Simulating a Simple NMR Experiment with Qiskit

This repository contains a simple Python script to simulate a very basic NMR experiment using Qiskit.

I simulated a quantum circuit with two qubits and two classical bits. The idea was to reproduce a simplified COSY experiment between $^{13}\mathrm{C}$ and $^{1}\mathrm{H}$.

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Pulse Sequence

1. A $\pi/2$ rotation around the x-axis is applied to both spins to prepare transverse magnetization.

2. The system evolves under a Hamiltonian containing chemical shifts and isotropic J-coupling:

$$ H = \hbar \omega _H I_{z}^{(H)} + \hbar \omega _C I_{z}^{(C)} + 2\pi J \left(\hat{I}^{(C)}\cdot \hat{I}^{(H)}\right) $$

where $I_\alpha ^{(H/C)}$ are the spin operators of hydrogen and carbon along $\alpha = x, y, z$.

3. The evolution time is chosen as:

$$ t_1 = \frac{1}{2J}, $$

which corresponds to the mixing period in a COSY-like experiment.

4. Another $\pi/2$ rotation around the x-axis is applied to both spins at the end of the sequence.

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1. Initial State

For a two-spin-½ system, the initial state is:

$$ |\psi _0 \rangle = |s_1\rangle \otimes |s_2\rangle $$

where each single-spin state is a 2-component vector (e.g. eigenstates of $(I_z)$). Thus the full Hilbert space is 4-dimensional, and all operators are constructed as Kronecker products of single-spin operators.

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2. Pulse as a Rotation

A pulse acting on spin $(k)$ corresponds to a rotation operator $(R^{(k)}(\theta,\hat{n}))$ acting only on that spin:

$$ R^{(k)} = \begin{cases} R(\theta ,\hat{n}) \otimes \mathbb{I}, & k = H, \\ \mathbb{I} \otimes R(\theta ,\hat{n}), & k = C, \end{cases} $$

where, for example, rotation along x-axis is described as:

$$ R_{x}(\theta) = \begin{pmatrix} cos(\theta / 2) & -i sin(\theta / 2) \\ -i sin(\theta /2) & cos(\theta / 2) \end{pmatrix} $$

The full rotation is again a (4 \times 4) operator constructed by Kronecker products.

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3. Density Matrix Evolution

After the pulse, the system evolves freely under the Hamiltonian:

$$ \rho_ {\text{final}} = U(t_1) \cdot \rho _0 \cdot U^\dagger(t_1) $$

with:

$$ U(t_1) = e^{-i H t_1 / \hbar}. $$

This is the standard unitary time evolution for closed quantum systems.

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4. Observable Magnetization

It is not recorded a FID, but the final state is studied by counts on two classical bits. For this reason, the model is very simple