Quantum Data Encoding Patterns and their Consequences

Overview

This repository contains the necessary artefacts to reproduce the results of the experiments carried out as part of the work "Quantum Data Encoding Patterns and their Consequences". The Jupyter Notebook covers the steps from a provided Sudoku graph to the final QAOA circuit for every encoding instance.

The code and data folders contain the implementations and results of experiments conducted on the binary boolean encoding for a $4\times 4$ Sudoku.

Framework

We implemented the encodings on the quantum simulator QLM. As the QLM has restricted access, it may not be possible to execute the algorithms on the simulated QPU.

Encodings.ipynb

The notebook starts of by introducing the Sudoku graph representation. AFter some required functions, the encodings are initiated using the SymPy library. For each encoding the amount of CNOTs and the circuit depth is outputted. The depth can vary, as some Observable to Ansatz strategies rely on heuristics.

QAOA implementation

Due to prior research, we assessed the binary encoding on standard QAOA and two adapted Warm-Starting QAOA, as proposed by Egger et al.

The algorithm parameters and experiment details are as follows:

• Algorithm executions per variant = $10$
• QAOA depth = $1$
• max optimizer iterations = $10000$
• number of shots = $0$ (if $0$ => maximum possible number of shots)

Warm-starting specic details:

• $\varepsilon$ = $0.25$

The python source code used in this project can be found in the code directory.

Standard QAOA (QAOA)

QAOA.py

Warm-starting QAOA with random initialization (WS-QAOA-RAND)

We simulated a relaxed solution to the general Sudoku problem by initializing the vector $c^*$ with random values $c_i = [0,1]$ and regulated it with $\varepsilon$. The $c^*$ was then fed as a parameter into the adapted initial state and mixer operator of the QAOA circuit.

Warm-starting QAOA with random row initialization (WS-QAOA-ROW)

For this variant, a random permuatation of the values from $1$ to $N$ is assgined to each row in the Sudoku. This produces a Sudoku, in which the row constraint is met. This Sudoku is then translated to the $c^*$ vector, $\varepsilon$-regulated and fed to the warm-starting adapted QAOA circuit.

Data

To conserve the results, the result objects have been serialised with the pickling based dill module. In each iteration folder the "minimize_result" object contains the results from the first QAOA phase, namely minimised final energy and optimal parameters $\gamma$ and $\beta$.

"success_probability" contains the total probability of retrieving a valid Sudoku.

"valid_samples" contain the set of valid binary encoded Sudokus with their measured probability after the number of shots.

To save on memory, we only evaluated the results for positive samples (valid Sudokus). With this, only $288$ Sample objects have to be process for the $4\times 4$ case instead of all possible $2^{32}$. The file valid_integers contains a pickled list with integers that correspond to binary encoded valid Sudokus.