Course background — why quantum computing matters →
The language of quantum mechanics. Kets (\(|\psi\rangle\)) are column vectors, bras (\(\langle\psi|\)) are row vectors, brakets (\(\langle\phi|\psi\rangle\)) are inner products. Every formula in this course uses this notation — it must be automatic.
Interactive reference for sine, cosine, and the key angles (\(\pi/6, \pi/4, \pi/3, \pi/2\)) that appear constantly in quantum state representations. Know these cold.
Two equivalent ways to compute measurement probabilities. The inner product \(|\langle x|\psi\rangle|^2\) is the pen-and-paper method. Gate-then-measure is the circuit implementation. Four worked examples showing they always agree.
A global phase (\(e^{i\theta}|\psi\rangle\)) is invisible — same measurement probabilities. A relative phase (different phases on \(|0\rangle\) and \(|1\rangle\)) changes the state physically. Trace the eigenvalues of X, Z, S, T to see the difference in action.
Step-by-step: take a messy state with 4 real numbers, factor out the global phase to make \(\alpha\) real and positive, arrive at 2 Bloch sphere coordinates (\(\theta, \phi\)). The recipe you'll use on every exam problem.
The 6 cardinal states (\(|0\rangle, |1\rangle, |+\rangle, |-\rangle, |{+i}\rangle, |{-i}\rangle\)) plotted on the Bloch sphere. Three axes = three measurement bases. The geometry that makes quantum state space click.
Non-reversible gates (SET, ZERO) cause amplitude pileup → norm exceeds 1 → probabilities exceed 100%. One number decides everything: \(\langle\text{out}_0|\text{out}_1\rangle\). If it's nonzero, the gate is broken. Interactive visualization of the mechanism.
The conjugate transpose (dagger), how to compute it, and the one-equation test for valid quantum gates. Worked pass/fail examples for real and complex matrices. If \(U^\dagger U = I\), the gate preserves orthonormality.
Meet the six standard single-qubit gates. Each one: matrix, what it does to basis states, eigenstates, and eigenvalues. Plus the key relationships (HXH = Z, T² = S, S² = Z) and interactive eigenstate visualizer.
States that survive a gate unchanged (up to phase). Why eigenstates matter: they're the fixed points that let you predict gate behavior on any input. Worked eigendecomposition for all six standard gates.
Every single-qubit unitary is a rotation of the Bloch sphere. The rotation axis = the gate's eigenstates. The rotation angle = the phase difference between eigenvalues. X rotates around x-axis, Z around z-axis, H around the XZ diagonal.
Full 3D Bloch sphere with drag-to-rotate, gate application with animated rotations, probability readout, and gate history. See everything from W1–W2 come together in one interactive tool.
The Kronecker product builds multi-qubit state spaces. Two qubits live in \(\mathbb{C}^4\), three in \(\mathbb{C}^8\), \(n\) in \(\mathbb{C}^{2^n}\). Product states vs entangled states, the factoring test \(ad \neq bc\), and applying gates to one qubit of a multi-qubit system.
The four Bell states — maximally entangled, orthonormal basis for \(\mathbb{C}^4\). How H + CNOT creates entanglement from product states. Transforming between Bell states with single-qubit gates. Why entanglement has no classical explanation (Bell's theorem).
What happens when you measure one qubit of an entangled system. The generalized Born rule: rewrite as \(\alpha|v\rangle \otimes |\phi_0\rangle + \beta|v^\perp\rangle \otimes |\phi_1\rangle\), read off probabilities and post-measurement states. Worked examples in both Z-basis and X-basis.
The CNOT gate in depth: matrix, outer product form (\(\ket{0}\bra{0} \otimes I + \ket{1}\bra{1} \otimes X\)), self-inverse proof. General controlled-U gates, why global phase becomes relative when controlled. CZ (symmetric in control/target). Phase kickback: how a gate's eigenvalue kicks back onto the control qubit. The mixed product property in action.
Any \(2 \times 2\) matrix decomposes into \(\{I, X, Y, Z\}\). Pauli strings span all \(n\)-qubit operators. Clifford gates map Paulis to Paulis — classically simulable (Gottesman-Knill). The T gate breaks the pattern (\(TXT^\dagger \notin\) Paulis). Why \(\{H, T, \text{CNOT}\}\) is universal.
The Boolean oracle \(U_f\ket{x}\ket{y} = \ket{x}\ket{y \oplus f(x)}\) — why XOR makes classical functions reversible. The phase oracle \(P_f\ket{x} = (-1)^{f(x)}\ket{x}\) via phase kickback with \(\ket{-}\). The garbage problem and uncomputation: compute, copy, reverse.
The first quantum speedups. Three promise problems solved with the Hadamard sandwich: \(H^{\otimes n} \to P_f \to H^{\otimes n} \to\) measure. DJ distinguishes constant/balanced in 1 call. BV finds a hidden string in 1 call. Simon's achieves exponential speedup with \(\sim n\) calls + Gaussian elimination.
16 exercises on multi-qubit Hadamard transforms, the cancellation lemma, Deutsch-Jozsa circuit tracing (constant + balanced), Bernstein-Vazirani with hidden strings, Simon's algorithm (collapse, \(H^{\otimes n}\), GF(2) linear systems), and query complexity comparison.
11 exercises on DJ, BV, Simon's, oracle construction (bit-flip ↔ phase), Toffoli universality for classical circuits, and the meaning of quantum advantage. Includes the \(f(x) = x^3 \bmod 2\) oracle problem and phase/bit-flip oracle conversion.
The QFT generalises the Hadamard: \(\text{QFT}\ket{x} = \frac{1}{\sqrt{N}} \sum_y \omega_N^{xy}\ket{y}\). Circuit uses \(O(n^2)\) gates (H + controlled-\(R_k\) rotations). QPE estimates eigenvalues: controlled-\(U^{2^j}\) encodes phase into counting qubits, QFT\(^\dagger\) decodes it. The key subroutine behind Shor's algorithm.
10 exercises on the DFT matrix (\(F_3, F_4\)), QFT circuit construction (\(F_8\)), QFT-based addition, unitarity proof, quantum parallelism, DJ speedup analysis, QFT vs classical FFT comparison, and periodic state mapping.
The crown jewel: reducing factoring to period-finding, then using QPE to find the period. The algorithm that broke RSA (in theory).
Every topic in this course follows the same arc:
The exam tests all levels: can you compute with the formalism (matrix multiplication, inner products), can you reason about what gates do (eigenstates, Bloch sphere), and can you trace through a quantum algorithm (circuit → measurement probabilities)?