Quantum Computing Simulations

A collection of research-oriented Jupyter notebooks exploring quantum computing algorithms and protocols using Qiskit.

Notebooks

Notebook	Topics	Qubits	Level
01 — Quantum Computing Fundamentals	Single-qubit gates, Bell states, teleportation, Deutsch–Jozsa, Grover's search, noise simulation	1–3	Introductory
02 — QFT & Shor's Algorithm	Quantum Fourier Transform, phase estimation, modular exponentiation, Shor's factoring (N=15, 21, 33, 35)	4–14	Advanced
03 — Variational Quantum Eigensolver	VQE framework, Pauli Hamiltonians, ansatz design, H₂ & LiH ground states, noise-aware VQE, barren plateaus	2–6	Advanced
04 — QAOA & Quantum Adiabatic Algorithm	Combinatorial optimisation, MaxCut, QAOA ansatz & landscape, depth scaling, adiabatic theorem, transverse-field Ising sweep, minimum gap	5–6	Advanced
05 — Quantum Error Correction	Bit-flip & phase-flip codes, Shor [[9,1,3]], stabiliser formalism, Steane [[7,1,3]], threshold theorem, surface code, noisy syndromes	3–17	Advanced

Getting Started

Prerequisites

• Python ≥ 3.9
• Jupyter Notebook or JupyterLab

Installation

git clone https://github.com/<your-username>/quantum-computing-simulations.git
cd quantum-computing-simulations
pip install -r requirements.txt
jupyter lab

Quick Verification

from qiskit import QuantumCircuit
from qiskit_aer import AerSimulator

qc = QuantumCircuit(2, 2)
qc.h(0)
qc.cx(0, 1)
qc.measure([0, 1], [0, 1])

result = AerSimulator().run(qc, shots=1024).result()
print(result.get_counts())
# Expected: {'00': ~512, '11': ~512}

What's Covered

Fundamentals

• Bloch sphere visualisation
• Quantum entanglement (all four Bell states)
• Quantum teleportation protocol
• Deutsch–Jozsa algorithm (exponential speedup over classical)
• Grover's search with amplitude amplification analysis
• Depolarising noise models and fidelity degradation

QFT & Shor's Algorithm

• QFT circuit construction and verification against analytic DFT matrix
• Approximate QFT with Coppersmith truncation
• Quantum phase estimation (QPE) with precision analysis
• Full Shor's algorithm with continued-fraction post-processing
• Factoring N = 15, 21, 33, 35
• Circuit resource scaling analysis

Variational Quantum Eigensolver

• Hamiltonian encoding via Pauli decomposition
• Hardware-efficient ansätze (RealAmplitudes, EfficientSU2)
• Exact vs. shot-based expectation value estimation
• H₂ potential energy surface (14 bond lengths)
• Optimiser comparison (COBYLA, L-BFGS-B, Nelder-Mead, Powell)
• Noise-aware VQE under shot noise and gate errors
• Barren plateau demonstration (gradient variance scaling)
• LiH ground state (6-qubit benchmark)

QAOA & Quantum Adiabatic Algorithm

• MaxCut on a small graph with brute-force optimum and Z₂-symmetric solutions
• Cost Hamiltonian encoding as SparsePauliOp and circuit implementation of $e^{-i\gamma H_C}$
• QAOA ansatz at general depth $p$, exact statevector cost evaluation
• $p=1$ cost landscape over $(\gamma,\beta)$
• Depth scan $p=1,\ldots,6$ with COBYLA + random restarts, approximation ratio convergence
• Output bitstring distribution at optimal angles
• Quantum adiabatic theorem: $H(s)=(1-s)H_B + sH_C$ via second-order Trotter with pre-diagonalised propagators
• QAA on the same MaxCut: instantaneous-GS overlap and success probability vs annealing time $T$, head-to-head against QAOA
• Transverse-field Ising chain: spectrum vs $s$, disentangling the trivial $\mathbb{Z}_2$ ferromagnetic degeneracy from the relevant gap
• Adiabatic theorem in action: fidelity vs $T$, comparison of linear and $\sin^2$ schedules, motivation for gap-adapted (Roland–Cerf) annealing

Quantum Error Correction

• Pauli error model (bit-flip, phase-flip, depolarising)
• 3-qubit bit-flip and phase-flip codes with syndrome extraction
• Shor [[9,1,3]] code — first code correcting arbitrary single-qubit errors
• Stabiliser formalism and eigenvalue-based error detection
• CSS codes and the Steane [[7,1,3]] code from the Hamming code
• Concatenated codes and doubly-exponential error suppression
• Threshold theorem and its implications
• Surface code lattice structure and resource estimates
• Repeated syndrome extraction under noisy measurements

Tech Stack

• Qiskit 2.x — circuit construction, transpilation, quantum info
• Qiskit Aer — high-performance statevector and shot-based simulation
• SciPy — classical optimisation (COBYLA, L-BFGS-B, etc.) and linear algebra (expm, eigh)
• NetworkX — graph construction and layout for MaxCut instances
• Matplotlib — visualisation

References

• Nielsen, M. A. & Chuang, I. L. (2010). Quantum Computation and Quantum Information. Cambridge University Press.
• Shor, P. W. (1994). "Algorithms for quantum computation: discrete logarithms and factoring." FOCS.
• Peruzzo, A. et al. (2014). "A variational eigenvalue solver on a photonic quantum processor." Nature Communications.
• McClean, J. R. et al. (2018). "Barren plateaus in quantum neural network training landscapes." Nature Communications.
• Farhi, E., Goldstone, J. & Gutmann, S. (2014). "A Quantum Approximate Optimization Algorithm." arXiv:1411.4028.
• Farhi, E., Goldstone, J., Gutmann, S. & Sipser, M. (2000). "Quantum Computation by Adiabatic Evolution." arXiv:quant-ph/0001106.
• Roland, J. & Cerf, N. J. (2002). "Quantum search by local adiabatic evolution." Phys. Rev. A 65, 042308.
• Gidney, C. & Ekerå, M. (2021). "How to factor 2048 bit RSA integers in 8 hours using 20 million noisy qubits." Quantum.
• Fowler, A. G. et al. (2012). "Surface codes: Towards practical large-scale quantum computation." Phys. Rev. A.
• Google Quantum AI (2023). "Suppressing quantum errors by scaling a surface code logical qubit." Nature.

License

MIT — see LICENSE.