Quantum Echo Lab

Interactive lab: Loschmidt echo on small spin chains + a toy 2-spin COSY spectrum. Open portfolio code — not employer IP.

pip install -r requirements.txt
make test
make lab    # Streamlit UI

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Motivation

In my bachelor years I interned at TerraQuantum. The work sat at a junction I still find addictive:

• NMR — relaxation timescales, multidimensional spectra (COSY and friends)
• Small quantum registers — modelling dynamics at roughly 8 / 16 qubits

Practically, that meant asking how chemical-shift / J-coupling / T₂ language from spectroscopy informs what a few-dozen-qubit simulator can and cannot faithfully reverse. Echoes die for the same family of reasons a quantum circuit loses fidelity: imperfect control, unwanted couplings, decoherence.

I never stopped caring about quantum computing. This repo is a clean-room teaching lab that keeps that thread public without shipping TerraQuantum Hamiltonians, pulse libraries, or proprietary datasets.

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What happens in the lab

1. Loschmidt echo tab

1. Build an open XXZ Heisenberg chain on n = 2…4 spins (exact Hilbert space 2ⁿ).
2. Prepare a product of |+\rangle states (sensitive to dephasing).
3. Evolve forward with H for time t.
4. Attempt time reversal with -(H + εV) — εV is a staggered local-Z perturbation (imperfect reversal / coherent error).
5. Optionally apply a crude T₂ model: damp density-matrix off-diagonals by e^{-t/T₂} between the two halves.
6. Plot

F(t) = |⟨ψ₀| ψ_final(t)⟩|²

When ε = 0 and T₂ is off, F(t) ≈ 1. Turn either knob and the echo decays — NMR spin-echo intuition in a few lines of NumPy/SciPy.

2. Toy COSY tab

A minimal 2-spin fragment with chemical shifts ω₁, ω₂ and coupling J:

1. Thermal-ish toy density matrix → soft π/2 pulse → evolve t₁ → mixing pulses → evolve t₂.
2. Read a complex FID-like signal with a T₂ envelope.
3. 2D FFT → magnitude map.

Diagonal ridges track the shifts; cross-peaks grow with J. This is a teaching FFT picture, not a spectrometer processing pipeline — enough to see why COSY encodes correlated transitions, the same correlation idea that shows up when you embed few-spin pieces into a small QC model.

3. Scaling note (honest)

Spins	Dim	Here
2–4	4–16	Interactive dense expm
8	256	Exact offline OK, not for casual sliders
16	65 536	Sparse / tensor / circuit methods

The bachelor question (NMR relaxation & COSY ↔ small QC) still stands. Engineering a serious 16-qubit simulator is a different repository.

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Quick start

python3 -m venv .venv
source .venv/bin/activate
pip install -r requirements.txt

make test
make lab

Layout

Path	Role
qelab/pauli.py	Kronecker Pauli helpers
qelab/hamiltonian.py	XXZ chain + reversal perturbation
qelab/evolve.py	Unitaries + phenomenological T₂
qelab/loschmidt.py	Echo protocol
qelab/cosy_toy.py	Educational COSY-like map
app.py	Streamlit UI
BACKGROUND.md	Shorter NMR ↔ QC notes

Scope / non-goals

• Not a real NMR processing stack
• Not a production 16-qubit simulator
• Not TerraQuantum IP

MIT. If you only skim one file beyond this README, make it BACKGROUND.md.