Z-Pauli expectation values showing the preserved Charge Density Wave (CDW) order with quantum fluctuations characteristic of the v = 1/3 Laughlin state.
This project utilizes IBM Qiskit to simulate the quantum circuit preparation of Fractional Quantum Hall (FQH) states on a 1D qubit chain. Specifically, it targets the Laughlin state.
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State Initialization: Prepares the root partition
$|100100\dots\rangle$ corresponding to the$\nu=1/3$ Charge Density Wave (CDW). -
Parameterized Ansatz: Uses layers of
$R_y(\theta)$ and$CX$ gates to introduce many-body entanglement. -
Measurement: Calculates the local magnetization
$\langle Z_i \rangle$ to verify the order parameter. - Visualization: Automatically generates circuit diagrams and expectation value plots.
The quantum circuit (below) implements the state preparation ansatz.
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Initialization (Blue X Gates): The qubits
$q_0, q_3, q_6, \dots$ are flipped to$|1\rangle$ , creating the initial pattern$|100100100\dots\rangle$ . - Entanglement (Rotations & CNOTs): The subsequent gates "melt" this crystalline order, introducing the necessary quantum superposition to approximate the topological ground state.
The results graph (shown at the top) plots the expectation value
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Period-3 Oscillation: The data follows a clear pattern of "Down-Up-Up" (
$\langle Z \rangle < 0$ followed by two$\langle Z \rangle > 0$ ), confirming that the system remains in the$\nu=1/3$ phase. -
Quantum Fluctuations: Crucially, the peaks do not reach the classical limits of
$\pm 1.0$ . The observed values of$\approx \pm 0.75$ indicate strong quantum fluctuations. This confirms that the system has evolved away from the trivial product state into a macroscopic superposition, a hallmark of the FQH liquid phase.
You will need Python installed along with Qiskit and Matplotlib:
pip install qiskit matplotlib numpy