Quantum Computing with Qiskit

A complete, runnable implementation of quantum algorithms, noise modelling, and quantum sensing simulations using IBM's Qiskit framework. All code runs on the Qiskit Aer local simulator — no quantum hardware access required.

Files

File	What it covers
1_quantum_circuits.py	Bell states, Grover's search, quantum teleportation
2_qaoa_vqe.py	QAOA for MaxCut, VQE for Ising ground state
3_noise_modelling.py	Depolarizing noise, T1/T2 decoherence, readout errors, fidelity
4_quantum_sensing.py	Ramsey interferometry, Trotter evolution, SNR scaling

How to run

pip install qiskit qiskit-aer numpy scipy matplotlib
python 1_quantum_circuits.py
python 2_qaoa_vqe.py
python 3_noise_modelling.py
python 4_quantum_sensing.py

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1. Quantum Circuits

Bell states — Four maximally entangled 2-qubit states. Measuring one qubit instantly determines the other regardless of distance.

Grover's algorithm — Finds a marked item in an unsorted database in O(√N) queries vs O(N) classically. On 3 qubits (8 items), achieves 94%+ probability on the target.

Quantum teleportation — Transmits a qubit state using a shared Bell pair and 2 classical bits. No quantum information is transmitted faster than light (no-cloning theorem respected).

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2. QAOA and VQE

QAOA — Quantum Approximate Optimization Algorithm

Solves combinatorial optimization problems (here: MaxCut on a 5-node graph). Uses alternating problem and mixer unitaries parameterised by (γ, β), optimised classically.

Depth p	Approximation ratio
1	0.641
2	0.766
3	0.785

Classical optimum (brute force) = 5.0. QAOA at p=3 achieves ~3.93 expected cut.

VQE — Variational Quantum Eigensolver

Finds ground state energy of the 1D transverse-field Ising Hamiltonian: H = -J·ΣZZ - h·ΣX using a hardware-efficient ansatz (Ry layers + CNOT entanglers). Minimises ⟨ψ(θ)|H|ψ(θ)⟩ via classical COBYLA optimizer.

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3. Noise Modelling

Three noise channels modelled:

Depolarizing noise: After each gate, the qubit state is replaced with the maximally mixed state I/2 with probability p. GHZ fidelity drops from ~0.996 (p=0.001) to ~0.834 (p=0.05).

T1/T2 decoherence:

• T1 (amplitude damping): qubit spontaneously decays from |1⟩ to |0⟩ with rate 1/T1
• T2 (dephasing): phase coherence lost with rate 1/T2, T2 ≤ 2·T1

Readout errors: Confusion matrix — P(measure 0 | state |1⟩) and P(measure 1 | state |0⟩). Even small readout errors (2-5%) significantly distort probabilities for multi-qubit circuits.

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4. Quantum Sensing

Ramsey interferometry — Core protocol for atomic clocks, magnetometers, and accelerometers. Sequence: π/2 pulse → free precession under field → π/2 pulse → measure. Phase accumulated encodes the field strength. Noise reduces fringe contrast (ideal: 0.999, noisy p=0.02: 0.974).

SNR scaling:

• Classical (shot noise limit): SNR ∝ √N
• Quantum (Heisenberg limit): SNR ∝ N
• At N=10: quantum advantage = 3.16×

Trotter evolution — Simulates time evolution under H = -J·ΣZZ - h·ΣX by decomposing U(t) into alternating ZZ and X rotations. Useful for quantum chemistry and materials simulation.

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Plots generated

• bell_states.png — measurement histograms for all 4 Bell states
• grover_search.png — probability distribution after Grover's algorithm
• qaoa_approximation_ratio.png — approximation ratio vs QAOA depth
• qaoa_distribution.png — QAOA output bitstring histogram
• vqe_convergence.png — VQE energy minimisation curve
• noise_analysis.png — fidelity vs noise, T1 decay, readout confusion
• quantum_sensing.png — Ramsey fringes, Trotter dynamics, SNR scaling

Skills demonstrated

Qiskit, Python, NumPy, SciPy, quantum circuits, variational algorithms, noise modelling, quantum sensing, Hamiltonian simulation