Efficient Hamiltonian encoding algorithms for extracting quantum control mechanism as interfering pathway amplitudes in the Dyson series
1Princeton University
2Massachusetts Institute of Technology
| Published: | 2025-02-10, volume 9, page 1626 |
| Editor: | Jin-Peng Liu |
| Eprint: | arXiv:2406.05585v3 |
| Doi: | https://doi.org/10.22331/q-2025-02-10-1626 |
| Citation: | Quantum 9, 1626 (2025). |
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Abstract
Hamiltonian encoding is a methodology for revealing the mechanism behind the dynamics governing controlled quantum systems. In this paper, following Mitra and Rabitz [9], we define mechanism via pathways of eigenstates that describe the evolution of the system, where each pathway is associated with a complex-valued amplitude corresponding to a term in the Dyson series. The evolution of the system is determined by the constructive and destructive interference of these pathway amplitudes. Pathways with similar attributes can be grouped together into pathway classes. The amplitudes of pathway classes are computed by modulating the Hamiltonian matrix elements and decoding the subsequent evolution of the system rather than by direct computation of the individual terms in the Dyson series. The original implementation of Hamiltonian encoding was computationally intensive and became prohibitively expensive in large quantum systems. This paper presents two new encoding algorithms that calculate the amplitudes of pathway classes by using techniques from graph theory and algebraic topology to exploit patterns in the set of allowed transitions, greatly reducing the number of matrix elements that need to be modulated. These new algorithms provide an exponential decrease in both computation time and memory utilization with respect to the Hilbert space dimension of the system. To demonstrate the use of these techniques, they are applied to two illustrative state-to-state transition problems.
Featured image: By representing the quantum system as a graph and encoding only the edges not in a chosen spanning tree, Hermitian pathway classes can be viewed as deformations of a default pathway via topologically-nontrivial cycles. Reinterpreting the $[| 1 \rangle \to | 2 \rangle \to | 6 \rangle \to | 8 \rangle \to | 7 \rangle \to | 5 \rangle]^{\text H}$ Hermitian pathway class as a deformation of the $|1\rangle\to|5\rangle$ default pathway by counterclockwise cycles on the front and top faces enables the pathway class to be readily reconstructed from only its non-spanning-tree edges—namely, a forward traversal of $e_5$ and a backward traversal of $e_2$.
Popular summary
Mechanism analysis aims to give its user a quantitative and qualitative understanding of the mechanism underlying the evolution of a quantum system under an external control field. The Dyson perturbation series expansion enables us to interpret mechanism by decomposing the evolution of the system into individual discrete pathways between the states and ascribing a complex amplitude for each pathway; these pathway amplitudes constructively and destructively interfere to drive the actual evolution of the system. The Hamiltonian encoding mechanism analysis techniques of Mitra and Rabitz [Phys. Rev. A 67, 033407 (2003)] provide a tractable way to extract important groupings of pathway amplitudes (i.e. pathway class amplitudes) from the evolution of a modulated quantum system without direct computation of the individual amplitudes; unfortunately, these techniques are too computationally intensive to apply to all but the simplest of quantum systems.
This paper improves on Hamiltonian encoding by using ideas from algebraic graph theory to remove redundant information in the original algorithm; this both dramatically reduces computation time and provides a deeper, more topological interpretation of the resulting pathway class amplitudes. By proving that pathway classes encode the ways in which the quantum system winds around the quantum system’s state graph, this paper introduces two novel algorithms for computing pathway class amplitudes, both of which offer exponential improvements in both computation time and memory utilization over prior Hamiltonian encoding techniques while provably yielding the exact same output. These dramatic improvements in computational time will enable Hamiltonian encoding to be applied in real-world controlled quantum systems and facilitate a deeper understanding of the mechanism behind the dynamics of their evolution.
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[1] Jiahui Chen and David Cory, "Engineering precise and robust effective Hamiltonians", Physical Review A 113 4, 042409 (2026).
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