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Classiq x DuQIS FLIQ Challenge

Science track challenge for FLIQ 2025 hackathon

Classifying Quantum Phases of Matter

A challenge hosted by Classiq Technologies and DuQIS (Duke Quantum Information Society) as part of the FLIQ Hackathon. Participants will apply quantum machine learning to distinguish between different phases of a Rydberg atom system using measurement data obtained in randomized bases.

Challenge Summary

The task is to build a Quantum Machine Learning (QML) model capable of classifying different phases of quantum matter from measurement data.

Unlike typical datasets, your inputs are classical shadows: compressed representations of quantum states constructed via randomized measurements. Your model should learn to identify the phase label of a quantum state based only on this information.

Background

Classical Shadows

A classical shadow is a compact representation of a quantum state constructed from local randomized measurements. Suppose an experiment prepares an $n$-qubit state $\rho$. Instead of reconstructing $\rho$ directly (which is exponentially costly), we perform $T$ rounds of random single-qubit Pauli measurements.

For each round $t$, each qubit is measured in a random Pauli basis, collapsing $\rho$ into a product state:

$$ \ket{s^{(t)}} = \bigotimes_{i=1}^n \ket{s_i^{(t)}}, \quad \ket{s_i^{(t)}} \in \lbrace \ket{g}, \ket{r}, \ket{+}, \ket{-}, \ket{+i}, \ket{-i} \rbrace. $$

We then form an estimate:

$$ \sigma_T(\rho) = \frac{1}{T} \sum_{t=1}^T \sigma_1^{(t)} \otimes \dots \otimes \sigma_n^{(t)}, \quad \sigma_i^{(t)} = 3\ketbra{s_i^{(t)}}{s_i^{(t)}} - I. $$

You are provided with $nT$ measurement results per data point. You may choose to compute reduced density matrices for specific subsystems, such as:

$$ \rho^A \approx \frac{1}{T} \sum_{t=1}^T \bigotimes_{i \in A} \sigma_i^{(t)}. $$

For more details, see:

  • Huang et al., Predicting Many Properties of a Quantum System from Very Few Measurements (2020), arXiv:2002.08953
  • Huang et al., Provably efficient machine learning for quantum many-body problems (2022), arXiv:2106.12627

Details on Rydberg Atoms

The Rydberg Hamiltonian for an atom chain reads

$$ H = \frac{\Omega}{2} \sum_{i=1}^N X_i - \delta \sum_{i=1}^N n_i + \sum_{i \lt j} \frac{\Omega R_b^6 }{(a|i-j|)^6} n_i n_j, $$

where $\Omega$ is the Rabi frequency; $\delta$ is the laser detuning; $a$ is the inter-atomic spacing; $R_b$ is the blockade radius; $n_i \equiv \ket{r_i}\bra{r_i}$ is the projector onto the Rydberg state on the $i^{\text{th}}$ qubit; and $X_i = \ket{g_i}\bra{r_i} + \ket{r_i}\bra{g_i}$ is a Pauli $X$ operator.

The Rydberg Hamiltonian contains three types of operators:

  1. Terms involving Pauli $X = \ket{r}\bra{g} + \ket{g}\bra{r}$ are responsible for driving atoms from $\ket{g}$ to $\ket{r}$.

  2. Terms involving the projector $n$ introduce the punishment (or reward) for being in the excited state: when $\delta > 0$, excitation is penalized; when $\delta < 0$, excitation is rewarded.

  3. The interaction terms $n_i \otimes n_j$ realize the Rydberg blockade mechanism.

The interaction terms prevent neighboring sites from being excited simultaneously, while the terms involving $n$ ensure that the number of excitations is maximized when $\delta \gg 0$. Thanks to this interplay of Hamiltonian terms, neutral-atom systems constitute interesting phases of matter even in a single spatial dimension. You can find the phase diagram for a 51-atom chain inside the challenge notebook. Your goal is to design a quantum model that can distinguish between the so-called $Z2$-ordered and $Z3$-ordered states.

For more information, please refer to:

QML Models

Participants are expected to build a parameterized quantum circuit to classify measurement data. The exact architecture — including encoding scheme, number of qubits, and circuit layers — is up to you.

The following reading might inspire your quantum circuit:

  • Sim et al., Expressibility and Entangling Capability of Parameterized Quantum Circuits for Hybrid Quantum‐Classical Algorithms (2019), arXiv:1905.10876

Dataset Format

Each data point consists of:

  • A list of $T$ measurement outcomes, each a list of $n$ elements $s_j^{(i,t)} \in \lbrace\text{"g"}, \text{"r"}, \text{"+"}, \text{"-"}, \text{"+i"}, \text{"-i"}\rbrace$
  • A label $y^{(i)}$ such as "Z2" indicating the phase

Example:

$$ x^{(i)} = \left[ \left[\text{"-"}, \text{"+i"}, \dots, \text{"g"}\right],\\ \left[\text{"g"}, \text{"-i"}, \dots, \text{"r"}\right],\\ \dots \right]; \quad y^{(i)} = \text{"Z2"} $$

It is up to you to choose which reduced density matrices to extract as features (e.g., 1-qubit, 2-qubit). Avoid reconstructing the full $\rho$, which is a $2^n \times 2^n$ object.

Your Task

Build a quantum circuit that:

  • Takes as input reduced density matrices constructed from the measurement data
  • Outputs a prediction of the quantum phase
  • Is optimized for both accuracy and efficiency

Grading Criteria

Each submission will be scored using the function:

$$ f(A, P, D, W) = A - 0.1 \cdot P - 0.0002 \cdot D - 0.1 \cdot W $$

Where:

  • $A$: accuracy on the test set
  • $P$: number of trainable parameters
  • $D$: circuit depth
  • $W$: number of qubits (circuit width)

Higher values of $f$ indicate better solutions.

This grading scheme is not rigorous - it only filters solutions. Even if your solution doesn't yield a high $f$, it may still be considered for manual grading. The best solutions will be manually graded.

This Repository

In this repo you can find the following files:

  • FLIQ_Challenge_ClassiqDuQIS.ipynb – the notebook with code snippets, in which you will show your solution
  • training_data.npz – file with randomized measurements. There are 10 data points corresponding to each of the two phases studied
  • phase_diagram.png – Rydberg phase diagram for a 51-particle neutral atom array

Getting Started

You may clone the repository locally and use your editor of choice. Alternatively, you may clone the repository into Classiq Studio - a web-based Classiq IDE. If you would like to use Classiq Studio, refer to the following guide: https://docs.classiq.io/latest/user-guide/classiq-studio/

Submission Instructions

On the submission platform, please upload the main Jupyter notebook along with any additional .py files used for data processing. You should also include:

  • The quantum program (qprog) saved as a .qprog file
  • The trained model parameters saved as a .npz file using NumPy

Ensure all files necessary to reproduce your results are included in the submission.

Submission deadline: Sunday, May 18 at 08:00 UTC — mark it, set alarms, summon caffeine.

Tips

  • Reduced density matrices of small subsystems may already carry enough information about the phase.
  • Consider different encoding strategies such as angle or amplitude encoding. You can find the pre-defined Classiq method for angle encoding here
  • You may apply classical preprocessing, but the model must ultimately be quantum.