Quantum Computing A-Z

Explore vectors in two-dimensional and complex spaces, decompose with a basis into x and y components, and use complex conjugates to normalize qubits in C^2 to the Euclidean norm.

Examine projection techniques, the concept of expectation value, and the singular value decomposition, revealing how projections use matrices, bra-ket forms, and a rotation–stretch–rotation interpretation.

Master the third postulate of quantum mechanics, the measurement process, and the born rule, exploring measurement operators m0 and m1, eigenvalues, and qubit state probabilities.

Explore how the cnot gate flips the second qubit when the first is one, represent the two-qubit state with a matrix, and create a bell state with a hadamard gate.

explores linear operators and density matrices, comparing pure and mixed states, and demonstrates density matrix calculations, traces, and how quantum gates act via matrix adjoint on density operators.

Explore quantum multiplication circuits using partial products and Toffoli gates, and implement a two-by-two quantum multiplier in Qiskit with adders. Verify results via measurements and carry handling.

Explore a quantum arithmetic logic unit that uses inputs A and B with an opcode to perform sum, minus, and xor, via a lookup table in a Python circuit.

Explore quantum phase estimation as a method to find eigenvalues of unitary gates, using phase kickback, controlled phase shifts, and the inverse quantum Fourier transform, with a Qiskit implementation.

Explore quantum teleportation using entanglement to transfer a qubit from Alice to Bob, including CNOT and Hadamard gates, classical bits, and conditional X and Z corrections; implemented with Qiskit.

Explore quantum least squares fitting and its reduction to a classical least squares problem using a basis matrix and pseudo inverse; compute slope and intercept from data.

Introduce quantum photonics and Q modes, visualize phase space of position and momentum, and demonstrate vacuum state manipulations with squeeze, displacement, rotation, cubic phase, and Kerr gates in strawberry fields.

Explore boson sampling in photonic circuits, using permanents and creation operators, and learn Gaussian boson sampling with squeezed states, beamsplitters, and Python code via Strawberry Fields.

Explore variational circuits, compute quantum gradients via the parameter shift rule, and evaluate multi-variable gradients by shifting parameters and comparing observable expectations.

Explore quantum feature maps as the precursor to support vector machines, using a kernel and qubit states to separate data by class through a mapped third dimension.

Explore the relevance of quantum machine learning by refreshing quantum algorithm concepts and applying new material like quantum embedding and Fisher information matrix.

Explore variational quantum classifiers (VQC) for classification, using amplitude embedding with two qubits and a variational ansatz to classify iris data, driven by cost functions and optimizers.

Explore quantum k-means clustering using the swap test to classify a point by its distance to cluster means, with amplitude or angle embedding in Pennylane.

Explore stabilizer codes and ancillas to map and correct error syndromes, then build a quantum noise model in Qiskit with gate and measurement errors.

Explore Wehrl entropy alongside von Neumann entropy, examine Husimi (Q) functions, and use Lieb’s theorem to prove strong subadditivity of quantum entropy via concavity and partial trace.

Explore Bennett's laws and the relation between qubits, bits, and phenomena like superdense coding and quantum teleportation. Learn the partial transpose, Peres-Horodecki criterion, and entanglement measures, focusing on logarithmic negativity.

Explore the no-hiding theorem and no-communication theorem, which describe quantum information conservation and the limits of communicating with entangled qubits, while noting superdense coding uses qubits plus classical bits.

Study quantum channels as processing of quantum information, including storage and transfer, using density matrices and cross operators. Compare classical, quantum, and entanglement-assisted capacities in channel communication.