Expanding Hardware-Efficiently Manipulable Hilbert Space via Hamiltonian Embedding

Jiaqi Leng; Joseph Li; Yuxiang Peng; Xiaodi Wu

doi:10.22331/q-2025-09-11-1857

Expanding Hardware-Efficiently Manipulable Hilbert Space via Hamiltonian Embedding

Jiaqi Leng^1,3,4, Joseph Li^2,3, Yuxiang Peng^2,3, and Xiaodi Wu^2,3

¹Department of Mathematics, University of Maryland, College Park, USA
²Department of Computer Science, University of Maryland, College Park, USA
³Joint Center for Quantum Information and Computer Science, University of Maryland
⁴Department of Mathematics and Simons Institute for the Theory of Computing, University of California, Berkeley

Published:	2025-09-11, volume 9, page 1857
Editor:	Jin-Peng Liu
Eprint:	arXiv:2401.08550v2
Doi:	https://doi.org/10.22331/q-2025-09-11-1857
Citation:	Quantum 9, 1857 (2025).

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Abstract

Many promising quantum applications depend on the efficient quantum simulation of an exponentially large sparse Hamiltonian, a task known as sparse Hamiltonian simulation, which is fundamentally important in quantum computation. Although several theoretically appealing quantum algorithms have been proposed for this task, they typically require a black-box query model of the sparse Hamiltonian, rendering them impractical for near-term implementation on quantum devices.
In this paper, we propose a technique named $\textit{Hamiltonian embedding}$. This technique simulates a desired sparse Hamiltonian by embedding it into the evolution of a larger and more structured quantum system, allowing for more efficient simulation through hardware-efficient operations. We conduct a systematic study of this new technique and demonstrate significant savings in computational resources for implementing prominent quantum applications. As a result, we can now experimentally realize quantum walks on complicated graphs (e.g., binary trees, glued-tree graphs), quantum spatial search, and the simulation of real-space Schrödinger equations on current trapped-ion and neutral-atom platforms. Given the fundamental role of Hamiltonian evolution in the design of quantum algorithms, our technique markedly expands the horizon of implementable quantum advantages in the NISQ era.

Featured image: An illustration of perturbative Hamiltonian embedding. The dynamics of a target Hamiltonian $A$ are embedded within a subspace $\mathcal{S}$ in a larger hardware Hilbert space. With respect to $\mathcal{S}$ and $\mathcal{S}^\perp$, $H$ is block-diagonal with $A$ being the upper-left block located by the projector $P_{\mathcal{S}}$.

Popular summary

A central goal for quantum computers is to efficiently simulate the dynamics of complex quantum systems, a task governed by large, sparse Hamiltonians. However, prominent quantum algorithms for this task are often impractical for current hardware, as they require deep, error-prone quantum circuits and complex input models.

In this paper, we introduce a more direct and resource-efficient technique called "Hamiltonian embedding". The strategy involves embedding the target sparse Hamiltonian as a block within a larger, more structured Hamiltonian that is easier to implement on a physical device. By evolving this larger system using the hardware's native operations, the desired simulation occurs naturally within a protected subspace. This method bypasses inefficient compilation steps and is tailored to the specific capabilities of the hardware, significantly reducing computational overhead.

Using this technique, we successfully demonstrated several applications on existing trapped-ion and neutral-atom quantum computers, including quantum walks on complex graphs and simulating real-space quantum dynamics. These experiments showcase the ability to tackle problems previously considered infeasible on near-term devices, expanding the horizon for achieving quantum advantage.

► BibTeX data

@article{Leng2025expandinghardware,
  doi = {10.22331/q-2025-09-11-1857},
  url = {https://doi.org/10.22331/q-2025-09-11-1857},
  title = {Expanding {H}ardware-{E}fficiently {M}anipulable {H}ilbert {S}pace via {H}amiltonian {E}mbedding},
  author = {Leng, Jiaqi and Li, Joseph and Peng, Yuxiang and Wu, Xiaodi},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1857},
  month = sep,
  year = {2025}
}

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