Quantum Computing

1. What is Quantum Computing?

Classical computers store information as bits: each bit is either 0 or 1. A qubit (quantum bit) exists in a superposition of both 0 and 1 simultaneously, described by a complex probability amplitude. When measured, the qubit collapses to 0 or 1 with probabilities determined by the amplitude.

This is not the same as "storing both 0 and 1" — a qubit before measurement is in a genuine quantum superposition, not an unknown classical value. The computational advantage comes from how quantum states interfere during computation: constructive interference amplifies correct answers, destructive interference suppresses wrong ones.

Entanglement creates correlations between qubits that have no classical analogue. Measuring one entangled qubit instantaneously determines the state of its partner, regardless of distance.

2. Why it matters

Quantum computing offers polynomial or exponential speedups for specific problem classes:

• Cryptography — Shor's algorithm (1994) factors large integers in polynomial time, breaking RSA encryption. This drives national quantum security programmes worldwide.
• Search and optimisation — Grover's algorithm achieves quadratic speedup for unstructured search. Quantum annealing and QAOA target combinatorial optimisation.
• Quantum simulation — simulating quantum systems (molecules, materials) is exponentially hard classically but natural for quantum computers. This is the most near-term scientific application.
• Machine learning — quantum kernels and variational circuits offer potential speedups for specific learning tasks, though the advantage is contested.

3. Fundamental building blocks

The qubit: mathematically described as |ψ⟩ = α|0⟩ + β|1⟩, where α and β are complex numbers satisfying |α|² + |β|² = 1. The Bloch sphere is the geometric representation.

Quantum gates: unitary matrices that transform qubit states reversibly. Key single-qubit gates:

• Hadamard (H): creates superposition from |0⟩ → (|0⟩+|1⟩)/√2
• Pauli-X: bit flip (classical NOT)
• Pauli-Z: phase flip
• T gate: π/8 rotation; universal when combined with H and CNOT

Multi-qubit gates: CNOT (controlled-NOT) entangles two qubits and, combined with single-qubit gates, forms a universal gate set.

Quantum circuits: sequences of gates applied to qubits, followed by measurement. Analogous to classical logic circuits but must be unitary (reversible) until measurement.

Quantum noise: real quantum hardware suffers from decoherence (quantum states decay into classical mixtures), gate errors, and readout errors. NISQ (Noisy Intermediate-Scale Quantum) devices have 50–1000 qubits but no error correction.

4. Key scientific questions

• What is the threshold for quantum advantage on practically useful problems, not synthetic benchmarks?
• How many logical qubits (after error correction) are needed to break RSA-2048 — and when will hardware reach that scale?
• Do quantum kernels offer genuine advantage over classical kernel methods on real datasets?
• What is the barren plateau problem in variational quantum circuits, and is it fundamental or solvable?
• Can quantum simulation of molecular systems (exact ground-state energies) achieve practical advantage before fault-tolerant quantum computers?
• How does quantum sensing differ from quantum computing in terms of near-term utility?

5. Current research frontier

6. Practical workflow

Getting started on real quantum hardware:

1. Qiskit or Pennylane — IBM's Qiskit and Xanadu's PennyLane are the two major open-source frameworks
2. IBM Quantum free tier — run circuits on real quantum hardware via IBM Quantum Platform (free access to 127-qubit processors)
3. Start with circuit basics — implement Bell state, GHZ state, quantum teleportation
4. Try VQE for H₂ — find the ground state energy of the hydrogen molecule (the hello world of quantum chemistry)
5. Explore quantum kernels — implement a quantum kernel classifier using PennyLane's qml.kernels module
6. Benchmark against classical — always compare with scikit-learn SVM on the same dataset

7. Key references

8. Cogninoid build direction

Quantum computing connects to Cogninoid's scientific AI work at two levels:

• Quantum kernels for materials — quantum feature maps for small datasets in materials property prediction; comparing classical vs. quantum kernel performance as a scientific benchmark
• Quantum simulation — VQE-based ground state estimation for candidate materials, complementing classical DFT
• Quantum sensing — future direction: NV-centre magnetometry for probing materials at the nanoscale

9. Beginner project

Visualise a qubit on the Bloch sphere:

• Tools: Qiskit, matplotlib, qiskit-terra's plot_bloch_multivector
• Goal: Create a circuit that applies H, then Ry(θ) for various θ; visualise how the state moves on the Bloch sphere
• Verification: At θ=0 after H, the state is |+⟩; at θ=π, it is |−⟩. Does the Bloch vector match your prediction?

10. Advanced project

Quantum kernel classifier vs. classical SVM:

• Tools: PennyLane, qml.kernels, scikit-learn, a benchmark dataset (Iris or 2-class MNIST subsets)
• Goal: Implement a quantum kernel using angle embedding (ZZFeatureMap); compute the kernel matrix; train an SVM; compare accuracy and training time with sklearn.svm.SVC with RBF kernel
• Verification: Report accuracy, kernel matrix computation time, and whether the quantum kernel shows any advantage. Be rigorous — it likely will not, but the comparison is the learning.