In this repository you can find the codes relative to the QHack 2024 submission of TheOffice215 team. Team members: Davide Cugini, Francesco Ghisoni, Angela Rosy Morgillo and Francesco Scala.

The Challange statement was:

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CHALLENGE: Bridging the Gap

Although in quantum chemistry we are always concerned with obtaining the ground state, this is not the only relevant information we can get from a molecule. Another important feature to calculate is what is known as the spectral gap. The spectral gap of a system is the energy difference between its ground state and its first excited state. There are many situations in which knowing this value is very important and your objective will be to design and implement a method that is able to obtain this quantity.

As always, you are free to choose your method. You can opt for techniques that use short circuits and approximate the value, or techniques designed for fault-tolerant computers that achieve high accuracy. We will value both the idea and the implementation.

Some resources

• https://pennylane.ai/qml/demos/tutorial_vqe_spin_sectors/
• https://journals.aps.org/prxquantum/pdf/10.1103/PRXQuantum.3.040327

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Project paper

You can find the full paper on ArXiv: arXiv:2402.17668

Project abstract

This work introduces a novel NISQ-friendly procedure for estimating spectral gaps in quantum systems. By leveraging Adiabatic Thermalization, we are able to create the Spectral Gap Superposition state, a newly defined quantum state exhibiting observable fluctuations in time that allow for the accurate estimation of any energy gap. Our method is tested by estimating the energy gap between the ground and the first excited state for: the Ising model, the Hydrogen molecule (H2) and Helium molecule (He2). Despite limiting our circuit design to have at most 40 Trotter steps, our numerical experiments of both noiseless and noisy devices for the presented systems give relative errors in the order of $10^{−2}$ and $10^{−1}$. Further experiments on the IonQ Aria device (with error mitigation) lead to spectral gap estimations with a relative error of $10^{−2}$ for a 4-site Ising chain, demonstrating the validity of the procedure for NISQ devices and charting a path towards a new way of calculating energy gaps.

Repository content

In this repository, a collection of files pertinent to the scientific investigation is provided.

Specifically, it contains our report Spectral_gap_superp_states.pdf and some Jupyter notebooks (given in order of priority):

• spectral_Ising_1D_ionq.ipynb: This notebook implements our proposed method for estimating the spectral gap in a 1D Ising chain. It covers both noiseless simulations and simulations with the noise model of the Aria IonQ device.
• molecules.ipynb: File for finding the spectral gap of different molecules. It works for general molecules, we applied it to H2 and He2.
• ionq_aria_launch_job.ipynb: Code to launch a hybrid job simulating three different spectral gaps for an Ising 1D chain of 4 sites, on the IonQ Aria 1 device available via AWS Braket.
• spectral_Ising_2D.ipynb: This notebook extends our approach for the spectral gap estimation to a two-dimensional Ising lattice.
• Find_Observable.ipynb: Notebook for finding the best observable for the 1D and 2D Ising models.

Folders:

• data: This folder stores all the raw data generated or used during our project.
• images: This folder contains all the images generated by our code.