time-evolution for quantum systems with rapid time dependence and guarantees accuracy at multiples of a characteristic time interval, called the Magnus interval. In contrast to sequential integration schemes, the Magnus expansion can be calculated in parallel for all intervals.Our numerical techniques are available as an open-source Python package, $\text{qCH}_\text{eff}$, which is available on GitHub and PyPI. Popular summaryAll but the simplest quantum systems require large amounts of computer memory and processing power for simulation and analysis. Many techniques have been developed to analyze such systems efficiently. Effective Hamiltonians capture the behavior of interest for quantum systems while reducing computational cost. Our work presents numerical implementations of two effective Hamiltonians calculation techniques, distributed as a Python package called $\text{qCH}_\text{eff}$. $\text{qCH}_\text{eff}$ uses NumPy and SciPy on CPU and CuPy on GPU efficient calculation. We leverage optimized matrix operations on GPUs to enable analysis of larger quantum systems than possible before. Many-body quantum systems grow exponentially in computer memory with their size. A complete description of such a system can be obtained using numerical diagonalization, but this becomes intractable for large systems. The Schrieffer-Wolff transformation (SWT) is a common but analytically tedious technique to compute a minimal effective description for such large systems. The first technique in our work, called Nonperturbative Analytical Diagonalization (NPAD) is an iterative SWT that leverages known symmetries to compute the solution efficiently. We show that NPAD can efficiently compute properties of the Jaynes-Cummings-Hubbard model, which describes the interaction between atom-cavity systems arranged in a ring. NPAD can be up to 100x faster than diagonalization even for modestly large quantum systems and runs up to 15x faster on GPU compared to CPU. The second technique, the Magnus expansion, applies to systems which vary rapidly in time. Typically, sequential integration schemes with a small time-step, which can be computationally expensive, are used to capture the rapid dynamics for such systems. The Magnus expansion guarantees an accurate solution at multiples of a characteristic time period, called the Magnus interval. Additionally, all Magnus intervals can be computed in parallel. The Magnus expansion is more accurate compared to some common analytical approximations and also sequential integration with large time steps. We optimize a control pulse for a linear chain of 10 spins using the Magnus expansion and observe up to a 42x speedup over conventional sequential integration.The GPU implementation is up to 300x faster compared to CPU for modest system sizes.► BibTeX data@article{Chakraborty2025gpuaccelerated, doi = {10.22331/q-2025-12-15-1946}, url = {https://doi.org/10.22331/q-2025-12-15-1946}, title = {{GPU}-accelerated {E}ffective {H}amiltonian {C}alculator}, author = {Chakraborty, Abhishek and Patti, Taylor L. and Khailany, Brucek and Jordan, Andrew N. and Anandkumar, Anima}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {9}, pages = {1946}, month = dec, year = {2025} }► References [1] Flemming Jørgensen. ``Effective hamiltonians''. Molecular Physics 29, 1137–1164 (1975). https://doi.org/10.1080/00268977500100971 [2] Georges Jolicard. ``Effective Hamiltonian Theory and Molecular Dynamics''. Annual Review of Physical Chemistry 46, 83–108 (1995). https://doi.org/10.1146/annurev.pc.46.100195.000503 [3] Daniel Zeuch, Fabian Hassler, Jesse J. Slim, and David P. DiVincenzo. ``Exact rotating wave approximation''. Annals of Physics 423, 168327 (2020). https://doi.org/10.1016/j.aop.2020.168327 [4] Easwar Magesan and Jay M. Gambetta. ``Effective Hamiltonian models of the cross-resonance gate''. Physical Review A 101, 052308 (2020). https://doi.org/10.1103/PhysRevA.101.052308 [5] Gioele Consani and Paul A Warburton. ``Effective Hamiltonians for interacting superconducting qubits: Local basis reduction and the Schrieffer–Wolff transformation''. New Journal of Physics 22, 053040 (2020). https://doi.org/10.1088/1367-2630/ab83d1 [6] Jayameenakshi Venkatraman, Xu Xiao, Rodrigo G. Cortiñas, Alec Eickbusch, and Michel H. Devoret. ``Static Effective Hamiltonian of a Rapidly Driven Nonlinear System''.

The European Physical Journal Plus 129, 12 (2014). https://doi.org/10.1140/epjp/i2014-14012-8 [35] William H. Press. ``Numerical Recipes 3rd Edition: The Art of Scientific Computing''.

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GPU-accelerated Effective Hamiltonian Calculator

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