Hamiltonian Embedding

Hamiltonian embedding is a hardware-efficient approach to sparse Hamiltonian simulation that does not assume access to a black-box query model. This technique leverages both the sparsity structure of the input data and the resource efficiency of the underlying quantum hardware, enabling the deployment of interesting quantum applications on current quantum computers.

This is a joint work by Jiaqi Leng, Joseph Li, Yuxiang Peng and Xiaodi Wu. The paper is published in Quantum.

Code organization

The source code is organized as follows:

• src/experiments contains scripts required for running the real-machine experiments. It contains a file ionq_circuit_utils.py that handles the circuit compilation and sending jobs to IonQ. In addition, the directory is subdivided into three subdirectories for each of the three computational tasks described in the paper.
  • quantum_walk
  • spatial_search
  • real_space_sim

These three directories contain a file run_experiments.ipynb which is used for running the IonQ experiments. For the real-space simulation on QuEra, an additional notebook titled QuEra_real_space_sim.ipynb within the real_space_sim/QuEra directory.

• src/resource_estimation contains scripts required for the resource comparison between the conventional approach (i.e., the standard binary encoding) and Hamiltonian embeddings. Within this directory, the files are divided as follows.
  • scripts provides the main scripts for running the resource estimation.
  • data contains all of estimated Trotter number and gate counts, stored as .npz files.
  • plot contains scripts for plotting the total gate counts.
  • figures stores the .png files for the resource analysis.

Finally, src/fig_1a.py is used to generate the matrix presented in Fig 1a, which represents the unary embedding of a 5-by-5 tridiagonal matrix.

Usage

The code has been tested with Python 3.10 but should also work with some earlier versions such as 3.8 or 3.9. There are several dependencies used in this project. Below are the relevant packages, along with the tested versions.

• numpy 1.23.5
• scipy 1.11.1
• networkx 3.2.1
• amazon-braket-default-simulator 1.18.3
• amazon-braket-schemas 1.19.0
• amazon-braket-sdk 1.51.0
• qiskit 0.44.1
• pytket 1.18.0

Experiments

Important: In order to use the provided source code, it is necessary to create a file .env with an IonQ API key. For example, if your API key is 00000000000000000000000000000000, the .env would contain the following.

IONQ_API_KEY=00000000000000000000000000000000

All Python scripts should be run from the project directory hamiltonian-embedding.

To run the experiments, Jupyter notebooks are provided for each task with the filename run_experiments.ipynb.

• src/experiments/quantum_walk/run_experiments.ipynb
• src/experiments/spatial_search/run_experiments.ipynb
• src/experiments/real_space_sim/IonQ/run_experiments.ipynb

These notebooks can be run without modification to reproduce the experimental results presented in the paper.

Resource analysis

The empirical gate count comparison corresponding to each of the experiments (as listed in the tables of the main body) are computed using src/resource_estimation/scripts/empirical_resource_comparison, in which the parameters are chosen to be the same as those used for the real-machine experiments. This notebook estimates the gate counts needed for standard binary to simulate the Hamiltonian to the same accuracy as in the experiments. The gate counts for Hamiltonian embedding are directly taken from the circuits run in the experiments (i.e. found in the run_experiments.ipynb files).

For the systematic resource analysis of varying system sizes, we use the scripts in the directory resource_estimation/scripts/resource_estimation_{name}.py, where {name} is replaced by the name of the task. To run these scripts without modification, it is highly suggested that they are run on an HPC cluster. The typical runtime needed to run these scripts to completion is a few days (1-3 days depending on the task). For small system sizes, the empirical resource comparison may be run on a laptop or PC, typically taking a few minutes.

After running these scripts, the data is saved in resource_estimation/data, and the scripts in resource_estimation/plot are used to generate the figures presented in the paper.

Citation

@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}
}