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Creates the Bell state |Phi+> = (|00> + |11>)/sqrt(2), the simplest demonstration of quantum entanglement. Two qubits are prepared such that measuring one instantly determines the other — the foundation for teleportation, superdense coding, and quantum key distribution.
Faithful reproduction of the Quantum Approximate Optimization Algorithm (QAOA) for MaxCut, as introduced in the foundational 2014 paper by Farhi, Goldstone, and Gutmann. Demonstrates approximation ratio improvement with increasing circuit depth p on triangle, square, and butterfly graphs (3-5 qubits). Includes automated verification checks against the paper's theoretical bounds.
Trains a classical Support Vector Machine on a kernel matrix computed from quantum circuits. The ZZ feature map encodes data into an exponentially large Hilbert space via entangling RZ(xi·xj·π) gates, and the inversion test measures K(x,y) = |⟨φ(x)|φ(y)⟩|² as the all-zeros probability. Includes synthetic data generation, SVM dual optimization, and classification evaluation.
Estimates the complex expectation value <psi|U|psi> of a unitary operator using quantum interference on an ancilla qubit. The real part is extracted from the default circuit; adding S-dagger before the final Hadamard gives the imaginary part.
Universal single-qubit classifier that re-encodes data between variational layers, proving 1 qubit suffices for any classification task
General-purpose VQD implementation for finding K eigenstates sequentially. Each state is found by minimizing the energy plus overlap penalties with all previously found states. Demonstrates on the 4-qubit Transverse Field Ising Model.
Finds ground and first excited states of lithium hydride (LiH) using VQD on a 4-qubit active-space Hamiltonian. LiH has richer electronic structure than H2, including ionic-covalent character mixing and an avoided crossing at stretched geometries (~3 A).
Computes the quantum fidelity |⟨φ(x)|φ(y)⟩|² directly using the SWAP test circuit — an interferometric technique that measures state overlap via an ancilla qubit and controlled-SWAP gates. Requires 2n+1 qubits for n-qubit data encoding.
Solves the Traveling Salesman Problem using QAOA with constraint penalties. Uses city-timestep encoding (n^2 qubits for n cities): qubit (c, t) = 1 means city c is visited at timestep t. Default: 2 cities with distance 5, 4 qubits. Cost Hamiltonian encodes distance terms plus row and column constraint penalties enforced via RZZ interactions.
Sends 2 classical bits using 1 qubit by leveraging a pre-shared Bell pair. The dual of quantum teleportation — demonstrates that entanglement can double the classical information capacity of a quantum channel.
Solves Graph 2-Coloring using the Quantum Approximate Optimization Algorithm (QAOA). Assigns one of two colors to each vertex to minimize edge conflicts (adjacent vertices sharing a color). Default: 4-vertex graph with triangle subgraph (chromatic number 3, minimum 1 conflict). Identical circuit structure to MaxCut — coloring minimizes conflicts while MaxCut maximizes cuts.
Finds the ground state energy of the hydrogen molecule (H2) using the Variational Quantum Eigensolver (VQE) — a hybrid quantum-classical algorithm that pairs a parameterized 2-qubit ansatz with COBYLA optimization. The qubit Hamiltonian is derived from the STO-3G basis via parity mapping with two-qubit reduction, giving an exact ground state of -1.1373 Hartree.
Encodes classical features as rotation angles on the Bloch sphere, with standard (RY/RZ/RX) and dense (2 features per qubit) variants
First quantum error correcting code (1995) — corrects any single-qubit error (X, Y, Z) via concatenated bit-flip and phase-flip codes
Projects quantum states to a finite-dimensional classical vector of expectation values instead of computing the full fidelity kernel. Uses an IQP (Instantaneous Quantum Polynomial) feature map with random Pauli or computational basis projections. More robust to noise and exponential concentration than full fidelity kernels.
Detects and quantifies vanishing gradients (barren plateaus) in variational quantum circuits by measuring gradient variance scaling with system size. Uses the parameter-shift rule and exponential decay fitting to determine if an ansatz is trainable.
Topological QEC code on a 2D lattice with ~1% error threshold — the leading candidate for practical fault-tolerant quantum computing
Reproduction of the first experimental VQE demonstration — finding the ground-state energy of H2 on a photonic quantum processor (Peruzzo et al., Nature Communications 2014)
Teleports an unknown quantum state from one qubit to another using a shared Bell pair and two classical bits. Demonstrates that quantum information can be transferred without physically sending the qubit — at the cost of consuming entanglement and requiring classical communication.
Searches an unstructured database of N=4 items using Grover's algorithm, finding the marked item with 100% probability in a single oracle query. Demonstrates amplitude amplification — the oracle marks the target with a phase flip, and the diffusion operator (inversion about the mean) boosts its probability from 25% to 100%. Provides a provably optimal quadratic speedup: O(sqrt(N)) vs O(N) classical.
Deleted runs still count, so these numbers can be too high.