Curriculum

The algebra every quantum state is written in: complex numbers and Euler's formula, the vector space $\mathbb{C}^n$ with bases and coordinates, inner products with Cauchy–Schwarz and Gram–Schmidt, and Dirac's bra–ket notation with outer products, projectors and the completeness relation.

Operators as matrices, adjoints and changes of basis; Hermitian and unitary operators with full proofs of their spectral properties; the spectral theorem and functions of operators such as $e^{-i\theta P/2}$; and tensor (Kronecker) products for composite quantum systems.

The four postulates that turn linear algebra into physics: states are unit rays in a Hilbert space ($|\psi\rangle\sim e^{i\gamma}|\psi\rangle$); closed systems evolve unitarily under the Schrödinger equation $i\hbar\,\partial_t|\psi\rangle=H|\psi\rangle$, so $|\psi(t)\rangle=e^{-iHt/\hbar}|\psi(0)\rangle$; projective measurement obeys the Born rule $P(i)=|\langle i|\psi\rangle|^2$ with post-measurement collapse; and composite systems live in the tensor product $\mathcal{H}_A\otimes\mathcal{H}_B$.

Density operators $\rho=\sum_i p_i|\psi_i\rangle\langle\psi_i|$ with $\rho=\rho^\dagger$, $\rho\succeq 0$ and $\mathrm{Tr}\,\rho=1$; purity $\mathrm{Tr}\,\rho^2$ and the qubit Bloch ball $\rho=\tfrac12(I+\vec r\cdot\vec\sigma)$; the partial trace $\rho_A=\mathrm{Tr}_B\,\rho_{AB}$ as the unique consistent description of a subsystem; and the von Neumann entropy $S(\rho)=-\mathrm{Tr}\,\rho\log_2\rho$ as the entanglement measure of pure bipartite states.

A qubit is a unit vector $\lvert\psi\rangle=\alpha\lvert0\rangle+\beta\lvert1\rangle\in\mathbb{C}^2$ with $\lvert\alpha\rvert^2+\lvert\beta\rvert^2=1$, defined up to a global phase; removing both redundancies leaves two real parameters, $\lvert\psi\rangle=\cos(\theta/2)\lvert0\rangle+e^{i\phi}\sin(\theta/2)\lvert1\rangle$, which place every pure state on the Bloch sphere at $\vec r=(\sin\theta\cos\phi,\ \sin\theta\sin\phi,\ \cos\theta)$, with mixed states filling its interior through $\rho=\tfrac12(I+\vec r\cdot\vec\sigma)$.

Every single-qubit gate is a $2\times2$ unitary, and up to a global phase every such unitary is a rotation $R_{\hat n}(\theta)=e^{-i\theta\,\hat n\cdot\vec\sigma/2}=\cos(\theta/2)I-i\sin(\theta/2)\,\hat n\cdot\vec\sigma$ of the Bloch sphere. The module builds this picture from the Pauli algebra $\sigma_j\sigma_k=\delta_{jk}I+i\epsilon_{jkl}\sigma_l$, through the Hadamard and phase gates $H,S,T$ with the conjugation identities $HXH=Z$, $S^2=Z$, $T^2=S$, to the rotations $R_x,R_y,R_z$ and the ZYZ Euler decomposition $U=e^{i\alpha}R_z(\beta)R_y(\gamma)R_z(\delta)$.

Controlled unitaries $CU=\lvert0\rangle\langle0\rvert\otimes I+\lvert1\rangle\langle1\rvert\otimes U$ couple qubits: the module derives $\mathrm{CNOT}=(I\otimes H)\,\mathrm{CZ}\,(I\otimes H)$, $\mathrm{SWAP}=\mathrm{CNOT}_{01}\mathrm{CNOT}_{10}\mathrm{CNOT}_{01}$ and phase kickback; decomposes the Toffoli gate into $\{H,T,T^\dagger,\mathrm{CNOT}\}$ and proves exact and approximate universality; constructs the four Bell states $\mathrm{CNOT}(H\otimes I)\lvert ab\rangle$, proves them entangled with $\rho_A=\rho_B=I/2$ and reaches the CHSH value $S=2\sqrt2>2$; proves the no-cloning theorem from $\langle\psi\vert\phi\rangle=\langle\psi\vert\phi\rangle^2$; teleports an unknown qubit with one Bell pair and two classical bits via the identity $(H\otimes I\otimes I)(\mathrm{CNOT}_{0\to1}\otimes I)\lvert\psi\rangle_0\lvert\Phi^+\rangle_{12}=\tfrac12\sum_{m_0,m_1}\lvert m_0m_1\rangle\otimes X^{m_1}Z^{m_0}\lvert\psi\rangle$ (with deferred measurement and no-signalling); and sends two classical bits through one qubit by superdense coding, bounded by Holevo's $I(X{:}Y)\le\chi$.

From oracles to Fourier analysis: the bit oracle $U_f\lvert x\rangle\lvert y\rangle=\lvert x\rangle\lvert y\oplus f(x)\rangle$ kicks the phase $(-1)^{f(x)}$ back onto the input register, and with $H^{\otimes n}\lvert x\rangle=2^{-n/2}\sum_z(-1)^{x\cdot z}\lvert z\rangle$ the Deutsch–Jozsa algorithm decides constant versus balanced with a single query against the classical deterministic bound $2^{n-1}+1$; the quantum Fourier transform $\lvert j\rangle\mapsto 2^{-n/2}\sum_k e^{2\pi i jk/2^n}\lvert k\rangle$ is derived from its product representation and built with $\mathcal{O}(n^2)$ Hadamard and controlled-phase gates; phase estimation reads an eigenphase $\varphi$ of $U$ into a $t$-qubit clock register through controlled powers $U^{2^k}$ and the inverse QFT, succeeding with probability at least $4/\pi^2$ on the nearest $t$-bit fraction; Grover search rotates $\lvert s\rangle$ by $2\theta$ per oracle-plus-diffusion step with $\sin\theta=\sqrt{M/N}$, reaching a marked item after $\lfloor\pi/(4\theta)\rfloor=\mathcal{O}(\sqrt{N/M})$ queries; and HHL composes phase estimation, an eigenvalue-controlled $R_y$ rotation and uncomputation to prepare $\lvert x\rangle\propto A^{-1}\lvert b\rangle$.

The classical machinery that quantum machine learning plugs into: empirical-risk minimisation $\hat f=\arg\min_{f\in\mathcal F}\frac1m\sum_i\ell(f(x_i),y_i)$, the mean-squared-error gradient $\nabla_wL=\frac2m\sum_i(\hat y_i-y_i)x_i$ and the logistic cross-entropy gradient, and the convergence rate $f(x_k)-f^\ast\le\frac{L\lVert x_0-x^\ast\rVert^2}{2k}$ of gradient descent on $L$-smooth convex objectives; the soft-margin support vector machine, its Lagrangian dual $\max_\alpha\sum_i\alpha_i-\tfrac12\sum_{ij}\alpha_i\alpha_jy_iy_jk(x_i,x_j)$ and the kernel trick; and the theory behind kernels — positive-definite kernels, the Moore–Aronszajn reproducing-kernel Hilbert space with $f(x)=\langle f,k(\cdot,x)\rangle$, the representer theorem and Mercer's theorem.

How classical data enters a quantum computer and what each encoding costs. Basis encoding $x\in\{0,1\}^n\mapsto\lvert x\rangle=\bigotimes_jX^{x_j}\lvert0\rangle$ uses $n$ qubits and at most $n$ gates, but its kernel $\delta_{xx'}$ cannot generalise. Amplitude encoding $\lvert x\rangle=\lVert x\rVert^{-1}\sum_ix_i\lvert i\rangle$ packs $N$ numbers into $\lceil\log_2N\rceil$ qubits at the price of $\Theta(N)$ CNOTs for generic vectors (Möttönen's tree of uniformly controlled rotations versus the parameter-counting lower bound). Angle and phase encodings give $\langle Z_j\rangle=\cos x_j$, the product kernel $\prod_j\cos^2\frac{x_j-x'_j}2$ and, with re-uploading, trigonometric-polynomial models. Hamiltonian-evolution encoding $e^{-iH(x)t}$ compiles commuting Pauli-$Z$ terms exactly with CNOT ladders, non-commuting ones with Trotter product formulas of error $O(t^2/r)$ and $O(t^3/r^2)$, and yields the ZZ feature map.

Kernel methods with quantum feature maps: the fidelity kernel $K(x,x')=\lvert\langle\phi(x)\vert\phi(x')\rangle\rvert^2=\mathrm{Tr}\big(\rho(x)\rho(x')\big)$ as the linear kernel of the Hilbert–Schmidt space of density operators, its positive semidefiniteness and RKHS of expectation values $\mathrm{Tr}\big(O\rho(x)\big)$, exponential concentration $\mathbb E[K]=2^{-n}$ and the projected quantum kernel that avoids it; the two circuits that estimate a kernel entry — the SWAP test on $2n+1$ qubits with $P(0)=\tfrac12\big(1+\lvert\langle\psi\vert\phi\rangle\rvert^2\big)$ and the inversion test $U(x')^\dagger U(x)$ on $n$ qubits with $P(0^n)=K(x,x')$ — with their resource counts, shot variances and Hoeffding bounds; and an end-to-end quantum-kernel support vector machine on the two-moons data, trained on a precomputed Gram matrix and evaluated on a held-out split.

The variational quantum model $f_\theta(x)=\langle0\vert U^\dagger(x)W^\dagger(\theta)HW(\theta)U(x)\vert0\rangle$: a feature map $U(x)$, a trainable ansatz $W(\theta)$ and an observable $H$, shown to be a linear model $\mathrm{Tr}\big(\rho(x)\,W^\dagger HW\big)$ in the Hilbert–Schmidt feature space, with its output range, decision rule, losses, shot noise and the hybrid quantum–classical training loop; the hardware-efficient ansatz built from native $R_Y$, $R_Z$ rotations and nearest-neighbour CNOT ladders, with its parameter, CNOT and depth counts, the all-zero initialisation, GHZ reachability, and expressibility measured against the Haar fidelity density $P_{\text{Haar}}(F)=(d-1)(1-F)^{d-2}$; strongly entangling layers, a general rotation $\mathrm{Rot}(\phi,\theta,\omega)=R_Z(\omega)R_Y(\theta)R_Z(\phi)$ on every wire followed by the ring $\mathrm{CNOT}(i,(i+r_\ell)\bmod n)$ with range $r_\ell=(\ell\bmod(n-1))+1$, with the ZYZ decomposition, the $\mathbb F_2$ parity map of a CNOT block and the causal light cone; and matrix product states $\lvert\psi\rangle=\sum_sA^{[1]s_1}\cdots A^{[n]s_n}\lvert s\rangle$ obtained by successive SVD, with Schmidt spectra, bond dimensions $\chi_k$, the bound $S_k\le\log_2\chi_k$ and the sequential staircase circuits that prepare them, such as the five-gate $W_3$ circuit.

Why reverse-mode backpropagation cannot run on quantum hardware (no-cloning, measurement collapse and the exponential cost of reading out a state), and why shot-noise central finite differences $\big[\hat f(\theta+h)-\hat f(\theta-h)\big]/2h$ trade a variance $\propto1/(h^2N)$ against a bias $\propto h^2$, with the MSE-optimal step $h^\ast\approx(9/N)^{1/6}$; the exact parameter-shift rule $\partial_{\theta_j}\langle H\rangle=\tfrac12\big[\langle H\rangle_{\theta_j+\pi/2}-\langle H\rangle_{\theta_j-\pi/2}\big]$ derived from the spectrum of the gate generator, its general form $f'(\theta)=\tfrac{r}{\sin2rs}\big[f(\theta+s)-f(\theta-s)\big]$, the four-term rule for controlled rotations and the $2P$-evaluation cost of a full gradient; and the quantum natural gradient $\theta\leftarrow\theta-\eta\,g^{+}(\theta)\nabla L$ with the Fubini–Study metric $g_{ij}=\mathrm{Re}\big[\langle\partial_i\psi\vert\partial_j\psi\rangle-\langle\partial_i\psi\vert\psi\rangle\langle\psi\vert\partial_j\psi\rangle\big]$, its block-diagonal and diagonal approximations, and its invariance under reparametrisation, illustrated on the Bloch sphere.

Barren plateaus: Haar first and second moments, the McClean et al. argument that a 2-design on either side of a parameter gives $\mathbb{E}[\partial_kC]=0$ and $\mathrm{Var}[\partial_kC]\in O(2^{-n})$, the exactly solvable product model $V(\theta)=\bigotimes_jR_X(\theta_j)$ with $\mathrm{Var}[\partial_1C_G]=\tfrac18\big(\tfrac38\big)^{n-1}$ for the global cost against $\mathrm{Var}[\partial_1C_L]=\tfrac1{8n^2}$ for the local cost, and the mitigations local costs, Grant et al. identity-block initialisation and layerwise training; noise and decoherence as quantum channels $\rho\mapsto\sum_kK_k\rho K_k^\dagger$ with $\sum_kK_k^\dagger K_k=I$, Stinespring dilation, amplitude damping $\gamma=1-e^{-t/T_1}$, phase damping, depolarizing and thermal relaxation with $\tfrac1{T_2}=\tfrac1{2T_1}+\tfrac1{T_\varphi}$ on the Bloch ball; and error mitigation without encoding: zero-noise extrapolation by global folding $U(U^\dagger U)^k$ and Richardson weights $\gamma_i=\prod_{j\ne i}\tfrac{\lambda_j}{\lambda_j-\lambda_i}$ with the variance amplification $\sum_i\gamma_i^2$, tensored readout-matrix inversion of $\tilde p=Ap$ with simplex projection, and the sampling overhead of probabilistic error cancellation.

Three architectures beyond a single variational circuit. Quantum convolutional neural networks (Cong, Choi & Lukin 2019): translationally invariant convolution layers that apply one shared two-qubit unitary to every neighbouring pair, measurement-based pooling rewritten by the deferred-measurement principle as a controlled rotation followed by a partial trace over the control, the $O(\log n)$ parameter count of halving layers with shared weights, and the Pesah et al. light-cone argument that $\mathrm{Var}[\partial C]\in\Omega(1/\mathrm{poly}(n))$, so QCNNs avoid barren plateaus. Quantum reservoir computing (Fujii & Nakajima 2017): a fixed four-qubit reservoir $H=\sum_{i<j}J_{ij}X_iX_j+h\sum_qZ_q$ driven by input injection on qubit $0$, the proof that injection is a quantum channel, fading memory, virtual nodes, and a ridge-regression readout $w=(\Phi^{\mathsf T}\Phi+\lambda I)^{-1}\Phi^{\mathsf T}y$ as the only trained part, scored by the normalised mean-squared error on delay and NARMA tasks. The capstone: a two-qubit data re-uploading classifier whose $L$ uploads give a Fourier series with frequencies in $\{-L,\dots,L\}^2$, trained end to end with exact parameter-shift gradients and Adam at a cost of $M(2P+1)$ circuit evaluations per epoch, with a depth study of train and test accuracy.