Topics

• The tensor product: product states, entangled states, and the exponential size of a register
• CNOT, the four Bell states, and the preparation of a Bell state from the zero-zero input
• Joint probabilities and conditional states; entanglement versus classical correlation
• Why entanglement does not permit signalling
• Bell's theorem: local hidden-variable models, the CHSH inequality, and the quantum correlator that violates it

Readings

• Course notes, Chapter 2
• Nielsen and Chuang, Sections 1.3.6, 2.1.7, and 2.6

Topics

• The circuit model: composition rules, controlled-U gates, a reference table of further gates, and approximate universality of {H, T, CNOT}
• The no-cloning theorem, by linearity alone, and what it does not forbid
• Quantum teleportation: the protocol, its circuit, the teleportation identity, and the Bell-basis measurement
• Outcomes and Pauli corrections: what teleportation consumes and why it respects no-signalling

Homework

HW1 due; HW2 assigned (computational, submitted as a Jupyter notebook): order matters for circuits; the teleportation protocol; no-cloning, computationally; building an oracle from a Python function; the Deutsch-Jozsa circuit; why the promise matters; classical queries against a random balanced function. Parts A and B can be attempted after Lectures 3 and 4.

Readings

• Course notes, Chapter 3
• Nielsen and Chuang, Sections 1.3.5, 1.3.7, and 4.1 to 4.3

Topics

• Oracles: reversible function evaluation, one-bit oracles, evaluation on a superposition, and the query model
• Phase kickback: writing a function value into a relative phase
• The Deutsch-Jozsa algorithm under the constant-versus-balanced promise: Deutsch's one-bit case, the Hadamard transform, the general algorithm, and the three-qubit circuit
• Classical query complexity: what the one-query separation does and does not show about speedups

Readings

• Course notes, Chapter 4
• Nielsen and Chuang, Section 1.4.3 to 1.4.4

Topics

• Unstructured search, the phase oracle, and why a single query on the uniform superposition is not a search algorithm
• The Grover iteration: inversion about the mean, two reflections as a rotation, the four-item case, and the iteration circuit
• Query complexity: the optimal number of iterations, overshooting, and several marked items
• The lower bound, oracle construction costs, and why the advantage is quadratic, not exponential

Homework

HW2 due; HW3 assigned: Grover search on eight items; two marked items; what the quadratic speedup means for a cipher key; coherences and purity under dephasing; the three-qubit code in action; when the code helps. Parts A and B can be attempted after Lectures 5 and 6.

Readings

• Course notes, Chapter 5
• Nielsen and Chuang, Sections 6.1 and 6.6

Topics

• The density matrix in minimal form: mixtures and coherences, dephasing, and the restricted bit-flip error model
• Bit-flip and phase-flip errors as X and Z applied at random
• The three-qubit bit-flip code: encoding without cloning, the parity checks Z1 Z2 and Z2 Z3, the syndrome, and the measurement circuit
• Recovery from the syndrome, the failure probability of order p², and the phase errors the code cannot see
• Commuting checks, and the assumptions of perfect encoding, checks, and recovery, removed in Lecture 9

Readings

• Course notes, Chapter 6
• Nielsen and Chuang, Sections 2.4.1 to 2.4.2 and 10.1 to 10.2

Topics

• The quantum Fourier transform as a change of basis: the Fourier basis, the product form, its O(n²) circuit, and why it is not a fast classical Fourier transform
• Phase estimation: controlled powers, the counting register, and exact versus inexact eigenphases
• Order finding as an eigenphase problem for modular multiplication, and why the eigenstates need not be prepared
• From the order to a factor: the classical reduction, the instance N = 15, and the structure of Shor's algorithm
• What the compact description hides: arithmetic circuits, error correction, and current hardware

Homework

HW3 due; HW4 assigned (computational, submitted as a Jupyter notebook): the quantum Fourier transform, by matrix and by circuit; phase estimation, exact and inexact; order finding for N = 21; entanglement swapping with a different resource; BB84 with an intercept-resend eavesdropper; what quantum computers threaten. Parts A and B can be attempted after Lectures 7 and 8.

Readings

• Course notes, Chapter 7
• Nielsen and Chuang, Sections 5.1 to 5.3

Topics

• Quantum networks: entanglement distribution, the entanglement-swapping identity, and quantum repeaters built from swapping, quantum memory, and purification
• Quantum key distribution: BB84 with an example run, the intercept-resend attack, classical postprocessing and authentication, and the entanglement-based variant
• Other uses of a network: distributed quantum computing, networked sensing, and blind computation
• What Shor's and Grover's algorithms threaten: public-key cryptography versus symmetric ciphers and hashes
• Harvest now, decrypt later; post-quantum cryptography and its standardized algorithms versus quantum key distribution
• Dated resource estimates for RSA-2048 against current hardware

Readings

• Course notes, Chapter 8
• Nielsen and Chuang, Section 12.6
• Wehner, Elkouss, and Hanson, Quantum internet: A vision for the road ahead (Science, 2018)
• NIST Post-Quantum Cryptography project

Topics

• Faults versus errors: propagation of bit and phase flips through CNOT, and fault amplification in the syndrome-extraction circuit
• The fault-tolerance requirement: transversal gates, verified ancillas, repeated syndrome measurement, and ancillas with the same connectivity as the data
• Why transversal gates alone are not universal
• The threshold theorem: concatenation overhead, its claim, its assumptions, and threshold estimates
• Physical versus logical qubits, the overhead of the surface code, and what the theorem does not promise

Homework

HW4 due.

Readings

• Course notes, Chapter 9
• Nielsen and Chuang, Section 10.6
• Preskill, Lecture Notes for Physics 219, Chapter 7 (fault tolerance)